WO2021254291A1 - 一种Polar码速率匹配方法及装置 - Google Patents

一种Polar码速率匹配方法及装置 Download PDF

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WO2021254291A1
WO2021254291A1 PCT/CN2021/099903 CN2021099903W WO2021254291A1 WO 2021254291 A1 WO2021254291 A1 WO 2021254291A1 CN 2021099903 W CN2021099903 W CN 2021099903W WO 2021254291 A1 WO2021254291 A1 WO 2021254291A1
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sequence
rate matching
bits
sub
code length
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PCT/CN2021/099903
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English (en)
French (fr)
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戴胜辰
李榕
童佳杰
张华滋
王献斌
王俊
黄威
皇甫幼睿
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华为技术有限公司
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Priority to EP21826123.8A priority Critical patent/EP4156524A4/en
Publication of WO2021254291A1 publication Critical patent/WO2021254291A1/zh
Priority to US18/065,479 priority patent/US20230113448A1/en

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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/63—Joint error correction and other techniques
    • H03M13/635—Error control coding in combination with rate matching
    • H03M13/6362—Error control coding in combination with rate matching by puncturing
    • H03M13/6368—Error control coding in combination with rate matching by puncturing using rate compatible puncturing or complementary puncturing
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/13—Linear codes
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/27—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes using interleaving techniques
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/61—Aspects and characteristics of methods and arrangements for error correction or error detection, not provided for otherwise
    • H03M13/618—Shortening and extension of codes
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/63—Joint error correction and other techniques
    • H03M13/6306—Error control coding in combination with Automatic Repeat reQuest [ARQ] and diversity transmission, e.g. coding schemes for the multiple transmission of the same information or the transmission of incremental redundancy
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/63—Joint error correction and other techniques
    • H03M13/635—Error control coding in combination with rate matching
    • H03M13/6362—Error control coding in combination with rate matching by puncturing
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0041—Arrangements at the transmitter end
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H04L1/0057—Block codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H04L1/0067—Rate matching

Definitions

  • the embodiments of the present application relate to the field of communication technologies, and in particular, to a Polar code rate method and device.
  • channel coding plays a vital role in ensuring the reliable transmission of data.
  • turbo codes, low density parity check (LDPC) and polar codes are generally used for channel coding.
  • Turbo codes cannot support information transmission with too low or too high bit rates.
  • Turbo codes and LDPC codes are also difficult to achieve ideal performance under limited code length due to their own coding and decoding characteristics.
  • Turbo codes and LDPC codes have high computational complexity in the process of encoding and decoding.
  • Polar code is a good code that theoretically proves that Shannon capacity can be obtained, and has relatively simple encoding and decoding complexity, so it has been more and more widely used.
  • the Polar code is determined as the coding method of the control channel.
  • the embodiments of the present application provide a method and device for rate matching of Polar code encoding applied to satellite communications, which have the advantage of simple implementation.
  • This design fits the characteristics of satellite communications, and the nesting feature of the rate matching sequence also provides a variety of possible ways for specific implementation, which is easy to implement, simplifies the operation of rate matching, and reduces the difficulty of implementation.
  • the rate matching sequence is a subsequence of the first sequence
  • the first sequence is the sequence 1 or the sequence 2 or the sequence 3 in Appendix 1, wherein the sequence 1 includes N2 polarons Channel number, the sequence 2 includes N2/2 polarized subchannel numbers, the sequence 3 includes N2 values of different sizes, the polarized subchannel number starts from 0, and N2 is greater than or equal to N.
  • the rate matching sequence is sequence 1a or sequence 2a or sequence 3a in the specification.
  • the rate matching sequence is the sequence 1b or 2b or 3b in the specification.
  • the rate matching sequence is the sequence 1c or 2c or 3c in the specification.
  • the rate matching sequence is the sequence 1d or 2d or 3d in the specification.
  • the rate matching sequence is the sequence 1e or 2e or 3e in the specification;
  • sequence 1a, the sequence 1b, the sequence 1c, the sequence 1d, and the sequence 1e are subsequences of the sequence 1;
  • sequence 2a, the sequence 2b, the sequence 2c, the sequence 2d, and the sequence 2e are subsequences of the sequence 2;
  • sequence 3a, the sequence 3b, the sequence 3c, the sequence 3d, and the sequence 3e are subsequences of the sequence 3.
  • the rate matching method is a puncturing operation
  • the puncturing operation is: taking the bits carried by the sub-channels corresponding to the last M polarization sub-channel numbers in the rate matching sequence according to the target code length M, or according to the The target code length M deletes the bits carried by the subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence; or
  • the puncturing operation is: deleting the bits carried by the subchannels corresponding to the first N-M polarized subchannel numbers in the rate matching sequence according to the target code length M; or
  • the puncturing operation is: deleting, according to the target code length M, the bits or bits carried by the polar subchannel where the NM values with the smallest value in the rate matching sequence are located. Take, according to the target code length M, the bits carried by the polarized subchannel where the M values with the largest value in the rate matching sequence are located.
  • the polarization encoding of the K bits to be coded according to the mother code length N is: the K bits to be coded are mapped to MP non-punctured polarized sub-channels
  • the K polarization sub-channels with higher reliability make the polarization sub-channel where the K bits to be coded are more reliable than the remaining MPK polarization sub-channels.
  • P is the pre-frozen polarization sub-channel. Number, P is greater than or equal to 0.
  • the encoded bit sequence is interleaved before performing rate matching on the encoded bit sequence according to the rate matching sequence to obtain the rate matching sequence.
  • the K bits to be coded include cyclic redundancy check bits.
  • the K bits to be encoded include parity bits.
  • the rate matching method is executed by a network device or a terminal.
  • a Polar code rate matching device which has the function of implementing the method described in the first aspect and any one of the possible designs of the first aspect.
  • the functions can be realized by hardware, or by hardware executing corresponding software.
  • the hardware or software includes one or more modules corresponding to the above-mentioned functions.
  • the Polar code rate matching device when part or all of the functions are implemented by hardware, includes: an input interface circuit for acquiring the bits to be encoded; a logic circuit for implementing the first aspect described above And the behavior described in any possible design of the first aspect; the output interface circuit is used to output the sequence after the rate matching.
  • the Polar code rate matching device may be a chip or an integrated circuit.
  • the Polar code rate matching device when part or all of the function is realized by software, includes: a memory, used to store a program; a processor, used to execute the program stored in the memory When the program is executed, the Polar code rate matching device can implement the method described in any one of the foregoing first aspect and the first aspect.
  • the foregoing memory may be a physically independent unit, or may be integrated with the processor.
  • the Polar code rate matching device when part or all of the functions are implemented by software, includes a processor.
  • the memory for storing the program is located outside the rate matching device, and the processor is connected to the memory through a circuit/wire for reading and executing the program stored in the memory.
  • the device is a network device or a terminal.
  • a communication system in a third aspect, includes a network device and a terminal, and the network device or terminal can execute the method described in the above-mentioned first aspect and its possible design.
  • a computer storage medium which stores a computer program, and the computer program includes instructions for executing the method described in the first aspect and any one of the possible designs of the first aspect.
  • the embodiments of the present application provide a computer program product containing instructions, which when run on a computer, cause the computer to execute the methods described in the foregoing aspects.
  • a wireless device including a rate matching device and a transceiver for implementing the first aspect and any possible design of the first aspect,
  • the transceiver is used to receive or transmit signals.
  • the wireless device is a terminal or a network device.
  • FIG. 1 is a schematic diagram of the architecture of a communication system applied in an embodiment of this application;
  • FIG. 3 is one of the structural schematic diagrams of the Polar code rate matching apparatus in an embodiment of the application.
  • FIG. 4 is the second structural diagram of a Polar code rate matching device in an embodiment of this application.
  • FIG. 5 is the third structural diagram of a Polar code rate matching device in an embodiment of this application.
  • FIG. 6 is the fourth structural diagram of a Polar code rate matching device in an embodiment of this application.
  • FIG. 7 is one of the structural schematic diagrams of the Polar code de-rate matching device in an embodiment of the application.
  • FIG. 8 is the second structural diagram of the Polar code de-rate matching device in an embodiment of the application.
  • FIG. 9 is the third structural diagram of the Polar code de-rate matching device in an embodiment of the application.
  • FIG. 10 is the fourth structural diagram of the Polar code derate-matching apparatus in an embodiment of this application.
  • the embodiment of the application provides a Polar code rate matching method and device. According to the reliability of the polarization sub-channel, online calculation or pre-stored reliability ranking is performed, and the sequence number of the polarization sub-channel for sending information bits is selected according to the reliability ranking. , Perform Polar code encoding according to the selected information bit sequence number, then perform rate matching operation on the encoded sequence according to the rate matching sequence, and output the rate matching sequence.
  • the embodiments of this application are more suitable for satellite communications, especially short message services in satellite communications.
  • Polar code uses a noise-free channel to transmit useful information for users, and an all-noise channel transmits agreed information or no information.
  • Polar code is also a linear block code, its coding matrix is G N , and the coding process is in It is a binary row vector with a length of N (that is, the code length. As you can see, the lengths of the sequence x and u before and after the encoding are both N, and N is also called the mother code length); G N is an N ⁇ N Matrix, and It is defined as the Kronecker product of log 2 N matrices F 2. The above matrix
  • a part of the bits are used to carry information, which is called the information bit set, and the index set of these bits is denoted as A; the other part of the bits is set to a fixed value pre-appointed by the receiving end and the sending end, which is called frozen bit set or frozen set of bits (frozen bits), index its complement a C a is represented by a set.
  • the encoding process of Polar code is equivalent to:
  • G N (A) is a sub-matrix G N obtained by those row index corresponding to the set A
  • G N (A C) are G N obtained from the set A C those rows corresponding to the index sub matrix.
  • u A is The set of information bits in, the number is K.
  • CRC Cyclic Redundancy Check
  • PC Parity Check
  • Bits can also be included in the information bit set; for The frozen bit set in, whose number is (NK), is a known bit. These freeze bits are usually set to 0, but as long as the receiving end and the sending end agree in advance, the freeze bits can be set arbitrarily.
  • u A is The set of information bits in
  • u A is a row vector of length K, that is
  • K,
  • K is the size of the information block
  • G N (A) is the matrix G N
  • the sub-matrix obtained from the rows corresponding to the index in the set A, G N (A) is a K ⁇ N matrix.
  • the construction process of the Polar code is the selection process of the set A, which determines the performance of the Polar code.
  • the construction process of the Polar code is usually: According to the code length N of the mother code, it is determined that there are N polarized sub-channels, corresponding to the N rows of the coding matrix. Without considering the rate matching, the top K with higher reliability is determined. a polarization index of the subchannel as an element of the set a, the remaining (NK) polarizations corresponding subchannel index set as a freeze-bit index C a elements.
  • the set A determines the position of the information bit, and the set A c determines the position of the frozen bit.
  • the sequence number of the polarized sub-channel is the position index of the information bit or frozen bit, which is Index of the position in.
  • the main consideration is to puncture or shorten.
  • M is the target code length , That is, the length of the bit sequence after rate matching.
  • the NM polarized sub-channels selected here are used to place frozen bits.
  • some additional polarizations called pre-freezing are also determined.
  • Sub-channels are also used to place frozen bits.
  • P is greater than or equal to 0 (for the shortened case, P is obviously 0, so you don’t need to consider P at this time.
  • the relative relationship of the reliability mentioned here is based on a given reliability calculation method. Different reliability calculation methods may cause the relative relationship of the polarimetric sub-channel reliability to change, but the information is chosen to be placed
  • the method of polarizing sub-channels of bits is the same.
  • information bits can also be placed in the last punctured or shortened polarized subchannel. This application does not restrict the selection of the polarized sub-channel for placing information bits in accordance with the 5G NR standard.
  • the Beidou satellite communications system there is a short message communication mode. It can be regarded as the commonly used "short message", but it is different. This type of published information can not only display the information itself, but also display the position of the publisher.
  • the Turbo code currently used in the Beidou second-generation short message system can use Polar codes to improve performance and reduce complexity.
  • the Beidou short message sends information bits to the ground center station, the information is first segmented. After the segmentation, the information bit length of each segment is 89 to 994 (including CRC bits), and the code rate used is 1/2. If subsequent consumer terminals want to access the short message system, they need to use a lower bit rate, such as 1/8.
  • the rate matching sequences corresponding to different mother code lengths also satisfy the nesting relationship, which can simplify the operation process and improve the processing efficiency.
  • This nesting feature is similar to the reliability ranking sequence corresponding to different mother code lengths in the 5G NR standard, that is, when the mother code length Na is less than or equal to the mother code length Nb, each polarized sub-channel in the reliability sequence corresponding to Na ( The relative reliability of polarized sub-channels 0,..., Na-1) is consistent with the relative reliability of these polarized sub-channels (polarized sub-channels 0,..., Na-1) in the reliability sequence corresponding to Nb In other words, in the reliability sequence corresponding to Nb, only the polarizing subchannel numbers smaller than Na are retained and the relative relationship between the positions of the Na polarizing subchannel numbers is not changed, and the reliability sequence corresponding to Na can be obtained.
  • either the reliability sequences of different mother code lengths can be stored in advance, or only the reliability sequence corresponding to the maximum mother code length N max can be stored in advance, and then the reliability sequences corresponding to other mother code lengths Na can be stored in advance. It is obtained from the reliability sequence corresponding to the maximum mother code length by only retaining the sequence numbers of the polarized sub-channels less than N max . This saves a little storage overhead on the one hand, and is also easy to implement on the other hand, without the need to consider sub-block interleaving and other operations .
  • Namax can also be equal to Nmax .
  • the rate matching (puncturing or shortening) of each polarized sub-channel (polarized sub-channel 0,..., Na-1) in the rate matching sequence corresponding to Na The relative relationship of the sequence is consistent with the relative relationship of the rate matching sequence in the rate matching sequence corresponding to Nb of these polarized sub-channels (polarized sub-channels 0,..., Na-1).
  • the rate matching sequence corresponding to Nb In the rate matching sequence corresponding to Na can be obtained by keeping only the polarizing sub-channel numbers less than Na and not changing the relative relationship between the positions of the Na polarizing sub-channel numbers.
  • either the rate matching sequences of different mother code lengths can be stored in advance, or only the rate matching sequence corresponding to the maximum mother code length N max can be stored in advance, and then the rate matching sequences corresponding to other mother code lengths Na can be used. It is obtained from the rate matching sequence corresponding to the maximum mother code length by retaining only the sequence numbers of the polarized sub-channels less than N max .
  • Na max can also be equal to N max .
  • Fig. 1 is a schematic structural diagram of a wireless communication network according to an embodiment of the present invention.
  • Fig. 1 is just an example, and other wireless networks that can use the rate matching method or device of the embodiment of the present invention are also within the protection scope of the present invention.
  • the wireless communication network 100 includes a network device 110 and a terminal 112.
  • the network device 110 may also be connected to the core network 102.
  • the network device 110 can also communicate with the IP network 104, such as the Internet, a private IP network, or other data networks.
  • Network equipment provides services for terminals within the coverage area.
  • the network device 110 provides wireless access to one or more terminals 112 within the coverage area of the network device 110.
  • the network devices may also communicate with each other.
  • the network device 110 may communicate with the network device 120.
  • the aforementioned network device may be a device used to communicate with a terminal device.
  • a terminal device may be an evolved base station (evolved Node B, eNB or eNodeB) in an LTE system, a gNB in a 5G network, a satellite in satellite communication, or a network-side device in a future communication system.
  • the network device may also be a relay station, an access point, a vehicle-mounted device, and so on.
  • D2D device-to-device
  • M2M machine-to-machine
  • the network device may also be a terminal that functions as a base station.
  • the aforementioned terminal may refer to User Equipment (UE), access terminal, user unit, mobile station, remote station, remote terminal, mobile equipment, user terminal, wireless communication equipment, user agent, or user device.
  • the access terminal can be a cellular phone, a cordless phone, a Session Initiation Protocol (SIP) phone, a wireless local loop (Wireless Local Loop, WLL) station, a personal digital processing (Personal Digital Assistant, PDA), with wireless communication Functional handheld devices, computing devices or other processing devices connected to wireless modems, in-vehicle devices, wearable devices, terminal devices in future communication networks, etc.
  • SIP Session Initiation Protocol
  • WLL Wireless Local Loop
  • PDA Personal Digital Assistant
  • the execution subject of the Polar code rate matching method may be the above-mentioned network device or terminal.
  • the network device or terminal When the network device or terminal is used as the sender to send data or information, it can be used This Polar code rate matching method.
  • the network device or terminal when the network device or terminal serves as the receiving end to receive data or information, it is also necessary to confirm the rate matching sequence first according to the method of the present invention.
  • the Polar code rate matching method provided by the embodiment of the present application will be described in detail below.
  • K bits to be coded here may also include, but are not limited to, one or more of other types of non-freezing bits such as CRC bits, PC bits, and so on.
  • the fixed code rate can also be a common code rate such as 1/2, 1/3, or 1/4. This approach can further reduce the complexity of the implementation. If a variable bit rate is used, it needs to be notified by means such as signaling to keep the receiving and sending ends consistent. The advantage of this method is that resources can be used more flexibly. In the specific application, it can be stipulated in advance through a standard agreement or a proprietary agreement.
  • Operation 220 Perform polarization encoding on the K bits to be encoded according to the length N of the mother code to obtain an encoded bit sequence;
  • the rate matching sequence needs to be used to determine which ones to puncture or shorten (ie delete) Polarized sub-channels and possible P pre-freeze polarized sub-channels (P greater than or equal to 0), and then select the K polarizers with higher reliability among the remaining MP polarized sub-channels according to their reliability
  • the channel is used to place K information bits.
  • the decoding side also determines the positions of the K information bits in the same way. To avoid repetition, the description of the rate matching sequence is placed in operation 230 for explanation.
  • interleaving is also performed on the encoded bit sequence.
  • This interleaving operation is slightly different from the sub-block interleaving in 5G NR. It does not need to be divided into 32 sub-blocks (or other numbers of sub-blocks). Instead, the encoded bit sequence is mapped to the circular buffer according to the rate matching sequence for subsequent steps. Punching operation (of course the shortening operation is similar).
  • Operation 230 Perform a rate matching operation on the encoded bit sequence according to the rate matching sequence to obtain a rate matching bit sequence of length M;
  • the minimum value of the polarized sub-channel number is 1 and the maximum value is N 2.
  • the first sequence can also be sequence 3.
  • the sequence 3 includes N 2 values of different sizes. The sequence number at the position of each value is the polarization subchannel number, and the size of the value is that the polarization subchannel where the value is located is rate-matched Priority.
  • sequences 1, 2, and 3 are essentially equivalent.
  • sequence 2 is essentially a sub-sequence of sequence 1, which directly takes the first half of sequence 1, and the value of each element in sequence 3 indicates the position of the element (that is, the polarized sub-channel) being punctured.
  • the priority of the hole the smaller the value, the higher the priority of being punched.
  • sequence 1 or sequence 2 if the target code length is M, then what needs to be punctured (or deleted) is the bit carried on the polarization sub-channel corresponding to the first NM polarization sub-channel numbers in sequence 1 or sequence 2.
  • sequence 3 what needs to be punctured is the bit carried by the polar subchannel where the smallest NM value in the puncturing sequence 3 is located, or in other words, the bit carried by the polar subchannel where the largest M value is reserved.
  • the priority of the hole is 3 (that is, the lowest), the priority of the second polarized sub-channel is punctured is the first, and the priority of the third polarized sub-channel is punctured is the second, similarly,
  • both methods indicate that the priority of the 0th polarized subchannel is shortened to the 0th (that is, the highest), and the first polarized subchannel is shortened to the 3rd priority ( That is, the highest), the shortened priority of the second polarized sub-channel is the first, and the shortened priority of the third polarized sub-channel is the second, so the nesting method of sequence 3 can be similar to the sequence on the one hand.
  • the nesting method of 1 selects a value less than or equal to N from a sequence greater than N.
  • sequence 1 and sequence 3 can be taken as the same sequence as sequence 1 and sequence 3 of the puncturing operation (so sequence 1 and sequence 3 in Appendix 1 can be used directly), and sequence 2 can be used.
  • sequence 2 can be used.
  • the value of each element in sequence 3 indicates the shortened priority of the position of the element (that is, the polarization sub-channel), the larger the value, the more The higher the priority of shortening.
  • sequence 1 or sequence 2 if the target code length is M, then what needs to be shortened is the bit carried on the polarized sub-channel corresponding to the last NM polarized sub-channel numbers of sequence 1 or sequence 2.
  • the rate matching sequence may be a subsequence of the first sequence.
  • a subsequence of a sequence may also include the sequence itself.
  • sequence A is a subsequence of sequence B, which refers to All elements in sequence A are taken from sequence B, and the relative order of these elements is exactly the same as their relative order in sequence B.
  • N 892
  • the rate matching sequence is 1a or 2a or 3a
  • sequence 1a is the same as sequence 1
  • sequence 2a is the same as sequence 2
  • sequence 3a is the same as sequence 3, so sequences 1a, 2a, and 3a will not be repeated.
  • the length is 1024
  • the rate matching sequences 1d, 2d, and 3d with a mother code length of 1024 the sequence 2d is 512 in length
  • the rate matching sequences 1e, 2e, and 3e with a mother code length of 512 the sequence 2e is 256 in length.
  • sequence 1a, sequence 1b, sequence 1c, sequence 1d, and sequence 1e are subsequences of sequence 1, which all meet the nesting characteristics described above; sequence 2a, sequence 2b, sequence 2c, sequence 2d, and sequence 2e are the subsequences of sequence 2.
  • sequence 3a, sequence 3b, sequence 3c, sequence 3d, and sequence 3e are subsequences of sequence 3
  • sequence 3b is the first half of sequence 3a
  • sequence 3c is the first half of sequence 3b (that is, the first 1/4 sequence of sequence 3a)
  • sequence 3d is the first half of sequence 3c (that is, the first 1/8 sequence of sequence 3a)
  • sequence 3e is the first half of sequence 3d Half of the sequence (that is, the first 1/16 sequence of sequence 3a) also meets the nesting characteristics described above.
  • the communication system will define a maximum mother code length N max .
  • the sending end and the receiving end can store only the puncturing sequence corresponding to N max (that is, the rate matching sequence), or the puncturing sequence corresponding to different mother code lengths less than or equal to N max . It can also store N max and commonly used numbers. Punching sequences corresponding to different mother code lengths.
  • appendix 2 gives another example of the first sequence.
  • rate matching sequences there are two possible rate matching sequences available. In actual applications, one of them can be used, unless otherwise specified.
  • This application takes Appendix 1 as an example.
  • Operation 240 output the sequence after rate matching.
  • the output here should be a sequence modulated based on the bit sequence after rate matching.
  • This sequence is a symbol sequence, and the length of the sequence is not necessarily M at this time.
  • the sequence after rate matching is the bit sequence after rate matching. Since modulation itself is not an indispensable technology for implementing the solution of the embodiments of this application, the terms "rate-matched sequence” and “rate-matched bit sequence” are combined in this application without affecting understanding. No strict distinction is made.
  • the acquisition includes K Determine the rate-matching sequence for the information sequence to be decoded of the bits to be decoded, the target code length M and the length of the mother code N, determine the rate matching sequence so as to perform rate matching and decoding and demodulation operations on the to-be-decoded information sequence sequence, thereby obtaining and outputting the decoding result, where
  • the solution rate matching is to restore the punctured position or the shortened position.
  • the likelihood ratio (LLR) can be considered to be 0 when decoding, and for the shortened position, the LLR can be considered to be infinite when decoding.
  • Detailed decoding algorithm There is no limitation here.
  • the positions of a few elements in the sequence are interchanged.
  • the position of the serial number can be adjusted within a set range, for example, if the set range is 5, the position of the element with the serial number 10 can be adjusted within 5 positions on the left and right;
  • the sequence contains N elements starting from 0 and ending at N-1, and the N elements starting from 0 and ending at N-1 represent the sequence numbers of the N polarized sub-channels.
  • the sequence numbers of the N polarized sub-channels can also start from 1 to N, just add 1 to each sequence number in the above sequence, which is also the sequence number format in the above calculation methods.
  • other methods may also be adopted to indicate the sequence number or identification of the above-mentioned polarized sub-channel, and the specific expression method does not affect the specific position of the polarized sub-channel indicated in the sequence.
  • the sequence 3b, 3c, 3d, and 3e may not all have elements between 0 and N-1, but it does not affect the priority ordering of the polarized sub-channels.
  • sequence numbers of the N polarized sub-channels in the above sequence are arranged from small to large according to the priority of the rate matching of the N polarized sub-channels.
  • sequence numbers of the N polarized sub-channels can also be arranged according to the N polar sub-channels.
  • the priority of the rate matching of the sub-channels is arranged in descending order, and the elements in the above sequence can be arranged in reverse order.
  • an embodiment of the present application also provides a Polar code rate matching device 300.
  • Part or all of the Polar code rate matching method shown in FIG. 3 can be implemented by hardware or software.
  • the Polar code rate matching device 300 For the Polar code rate matching device 300, based on the same inventive concept of the Polar code rate matching method shown in FIG. 2, the Polar code rate matching device 300 is used to implement the Polar code rate matching method shown in FIG. 2.
  • the Polar code rate matching device 300 includes: an input interface circuit 301 for obtaining the bits to be encoded; a logic circuit 302 for executing the Polar code rate matching method shown in FIG. 2 above, please refer to the previous method for details The description in the embodiment will not be repeated here; the output interface circuit 303 is used to output the sequence after the rate matching.
  • the rate-matched sequence is still a modulated sequence, this sequence is a modulated symbol sequence, which is output to the transceiver 320, and the transceiver 320 performs corresponding processing on the modulated sequence (including but not limited to digital-to-analog conversion and / Or processing such as frequency conversion) and then sent out through the antenna 330.
  • the Polar code rate matching device 300 may be a chip or an integrated circuit during specific implementation.
  • the Polar code rate matching apparatus 400 includes: a memory 401 for storing a program; and a processor 402 , For executing the program stored in the memory 401, when the program is executed, the Polar code rate matching apparatus 400 can implement the Polar code rate matching method provided in the foregoing embodiment.
  • the foregoing memory 401 may be a physically independent unit, or as shown in FIG. 5, the memory 501 and the processor 502 are integrated.
  • the Polar code rate matching apparatus 400 may also only include the processor 402.
  • the memory 401 for storing programs is located outside the Polar code rate matching device 400, and the processor 402 is connected to the memory 401 through a circuit/wire for reading and executing the programs stored in the memory 401.
  • an embodiment of the present application also provides a Polar code rate matching device 600, and the Polar code rate matching device 600 is used to execute the Polar code shown in FIG.
  • the Polar code rate matching device 600 includes:
  • the obtaining unit 601 is configured to obtain K bits to be coded and a target code length M;
  • the encoding unit 602 is configured to determine the mother code length N, and perform polarization encoding on the K to-be-encoded bits according to the mother code length N to obtain an encoded bit sequence. For a specific encoding process, refer to operation 220 of the foregoing embodiment;
  • the rate matching unit 603 is configured to perform a rate matching operation on the encoded bit sequence according to the rate matching sequence to obtain a rate matching bit sequence of length M (or a sequence obtained by further modulating the bit sequence, also called rate matching The following sequence).
  • a rate matching bit sequence of length M or a sequence obtained by further modulating the bit sequence, also called rate matching The following sequence.
  • the receiving end device can also be similarly designed.
  • the Polar code de-rate matching device 700 includes: an input interface circuit 701 for acquiring the information sequence to be decoded, and a logic circuit 702 for executing the aforementioned Polar code de-rate matching method to acquire decoded information bits; and an output interface circuit 703 Used to output decoded information bits.
  • the Polar code de-rate matching device 700 may further include a transceiver 720 to obtain the information sequence to be decoded through the antenna 730. In specific implementation, it can be a chip or an integrated circuit.
  • the Polar code de-rate matching device 800 includes: a memory 801 for storing a program; and processing; The device 802 is configured to execute a program stored in the memory 801, and when the program is executed, the Polar code rate matching device 800 can implement the Polar code rate matching method provided in the foregoing embodiment.
  • the foregoing memory 801 may be a physically independent unit, or as shown in FIG. 9, the memory 901 and the processor 902 are integrated.
  • the Polar code rate dematching apparatus 800 may also only include the processor 802.
  • the memory 01 for storing the program is located outside the Polar code derate matching device 800, and the processor 802 is connected to the memory 801 through a circuit/wire for reading and executing the program stored in the memory 801.
  • the processor 402 and/or the processor 802 may be a central processing unit (CPU), a network processor (NP), or a combination of a CPU and an NP.
  • CPU central processing unit
  • NP network processor
  • the processor 402 and/or the processor 802 may be a central processing unit (CPU), a network processor (NP), or a combination of a CPU and an NP.
  • the processor 402 and/or the processor 802 may further include a hardware chip.
  • the aforementioned hardware chip may be an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof.
  • ASIC application-specific integrated circuit
  • PLD programmable logic device
  • the above-mentioned PLD may be a complex programmable logic device (CPLD), a field-programmable gate array (FPGA), a generic array logic (GAL), or any combination thereof.
  • CPLD complex programmable logic device
  • FPGA field-programmable gate array
  • GAL generic array logic
  • the memory in the foregoing embodiments may include volatile memory (volatile memory), such as random-access memory (RAM); the memory may also include non-volatile memory (non-volatile memory), such as flash Flash memory, hard disk drive (HDD) or solid-state drive (SSD); the memory may also include a combination of the above types of memory.
  • volatile memory such as random-access memory (RAM)
  • non-volatile memory such as flash Flash memory, hard disk drive (HDD) or solid-state drive (SSD)
  • SSD solid-state drive
  • the memory may also include a combination of the above types of memory.
  • Polar code derate rate matching device 1000 includes:
  • the obtaining unit 1001 is configured to obtain the information sequence to be decoded
  • the processing unit 1002 is configured to determine the mother code length N, and perform rate matching and decoding demodulation operations on the information sequence to be decoded according to the mother code length N and the rate matching sequence, so as to obtain and output the decoding result;
  • the embodiment of the present application also provides a computer storage medium that stores computer program instructions.
  • the computer program instructions are executed by a computer, the aforementioned Polar code rate matching method or Polar code derate rate matching method is executed.
  • the embodiment of the present application also provides a computer program product containing instructions, which when running on a computer, enables the aforementioned Polar code rate matching method or Polar code derate rate matching method to be executed.
  • this application can be provided as a method, a system, or a computer program product. Therefore, this application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, this application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.
  • a computer-usable storage media including but not limited to disk storage, CD-ROM, optical storage, etc.
  • These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing equipment to work in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture including the instruction device.
  • the device implements the functions specified in one process or multiple processes in the flowchart and/or one block or multiple blocks in the block diagram.
  • These computer program instructions can also be loaded on a computer or other programmable data processing equipment, so that a series of operation steps are executed on the computer or other programmable equipment to produce computer-implemented processing, so as to execute on the computer or other programmable equipment.
  • the instructions provide steps for implementing the functions specified in one process or multiple processes in the flowchart and/or one block or multiple blocks in the block diagram.
  • Appendix 1 An example of the first sequence:
  • Punc_Q [0, 1, 2, 3. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, twenty one, twenty two, twenty three, twenty four, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
  • Punc_Q_half [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 ,
  • Punc_Z [0, 1, 2, 3. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, twenty one, twenty two, twenty three, twenty four, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
  • sequence 1a is sequence 1
  • sequence 2a is sequence 2
  • sequence 3a is sequence 3;
  • Sequence 1b is:
  • Punc_Q [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • Sequence 2b is:
  • Punc_Q_half [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 ,
  • Sequence 3b is:
  • Punc_Z [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • the sequence 1c is:
  • Punc_Q [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • the sequence 2c is:
  • Punc_Q_half [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 ,
  • Punc_Z [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • the sequence 1d is:
  • Punc_Q [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • the sequence 2d is:
  • Punc_Q_half [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 ,
  • Punc_Z [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 , 99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,
  • the sequence 1e is:
  • Punc_Q [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 , 124,
  • Sequence 2e is:
  • Punc_Q_half [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 ,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123 ,
  • Sequence 3e is:
  • Punc_Z [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48 ,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73 ,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98 , 99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,
  • Appendix 2 Another example of the first sequence:
  • Shorten_Q [0, 4096, 2048, 6144, 1024, 5120, 3072, 7168, 512, 4608, 2560, 6656, 1536, 5632, 3584, 7680, 256, 4352, 2304, 6400, 1280, 5376, 3328, 7424, 768, 4864, 2816, 6912, 1792, 5888, 3840, 7936, 128, 4224, 2176, 6272, 1152, 5248, 3200, 7296, 640, 4736, 2688, 6784, 1664, 5760, 3712, 7808, 384, 4480, 2432, 6528, 1408, 5504, 3456, 7552, 896, 4992, 2944, 7040, 1920, 6016, 3968, 8064, 64, 4160, 2112, 6208, 1088, 5184, 3136, 7232, 576, 4672, 2624, 6720, 1600, 5696, 3648, 7744, 320, 4416,
  • Shorten_Q_half [1,4097,2049,6145,1025,5121,3073,7169,513,4609,2561,6657,1537,5633,3585,7681,257,4353,2305,6401,1281,5377,3329,7425 ,769,4865,2817,6913,1793,5889,3841,7937,129,4225,2177,6273,1153,5249,3201,7297,641,4737,2689,6785,1665,5761,3713,7809,385 ,4481,2433,6529,1409,5505,3457,7553,897,4993,2945,7041,1921,6017,3969,8065,65,4161,2113,6209,1089,5185,3137,7233,577,4673 ,2625,6721,1601,5697,3649,7745,321,4417,2369,6465,1345,5441,3393,7489,
  • Shorten_Z [0, 4096, 2048, 6144, 1024, 5120, 3072, 7168, 512, 4608, 2560, 6656, 1536, 5632, 3584, 7680, 256, 4352, 2304, 6400, 1280, 5376, 3328, 7424, 768, 4864, 2816, 6912, 1792, 5888, 3840, 7936, 128, 4224, 2176, 6272, 1152, 5248, 3200, 7296, 640, 4736, 2688, 6784, 1664, 5760, 3712, 7808, 384, 4480, 2432, 6528, 1408, 5504, 3456, 7552, 896, 4992, 2944, 7040, 1920, 6016, 3968, 8064, 64, 4160, 2112, 6208, 1088, 5184, 3136, 7232, 576, 4672, 2624, 6720, 1600, 5696, 3648, 7744, 320, 4416,
  • sequence 1a is sequence 1
  • sequence 2a is sequence 2
  • sequence 3a is sequence 3;
  • Sequence 1b is:
  • Shorten_Q [0,2048,1024,3072,512,2560,1536,3584,256,2304,1280,3328,768,2816,1792,3840,128,2176,1152,3200,640,2688,1664,3712 ,384,2432,1408,3456,896,2944,1920,3968,64,2112,1088,3136,576,2624,1600,3648,320,2368,1344,3392,832,2880,1856,3904,192 ,2240,1216,3264,704,2752,1728,3776,448,2496,1472,3520,960,3008,1984,4032,32,2080,1056,3104,544,2592,1568,3616,288,2336 ,1312,3360,800,2848,1824,3872,160,2208,1184,3232,672,2720,1696,3744,416,2464,1440,3488,928,2976,1952,4000,96,214
  • Sequence 2b is:
  • Shorten_Q_half [1,2049,1025,3073,513,2561,1537,3585,257,2305,1281,3329,769,2817,1793,3841,129,2177,1153,3201,641,2689,1665,3713 ,385,2433,1409,3457,897,2945,1921,3969,65,2113,1089,3137,577,2625,1601,3649,321,2369,1345,3393,833,2881,1857,3905,193 ,2241,1217,3265,705,2753,1729,3777,449,2497,1473,3521,961,3009,1985,4033,33,2081,1057,3105,545,2593,1569,3617,289,2337 ,1313,3361,801,2849,1825,3873,161,2209,1185,3233,673,2721,1697,3745,417,2465,1441,3489,929,
  • Sequence 3b is:
  • Shorten_Z [0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424 ,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384 ,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672 ,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,
  • the sequence 1c is:
  • Shorten_Q [0,1024,512,1536,256,1280,768,1792,128,1152,640,1664,384,1408,896,1920,64,1088,576,1600,320,1344,832,1856 ,192,1216,704,1728,448,1472,960,1984,32,1056,544,1568,288,1312,800,1824,160,1184,672,1696,416,1440,928,1952,96 ,1120,608,1632,352,1376,864,1888,224,1248,736,1760,480,1504,992,2016,16,1040,528,1552,272,1296,784,1808,144,1168 ,656,1680,400,1424,912,1936,80,1104,592,1616,336,1360,848,1872,208,1232,720,1744,464,1488,976,2000,48,1072,560 ,1584,304,1328,816,1840,176,1200,688,1712,432,
  • the sequence 2c is:
  • Shorten_Q_half [1,1025,513,1537,257,1281,769,1793,129,1153,641,1665,385,1409,897,1921,65,1089,577,1601,321,1345,833,1857 ,193,1217,705,1729,449,1473,961,1985,33,1057,545,1569,289,1313,801,1825,161,1185,673,1697,417,1441,929,1953,97 ,1121,609,1633,353,1377,865,1889,225,1249,737,1761,481,1505,993,2017,17,1041,529,1553,273,1297,785,1809,145,1169 ,657,1681,401,1425,913,1937,81,1105,593,1617,337,1361,849,1873,209,1233,721,1745,465,1489,977,2001,49,1073,561 ,1585,305
  • Shorten_Z [0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424 ,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384 ,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672 ,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,
  • the sequence 1d is:
  • Shorten_Q [0,512,256,768,128,640,384,896,64,576,320,832,192,704,448,960,32,544,288,800,160,672,416,928,96,608,352,864,224,736,480,992,16,528,272,784,144,656,400,912,80,592,336,848,208,720,464,976,48,560,304,816,176,688,432,944,112,624,368,880,240,752,496,1008,8,520,264,776,136,648,392,904,72,584,328,840,200,712,456,968,40,552,296,808,168,680,424,936,104,616,360,872,232,744,488,1000,24,536,280,792,152,664,408,920,88,600,344,856,216,728,472,984,56,568,312,824,184,696,440,952,120,632,376,888,
  • the sequence 2d is:
  • Shorten_Q_half [1,513,257,769,129,641,385,897,65,577,321,833,193,705,449,961,33,545,289,801,161,673,417,929,97,609,353,865,225,737,481,993,17,529,273,785,145,657,401,913,81,593,337,849,209,721,465,977,49,561,305,817,177,689,433,945,113,625,369,881,241,753,497,1009,9,521,265,777,137,649,393,905,73,585,329,841,201,713,457,969,41,553,297,809,169,681,425,937,105,617,361,873,233,745,489,1001,25,537,281,793,153,665,409,921,89,601,345,857,
  • Shorten_Z [0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424 ,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384 ,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672 ,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,
  • the sequence 1e is:
  • Shorten_Q [0,256,128,384,64,320,192,448,32,288,160,416,96,352,224,480,16,272,144,400,80,336,208,464,48,304,176,432,112,368,240,496,8,264,136,392,72,328,200,456,40,296,168,424,104,360,232,488,24,280,152,408,88,344,216,472,56,312,184,440,120,376,248,504,4,260,132,388,68,324,196,452,36,292,164,420,100,356,228,484,20,276,148,404,84,340,212,468,52,308,180,436,116,372,244,500,12,268,140,396,76,332,204,460,44,300,172,428,108,364,236,492,28,284,156,412,92,348,220,476 , 60,316,188,444,
  • Sequence 2e is:
  • Shorten_Q_half [1,257,129,385,65,321,193,449,33,289,161,417,97,353,225,481,17,273,145,401,81,337,209,465,49,305,177,433,113,369,241,497,9,265,137,393,73,329,201,457,41,297,169,425,105,361,233,489,25,281,153,409,89,345,217,473,57,313,185,441,121,377,249,505,5,261,133,389,69,325,197,453,37,293,165,421,101,357,229,485,21,277,149,405,85,341,213,469,53,309,181,437,117,373,245,501,13,269,141,397,77,333,205,461,45,301,173,429,109,365,237,
  • Sequence 3e is:
  • Shorten_Z [0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424 ,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384 ,4480,2432,6528,1408,5504,3456,7552, 896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672 ,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880

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Abstract

一种Polar码速率匹配的方法、装置和介质，能够提高系统的可靠性。该方法包括：获取K个待编码比特，获取母码长度N，N＝2 n（210），根据所述母码长度N对所述K个待编码比特进行极化编码，得到编码后的比特序列（220），根据速率匹配序列对极化编码后的比特序列进行速率匹配，获取长度为M的速率匹配后的比特序列（230），输出速率匹配后的序列（240）。根据给定的速率匹配序列，可以简化速率匹配的操作。

Description

一种Polar码速率匹配方法及装置 技术领域
本申请实施例涉及通信技术领域，尤其涉及一种Polar码速率方法及装置。
背景技术
信道编码作为最基本的无线接入技术，在保证数据的可靠性传输方面起到至关重要的作用。在现有的无线通信系统中，一般采用Turbo码、低密度奇偶校验码(low density parity check，LDPC)和极化(Polar)码进行信道编码。Turbo码不能够支持过低或过高码率的信息传输。而对于中短包传输，Turbo码和LDPC码也由于自身编译码的特点，在有限码长下很难达到理想的性能。在实现方面，Turbo码和LDPC码在编译码实现过程中具有较高的计算复杂度。Polar码是理论上证明可以取得香农容量，且具有相对简单的编译码复杂度的好码，因而得到了越来越广泛的应用。在第五代(5th generation，5G)通信系统中，将Polar码确定为控制信道的编码方式。
但在卫星通信中，尚未明确到底采用何种信道编码方式，如果要将Polar码引入卫星通信系统，需要进行怎样的设计才能满足需求，就成为一个需要解决的问题。
发明内容
本申请实施例提供一种应用于卫星通信的Polar码编码的速率匹配方法及装置，具有实现简单的优点。
本申请实施例提供的具体技术方案如下：
第一方面，提供一种Polar码编码的速率匹配方法，
获取K个待编码比特、目标码长M、母码长度N，N＝2 n，M、K、n和N为正整数；
根据所述母码长度N对所述K个待编码比特进行极化编码，得到编码后的比特序列；
根据速率匹配序列对所述编码后的比特序列进行速率匹配，获取速率匹配后的序列；
输出所述速率匹配后的序列。
这种设计切合了卫星通信的特点，速率匹配序列的嵌套特性也为具体实现提供了多种可能的方式，易于实现，简化了速率匹配的操作，减小了实现难度。
在一个可能的设计中，所述速率匹配序列为第一序列的子序列，所述第一序列为附录一中的序列1或序列2或序列3，其中所述序列1包括N2个极化子信道号，所述序列2包括N2/2个极化子信道号，所述序列3包括N2个大小不同的值，所述极化子信道号由0开始，N2大于或等于N。
在一个可能的设计中，当N＝8192，所述速率匹配序列为说明书中的序列1a或序列2a或序列3a；或者
当N＝4096，所述速率匹配序列为说明书中的序列1b或2b或3b；或者
当N＝2048，所述速率匹配序列为说明书中的序列1c或2c或3c；或者
当N＝1024，所述速率匹配序列为说明书中的序列1d或2d或3d；或者
当N＝512，所述速率匹配序列为说明书中的序列1e或2e或3e；
其中所述序列1a、所述序列1b、所述序列1c、所述序列1d、所述序列1e为所述序列1的子序列；
所述序列2a、所述序列2b、所述序列2c、所述序列2d、所述序列2e为所述序列2的子序列；
所述序列3a、所述序列3b、所述序列3c、所述序列3d、所述序列3e为所述序列3的子序列。
在一个可能的设计中，所述速率匹配的方式为打孔操作，
若所述第一序列为所述序列1，所述打孔操作为：根据所述目标码长M取所述速率匹配序列中后M个极化子信道号对应子信道承载的比特或根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
若所述第一序列为所述序列2，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
若所述第一序列为所述序列3，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中取值最小的N-M个值所在的极化子信道承载的比特或根据所述目标码长M取所述速率匹配序列中取值最大的M个值所在的极化子信道承载的比特。
在一个可能的设计中，所述根据所述母码长度N对所述K个待编码比特进行极化编码为：所述K个待编码比特映射到M-P个非打孔的极化子信道中可靠度较高的K个极化子信道，使得K个待编码比特所在的极化子信道的可靠度高于其余M-P-K个极化子信道的可靠度，P为预冻结的极化子信道的数目，P大于或等于0。
在一个可能的设计中，所述根据速率匹配序列对所述编码后的比特序列进行速率匹配获取速率匹配后的序列之前，对所述编码后的比特序列进行交织。
在一个可能的设计中，所述K个待编码比特中，包括循环冗余校验比特。
在一个可能的设计中，所述K个待编码比特中，包括奇偶校验比特。
在一个可能的设计中，所述速率匹配方法由网络设备或终端执行。
第二方面，提供一种Polar码速率匹配装置，该装置具有实现上述第一方面和第一方面的任一种可能的设计中所述的方法的功能。所述功能可以通过硬件实现，也可以通过硬件执行相应的软件实现。所述硬件或软件包括一个或多个与上述功能相对应的模块。
在一个可能的设计中，当所述功能的部分或全部通过硬件实现时，所述Polar码速率匹配装置包括：输入接口电路，用于获取待编码比特；逻辑电路，用于执行上述第一方面和第一方面的任一种可能的设计中所述的行为；输出接口电路，用于输出速率匹配后的序列。
可选的，所述Polar码速率匹配装置可以是芯片或者集成电路。
在一个可能的设计中，当所述功能的部分或全部通过软件实现时，所述Polar码速率匹配装置包括：存储器，用于存储程序；处理器，用于执行所述存储器存储的所述程序，当所述程序被执行时，所述Polar码速率匹配装置可以实现如上述第一方面和第一方面的任一种可能的设计中所述的方法。
可选的，上述存储器可以是物理上独立的单元，也可以与处理器集成在一起。
在一个可能的设计中，当所述功能的部分或全部通过软件实现时，所述Polar码速率匹配装置包括处理器。用于存储程序的存储器位于所述速率匹配装置之外，处理器通过电路/ 电线与存储器连接，用于读取并执行所述存储器中存储的程序。
在一个可能的设计中，所述装置为网络设备或终端。
第三方面，提供了一种通信系统，该通信系统包网络设备和终端，所述网络设备或终端可以执行如上述第一方面及其可能的设计所述的方法。
第四方面，提供了一种计算机存储介质，存储有计算机程序，该计算机程序包括用于执行第一方面和第一方面的任一可能设计中任一种所述的方法的指令。
第五方面，本申请实施例提供了一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行上述各方面所述的方法。
第六方面，提供了一种无线设备，包括用于实施第一方面和第一方面的任一可能设计的速率匹配装置和收发器，
所述收发器用于接收或发送信号。
在一个可能的设计中，所述无线设备为终端或者网络设备。
附图说明
图1为本申请实施例中应用的通信系统架构示意图；
图2为本申请实施例中Polar码速率匹配方法的流程示意图；
图3为本申请实施例中Polar码速率匹配装置结构示意图之一；
图4为本申请实施例中Polar码速率匹配装置结构示意图之二；
图5为本申请实施例中Polar码速率匹配装置结构示意图之三；
图6为本申请实施例中Polar码速率匹配装置结构示意图之四；
图7为本申请实施例中Polar码解速率匹配装置结构示意图之一；
图8为本申请实施例中Polar码解速率匹配装置结构示意图之二；
图9为本申请实施例中Polar码解速率匹配装置结构示意图之三；
图10为本申请实施例中Polar码解速率匹配装置结构示意图之四。
具体实施方式
下面将结合附图，对本申请实施例进行详细描述。
如果不考虑卫星通信的特点，直接将现有5G标准中的Polar码编码方式照搬到卫星通信中，会有至少以下几个问题：1)目前的5G标准中Polar码只用于控制信道，所以不一定能完全适用于数据信道；2)目前的5G标准中，Polar码的母码长度最长为1024，所以能支持的数据包长度有限。因此有必要重新设计Polar码的编码方式，以便更适应于卫星通信，同时兼顾适用于地面通信。
本申请实施例提供一种Polar码速率匹配方法及装置，根据极化子信道的可靠度，在线计算或者预先存储可靠度排序，按照可靠度排序选择用于发送信息比特的极化子信道的序号，按照所选择的信息比特序号进行Polar码编码，然后根据速率匹配序列对编码后的序列进行速率匹配操作，并输出速率匹配后的序列。本申请实施例更适用于卫星通信，尤其是卫星通信中的短消息业务。下面结合附图详细说明本发明实施例提供的速率匹配方法和装置。
为方便对本申请实施例的理解，下面对Polar码作简单介绍。
Polar码的编码策略利用无噪信道传输用户有用的信息，全噪信道传输约定的信息或者 不传信息。Polar码也是一种线性块码，其编码矩阵为G N，编码过程为
Figure PCTCN2021099903-appb-000001
其中
Figure PCTCN2021099903-appb-000002
是一个二进制的行矢量，长度为N(即码长，可以看到，编码前后的序列x和u的长度均为N，N也称为母码码长)；G N是一个N×N的矩阵，且
Figure PCTCN2021099903-appb-000003
Figure PCTCN2021099903-appb-000004
定义为log 2N个矩阵F 2的克罗内克(Kronecker)乘积。上述矩阵
Figure PCTCN2021099903-appb-000005
Polar码的编码过程中，
Figure PCTCN2021099903-appb-000006
中的一部分比特用来携带信息，称为信息比特集合，这些比特的索引的集合记作A；另外的一部分比特设置为接收端和发送端预先约定的固定值，称之为冻结比特集合或冻结比特集合(frozen bits)，其索引的集合用A的补集A c表示。Polar码的编码过程相当于：
Figure PCTCN2021099903-appb-000007
这里，G N(A)是G N中由集合A中的索引对应的那些行得到的子矩阵，G N(A C)是G N中由集合A c中的索引对应的那些行得到的子矩阵。u A为
Figure PCTCN2021099903-appb-000008
中的信息比特集合，数量为K，一般地，包括但不限于循环冗余校验(Cyclic Redundancy Check，简称CRC)比特和奇偶校验(Parity Check，简称PC)比特在内的各类校验比特也可以都包括在信息比特集合中；
Figure PCTCN2021099903-appb-000009
为
Figure PCTCN2021099903-appb-000010
中的冻结比特集合，其数量为(N-K)，是已知比特。这些冻结比特通常被设置为0，但是只要接收端和发送端预先约定，冻结比特可以被任意设置。从而，Polar码的编码输出可简化为：
Figure PCTCN2021099903-appb-000011
这里u A为
Figure PCTCN2021099903-appb-000012
中的信息比特集合，u A为长度K的行矢量，即|A|＝K，|·|表示集合中元素的个数，K为信息块大小，G N(A)是矩阵G N中由集合A中的索引对应的那些行得到的子矩阵，G N(A)是一个K×N的矩阵。
在确定母码码长N后，Polar码的构造过程即集合A的选取过程，决定了Polar码的性能。Polar码的构造过程通常是，根据母码码长N确定共存在N个极化子信道，分别对应编码矩阵的N个行，在不考虑速率匹配的情况下，将可靠度较高的前K个极化子信道的索引作为集合A的元素，剩余(N-K)个极化子信道对应的索引作为冻结比特的索引集合A c的元素。集合A决定了信息比特的位置，集合A c决定了冻结比特的位置。极化子信道的序号即为信息比特或者冻结比特的位置索引，也即在
Figure PCTCN2021099903-appb-000013
中的位置索引。
当考虑速率匹配的时候，主要是考虑打孔(puncture)或缩短(shorten)的时候，一般是先确定需要打孔或缩短(即删除)的N-M个极化子信道，这里M是目标码长，即速率匹配后的比特序列长度，这里选出的N-M个极化子信道用于放置冻结比特，在5G NR标准中，针对打孔的情形，还额外确定了一些称为预冻结的极化子信道，也用于放置冻结比特，这里不妨将预冻结的极化子信道数量定义为P，P大于或者等于0(对于缩短的情形，P显然为0，这时候是可以不需要考虑P的)，然后在剩下的M-P个极化子信道中按照可靠度挑选其中可靠度较高的K个极化子信道用于放置K个信息比特。当然，也可以先选择可靠度较低的M-P-K个子信道用于放置冻结比特，而余下的K个子信道则用于放置信息比特。放置K个信息比特的K个极化子信道中的任意一个极化子信道的可靠度都高于用于放置冻结比特的M-P-K个子信道中的任意一个子信道的可靠度。在本申请中，对P的取值不做限制，即使对于打孔的情形，P也可以为0，即使对于缩短的情形，P也可以取为大于0，并不影响本申请技术方案的实施，无论是编码端还是解码端，确认K个子信道用于放置K个信息比特 的原则和方法是一致的。同时，5G NR标准中，对于N-M个极化子信道的选取，是将子块交织(分成32个子块)后的序列放到循环缓存(英文为circular buffer，相当于速率匹配序列)上，如果是打孔，则在循环缓存上从第N-M个位置开始读取，丢弃从0到N-M-1位置上的比特，如果是缩短，则在循环缓存上从第0个位置开始读取直至第M-1个位置，而丢弃从第M到N-1位置上的比特。这种方式并未考虑不同母码长度之间的速率匹配序列之间的关系。
需要说明的是，这里说的可靠度的相对关系是基于一个给定的可靠度的计算方式，不同的可靠度计算方式可能会导致极化子信道的可靠度相对关系发生变化，但选择放置信息比特的极化子信道的方法是一样的。在5G NR标准之外，信息比特也可以考虑放置于最后被打孔或者缩短的极化子信道。本申请并不限制必须按照5G NR的标准选择放置信息比特的极化子信道。
在当前的卫星通信中，例如北斗卫星通信系统中，有个短报文通信模式。可以看作是现在常用的“短信息”，但又有不同，具体体现在这类发布的信息除了显示信息本身外，既能够定位，又能显示发布者的位置。但是，当前北斗二代短报文系统采用的Turbo码，可以采用Polar码来提升性能和降低复杂度。北斗短报文发送信息比特到地面中心站时，先对信息进行分段，分段后每段的信息比特长度为89到994(包含CRC比特)，采用的码率为1/2。后续消费类终端若要接入短报文系统，需要采用更低的码率，比如1/8。
本申请实施例中，通过预先给定速率匹配序列的方式，不同母码长度对应的速率匹配序列之间也满足嵌套关系，这样可以简化操作流程，提高处理效率。这个嵌套特性，类似于5G NR标准不同母码长度对应的可靠度排序序列类似，即母码长度Na小于或等于母码长度Nb时，Na对应的可靠度序列中的各极化子信道(极化子信道0，…，Na-1)的可靠度相对关系与这些极化子信道(极化子信道0，…，Na-1)在Nb对应的可靠度序列中的可靠度相对关系一致，换言之，在Nb对应的可靠度序列中，只保留小于Na的极化子信道号且不改变该Na个极化子信道号位置的前后相对关系，就可以得到Na对应的可靠度序列了。因此具体实现中，既可以把不同母码长度的可靠度序列都预先存储下来，也可以只预先存储最大母码长度N max对应的可靠度序列，然后其他母码长度Na对应的可靠度序列就从最大母码长度对应的可靠度序列中只保留小于N max的极化子信道的序号而得到，这样一方面可以节省一点存储开销，另一方面也便于实施，无需再考虑子块交织等操作。特别地，Na最大也可以等于N max。
类似地，母码长度Na小于或等于母码长度Nb时，Na对应的速率匹配序列中的各极化子信道(极化子信道0，…，Na-1)的速率匹配(打孔或缩短)顺序的相对关系与这些极化子信道(极化子信道0，…，Na-1)在Nb对应的速率匹配序列中的速率匹配顺序的相对关系一致，换言之，在Nb对应的速率匹配序列中，只保留小于Na的极化子信道号且不改变该Na个极化子信道号位置的前后相对关系，就可以得到Na对应的速率匹配序列了。因此具体实现中，既可以把不同母码长度的速率匹配序列都预先存储下来，也可以只预先存储最大母码长度N max对应的速率匹配序列，然后其他母码长度Na对应的速率匹配序列就从最大母码长度对应的速率匹配序列中只保留小于N max的极化子信道的序号而得到，特别地，Na最大也可以等于N max。
图1为本发明实施例无线通信网络的结构示意图。图1只是一种示例，其它能用到本 发明实施例的速率匹配方法或装置的无线网络也都在本发明的保护范围内。
如图1所示，无线通信网络100包括网络设备110，和终端112。当无线通信网络100包括核心网102时，该网络设备110还可以与核心网102相连。网络设备110还可以与IP网络104进行通信，例如，因特网(internet)，私有的IP网，或其它数据网等。网络设备为覆盖范围内的终端提供服务。例如，参见图1所示，网络设备110为网络设备110覆盖范围内的一个或多个终端112提供无线接入。除此之外，网络设备之间的覆盖范围可以存在重叠的区域，例如网络设备110和120。网络设备之间还可以可以互相通信，例如，网络设备110可以与网络设备120之间进行通信。
上述网络设备可以是用于与终端设备进行通信的设备。例如，可以是LTE系统中的演进型基站(Evolved Node B，eNB或eNodeB)，还可以是5G网络中的gNB，还可以是卫星通信中的卫星，或者未来通信系统中的网络侧设备等。或者该网络设备还可以是中继站、接入点、车载设备等。在设备对设备(Device to Device，D2D)通信系统、机器对机器(Machine to Machine，M2M)通信系统和车联网系统中，该网络设备还可以是担任基站功能的终端。
上述终端可以指用户设备(User Equipment，UE)、接入终端、用户单元、移动站、远方站、远程终端、移动设备、用户终端、无线通信设备、用户代理或用户装置。接入终端可以是蜂窝电话、无绳电话、会话启动协议(Session Initiation Protocol，SIP)电话、无线本地环路(Wireless Local Loop，WLL)站、个人数字处理(Personal Digital Assistant，PDA)、具有无线通信功能的手持设备、计算设备或连接到无线调制解调器的其它处理设备、车载设备、可穿戴设备，未来通信网络中的终端设备等。
基于图1所示的通信系统架构，本申请实施例中，执行Polar码速率匹配方法的执行主体可以为上述网络设备或终端，当网络设备或终端作为发送端发送数据或信息时，均可以采用此Polar码速率匹配方法。相应的，当网络设备或终端作为接收端接收数据或信息时，也需要根据本发明所述的方法先确认速率匹配序列。下面将对本申请实施例提供的Polar码速率匹配方法做详细介绍。
基于图1所示的通信系统架构，如图2所示，本申请实施例提供的Polar码速率匹配方法具体流程如下所述。
操作210：获取K个待编码比特、目标码长M、母码长度N，N＝2 n，M、K、n和N为正整数；
这里的K个待编码比特中，除了载荷比特外，还可以包括但不限于CRC比特、PC比特等等其他各种类型的非冻结比特中的一类或者多类。
为了系统简单起见，本发明不考虑重复这种速率匹配方式，因此母码长度N可以根据目标码长获取：N＝2^ceil(log 2(M))，即n＝ceil(log 2(M))，这里的ceil操作是指上取整操作，可以看出，N为不小于M的最小的满足2的整数次方的数。
另外，如果采用固定码率，那么在获取K个待编码比特的时候，目标码长M即可换算得到，或者在获取目标码长M后就可以知道该获取多少个待编码比特。例如采用固定码率1/8，则M＝8K，或者K＝M/8。固定码率还可以是1/2、1/3或1/4等常用码率。这种方式是可以进一步减少实现的复杂度。如果采用可变码率，则需要通过信令等方式通知，以使得收发两端保持一致，这种方式的好处是可以更灵活的使用资源。具体应用中，事先可以通过标准协议或者专有协议予以规定即可。
操作220：根据所述母码长度N对所述K个待编码比特进行极化编码，得到编码后的比特序列；
具体的编码操作可以参见前述的描述。如果是按照与5G NR标准类似的操作，先确定打孔或缩短的极化子信道，那么在这一步操作中就需要用到速率匹配序列，确定到底打孔或者缩短掉(即删除掉)哪些极化子信道以及可能的P个预冻结极化子信道(P大于或等于0)，然后在剩下的M-P个极化子信道中按照可靠度挑选其中可靠度较高的K个极化子信道用于放置K个信息比特。解码侧也通过同样的方式确定K个信息比特的位置。为免赘述，速率匹配序列的描述放在操作230中予以说明。需要注意的是，在5G NR系统中，需要在每次速率匹配的时候，确认到底是采用打孔还是缩短的速率匹配方式。在本申请中，为实现简单起见，可以固定采用打孔或者固定采用缩短作为速率匹配方式。
可选的，对编码后的比特序列，还进行交织。这个交织操作与5G NR中的子块交织略有不同，无需分成32个子块(或者其他数目的子块)，而是根据速率匹配序列将编码后的比特序列映射到循环缓存上以进行后续的打孔操作(当然缩短操作也是类似的)。
操作230：根据速率匹配序列对所述编码后的比特序列进行速率匹配操作获取长度为M的速率匹配后的比特序列；
对于打孔操作，在附录一中，提供了示例性的第一序列，第一序列为序列1或序列2或序列3，其中序列1包括N 2个极化子信道号，序列2包括N 2/2个极化子信道号，极化子信道号的最小值为0、最大值为N 2-1，N 2＝8192，N 2大于或等于N。当然，也可以是极化子信道号的最小值为1、最大值为N 2，这时候涉及的极化子信道号都相应加1即可。第一序列还可以是序列3，序列3中包括N 2个大小不同的值，每个值所在位置的序号就是极化子信道号，其中值的大小就是该值所在极化子信道被速率匹配的优先级。
需要注意的是，序列1、2、3实质上是等价的。对于打孔操作，序列2实质上就是序列1的子序列，直接取了序列1的前一半，而序列3中的各元素的取值表示该元素所在位置(即极化子信道)的被打孔的优先级，值越小，被打孔的优先级越高。根据序列1或序列2，如果目标码长为M，那么需要打孔(或者说删除)的就是序列1或者序列2中前N-M个极化子信道号对应的极化子信道上承载的比特，对于序列3而言，需要打孔的就是打孔序列3中最小的N-M个值所在的极化子信道承载的比特，或者说保留最大的M个值所在的极化子信道承载的比特。
通信系统通常会定义一个最大母码长度N max，假如N max＝8，序列1为[0,4,6,2,3,1,7,5]，相应地，序列2为[0,4,6,2]，即为序列1的前半段，序列3为[0,5,3,4,1,7,2,6]，序列1中的‘5’表示极化子信道5，它被打孔的优先级为7，是最迟被打孔的，也就是对应于序列3中的‘7’，即序列3中极化子信道5(从第0个元素开始排的第5个元素)的打孔优先级为‘7’。假如M＝6，则在序列1或序列2中，打孔的就是序列最前N-M＝2个极化子信道(极化子信道0和4)承载的比特，则打孔的就是小于或等于N-M＝2的值(即为0和1)所在的极化子信道(极化子信道0和4)。根据嵌套的特性，当N＝4的时候，序列1对应的序列为[0,2,3,1]，序列2对应的序列为[0,2]，注意在序列3中，由于其中的值的大小仅代表优先级，因此取值不一定非得受限于最大的极化子信道号，在不同的取值方法下，只需要确认不同的极化子信道对应的优先级的先后关系一致即可，因此在N＝4时，对序列3而言，可以简单的取N max＝8的序列的前一半，即此时序列3对应的序列为[0,5,3,4]，其中的5已经 比N大了，但这个取法和根据N＝4时的序列1(即[0,2,3,1])推导得到的序列3(即[0,3,1,2])是等价的，即当速率匹配方式为打孔时，两种方式都表示第0个极化子信道被打孔的优先级为第0个(即最高)，第1个极化子信道被打孔的优先级为第3(即最低)，第2个极化子信道被打孔的优先级位为第1，第3个极化子信道被打孔的优先级为第2，类似的，当速率匹配方式为缩短时，两种方式都表示第0个极化子信道被缩短的优先级为第0个(即最高)，第1个极化子信道被缩短的优先级为第3(即最高)，第2个极化子信道被缩短的优先级位为第1，第3个极化子信道被缩短的优先级为第2，因此序列3的嵌套方式一方面可以类似于序列1的嵌套方式从大于N的序列中挑选小于或等于N的值，另一方面也可以更简单的从长度为大于N的第3序列中取前N个位置的值即可。事实上，由于序列1、序列2、序列3都要满足嵌套特性，而序列2又是序列1的前一半，序列3则是序列1的以打孔优先级表达的方式，所以并非所有的序列都能满足这样的要求的，仍以N max＝8为例，假如序列1为[0,2,4,3,6,1,7,5]时就无法满足序列2在N＝4时的嵌套，因此在N 2较大时，嵌套的速率匹配序列的设计是非常有难度的。
当速率匹配方式是缩短时，序列1和序列3可以取为与打孔操作的序列1和序列3相同的序列(因此附录一中的序列1和序列3可以直接采用)，而序列2就可以取序列1的后一半序列，它们也同样都满足嵌套特性，序列3中的各元素的取值表示该元素所在位置(即极化子信道)的被缩短的优先级，值越大，被缩短的优先级越高。根据序列1或序列2，如果目标码长为M，那么需要缩短的就是序列1或者序列2最后的N-M个极化子信道号对应的极化子信道上承载的比特，对于序列3而言，需要缩短的就是缩短序列3中所有大于或等于M的值所在的极化子信道承载的比特。还以上面的序列[0,4,6,2,3,1,7,5]作为序列1的话，相应的序列2则为[3,1,7,5]，具体进行缩短操作时，需要缩短的就是序列1或者序列2最后的N-M个极化子信道号对应的极化子信道上承载的比特，对于序列3而言，需要缩短的就是缩短序列3中取值最大的N-M个值所在的极化子信道承载的比特，或者说，是保留缩短序列3中取值最小的M个值所在的极化子信道承载的比特。为避免混淆，本申请其他部分均以打孔为例，但整体的原则也适用于缩短的操作，不再赘述。
实际应用中，速率匹配序列可以为该第一序列的子序列，注意在本申请中，一个序列的子序列也可以包括该序列本身，同时，所谓序列A为序列B的子序列，指的是序列A中所有元素均取自于序列B，而且这些元素之间的先后相对顺序与它们在序列B中的先后相对顺序完全一致。当N＝8192时，速率匹配序列为1a或2a或3a，序列1a与序列1相同，序列2a与序列2相同，序列3a与序列3相同，因此不再重复给出序列1a、2a和3a。附录一中，还提供了母码长度为N＝4096的速率匹配序列1b、2b和3b(序列2b长为2048)，母码长度为N＝2048的速率匹配序列1c、2c和3c(序列2c长为1024)，母码长度为1024的速率匹配序列1d、2d和3d(序列2d长为512)，母码长度为512的速率匹配序列1e、2e和3e(序列2e长为256)。其中序列1a、序列1b、序列1c、序列1d、序列1e为序列1的子序列，都满足前文所述的嵌套特性；序列2a、序列2b、序列2c、序列2d、序列2e为序列2的子序列，也都满足前文所述的嵌套特性；序列3a、序列3b、序列3c、序列3d、序列3e为所述序列3的子序列，且序列3b即为序列3a的前一半序列，序列3c为序列3b的前一半序列(也即序列3a的前1/4序列)，序列3d为序列3c的前一半序列(也即序列3a的前1/8序列)，序列3e为序列3d的前一半序列(也即序列3a的前1/16序列)，同样也都满足前文 所述的嵌套特性。
一般地，通信系统会定义一个最大母码长度N max，一般地，N max小于或等于N 2，也即满足N<＝N max<＝N 2。发送端和接收端可以只存储N max对应的打孔序列(即速率匹配序列)，也可以存储小于或等于N max的不同母码长度对应的打孔序列，还可以存储N max和常用的几个不同母码长度对应的打孔序列。
除了附录一中的例子，附录二给出了另一个第一序列的例子，相应的，也就有两套可能的速率匹配序列可用，实际应用中取其中一套应用即可，如无特别说明，本申请均以附录一为例。
操作240：输出速率匹配后的序列。
注意如果考虑调制操作，这里输出的就应是基于速率匹配后的比特序列进行调制后的序列，该序列为符号序列，此时序列的长度也就不一定还是M了。如果不考虑调制，则速率匹配后的序列即为速率匹配后的比特序列。由于调制本身并非实现本申请实施例方案必不可少的技术，在不影响理解的情况下，在本申请中对“速率匹配后的序列”和“速率匹配后的比特序列”这两个术语并不加以严格区分。
上述图2所示的实施例是发送端的操作，接收端的操作也是类似的，接收机(或者接收装置)在接收到待解码信息序列后，也需要根据类似的方法，具体而言，获取包括K个待解码比特的待解码信息序列、目标码长M和母码长度N，确定速率匹配序列从而对待解码信息序列序列进行解速率匹配的操作和解码解调操作，从而得到并输出解码结果，其中解速率匹配为恢复打孔的位置或缩短的位置，对于打孔位置，解码时可以认为似然比(likelihood，LLR)为0，对于缩短位置，解码时可以认为LLR为无穷大，详细的解码算法这里不做限定。
需要说明的是，附录一和附录二中的第一序列是优选的两个例子，其应用到Polar编码过程中会有助于提高Poalr码编译码性能。任一种举例的序列中，在不影响其整体效果的前提下，可以做包括但不限于如下几方面的调整或者等同替换：
1、序列中少数元素之间的位置互换。例如，序号位置可以在设定幅度内调整，例如，设定幅度为5，将序号为10的元素位置在左右5个位置内调整均可；
2、序列中的部分元素进行调整，但是根据该序列选择出用于传输T比特信息的信道集合是一致或相似的。
3、序列中包含从0开始到N-1结束的N个元素，从0开始到N-1结束的N个元素代表N个极化子信道的序号。实际上，N个极化子信道的序号也可以从1开始到N结束，将上述序列中各序号均加1即可，这也是上述各计算方式中的序号形式。当然，也可以采取其他方式表示上述极化子信道的序号或者标识，该具体表达方式不影响序列中所表示的极化子信道的具体位置。特别是序列3b、3c、3d和3e，如前所述，其中的元素未必都在0～N-1之间，但并不影响各个极化子信道的优先级先后排序。
4、上述序列中的N个极化子信道的序号是按照N个极化子信道速率匹配的优先级从小到大排列的，实际上，N个极化子信道的序号也可以按照N个极化子信道的速率匹配优先级从大到小排列，将上述序列中的元素反序排列即可。
如图3所示，本申请实施例中还提供一种Polar码速率匹配装置300。图3所示的Polar码速率匹配方法中的部分或全部可以通过硬件来实现也可以通过软件来实现。
对于Polar码速率匹配装置300，基于图2所示的Polar码速率匹配方法的同一发明构思，该Polar码速率匹配装置300用于执行图2所示的Polar码速率匹配方法。当通过硬件实现时，Polar码速率匹配装置300包括：输入接口电路301，用于获取待编码比特；逻辑电路302，用于执行上述图2所示的Polar码速率匹配方法，具体请见前面方法实施例中的描述，此处不再赘述；输出接口电路303，用于输出速率匹配后的序列。进一步地，速率匹配后的序列还是经过调制的序列，这个序列是调制后的符号序列，它们输出给收发器320，收发器320对调制后的序列进行相应处理(包括但不限于数模变换和/或变频等处理)后通过天线330发送出去。可选的，Polar码速率匹配装置300在具体实现时可以是芯片或者集成电路。
可选的，当上述实施例的Polar码速率匹配方法中的部分或全部通过软件来实现时，如图4所示，Polar码速率匹配装置400包括：存储器401，用于存储程序；处理器402，用于执行存储器401存储的程序，当程序被执行时，使得Polar码速率匹配装置400可以实现上述实施例提供的Polar码速率匹配方法。
可选的，上述存储器401可以是物理上独立的单元，也可以如图5所示，存储器501与处理器502集成在一起。
可选的，当上述实施例的速率匹配方法中的部分或全部通过软件实现时，Polar码速率匹配装置400也可以只包括处理器402。用于存储程序的存储器401位于Polar码速率匹配装置400之外，处理器402通过电路/电线与存储器401连接，用于读取并执行存储器401中存储的程序。
基于图2所示的Polar码速率匹配方法，如图6所示，本申请实施例还提供了一种Polar码速率匹配装置600，Polar码速率匹配装置600用于执行图2所示的Polar码速率匹配方法，Polar码速率匹配装置600包括：
获取单元601，用于获取K个待编码比特、目标码长M；
编码单元602，用于确定母码长度N，根据母码长度N对该K个待编码比特进行极化编码，得到编码后的比特序列，具体编码过程可参见前述实施例的操作220；
速率匹配单元603，用于根据速率匹配序列对编码后的比特序列进行速率匹配操作获取长度为M的速率匹配后的比特序列(或者是对该比特序列进一步通过调制的序列，也称为速率匹配后的序列)，具体的速率匹配方法可参见前述实施例的操作230。
与发送端相对应，接收端的装置也可以进行类似的设计。
对于Polar码解速率匹配装置700包括：输入接口电路701用于获取待解码信息序列，逻辑电路702用于执行前述的Polar码解速率匹配的方法，以获取解码后的信息比特；输出接口电路703用于输出解码后的信息比特。Polar码解速率匹配装置700还可以包括收发器720通过天线730获取待解码信息序列。在具体实现时可以是芯片或者集成电路。
可选的，当上述实施例的Polar码解速率匹配方法中的部分或全部通过软件来实现时，如图8所示，Polar码解速率匹配装置800包括：存储器801，用于存储程序；处理器802，用于执行存储器801存储的程序，当程序被执行时，使得Polar码速率匹配装置800可以实现上述实施例提供的Polar码解速率匹配方法。
可选的，上述存储器801可以是物理上独立的单元，也可以如图9所示，存储器901与处理器902集成在一起。
可选的，当上述实施例的解速率匹配方法中的部分或全部通过软件实现时，Polar码解速率匹配装置800也可以只包括处理器802。用于存储程序的存储器01位于Polar码解速率匹配装置800之外，处理器802通过电路/电线与存储器801连接，用于读取并执行存储器801中存储的程序。
处理器402和/或处理器802可以是中央处理器(central processing unit，CPU)，网络处理器(network processor，NP)或者CPU和NP的组合。
处理器402和/或处理器802还可以进一步包括硬件芯片。上述硬件芯片可以是专用集成电路(application-specific integrated circuit，ASIC)，可编程逻辑器件(programmable logic device，PLD)或其组合。上述PLD可以是复杂可编程逻辑器件(complex programmable logic device，CPLD)，现场可编程逻辑门阵列(field-programmable gate array，FPGA)，通用阵列逻辑(generic array logic,GAL)或其任意组合。
上述实施例中的存储器可以包括易失性存储器(volatile memory)，例如随机存取存储器(random-access memory，RAM)；存储器也可以包括非易失性存储器(non-volatile memory)，例如快闪存储器(flash memory)，硬盘(hard disk drive，HDD)或固态硬盘(solid-state drive，SSD)；存储器还可以包括上述种类的存储器的组合。
基于上述的Polar码解速率匹配方法，如图10所示，本申请实施例还提供了一种Polar码解速率匹配装置1000，Polar码速率匹配装置1000用于执行前述的Polar码解速率匹配方法，Polar码解速率匹配装置1000包括：
获取单元1001，用于获取待解码信息序列；
处理单元1002，用于确定母码长度N，根据母码长度N和速率匹配序列对待解码信息序列序列进行解速率匹配的操作和解码解调操作，从而得到并输出解码结果；
本申请实施例还提供了一种计算机存储介质，存储有计算机程序指令，该计算机程序指令由计算机执行时，使得前述的Polar码速率匹配方法或者Polar码解速率匹配方法被执行。
本申请实施例还提供了一种包含指令的计算机程序产品，当其在计算机上运行时，使得前述的Polar码速率匹配方法或者Polar码解速率匹配方法被执行。
本领域内的技术人员应明白，本申请的实施例可提供为方法、系统、或计算机程序产品。因此，本申请可采用完全硬件实施例、完全软件实施例、或结合软件和硬件方面的实施例的形式。而且，本申请可采用在一个或多个其中包含有计算机可用程序代码的计算机可用存储介质(包括但不限于磁盘存储器、CD-ROM、光学存储器等)上实施的计算机程序产品的形式。
本申请是参照根据本申请实施例的方法、设备(系统)、和计算机程序产品的流程图和/或方框图来描述的。应理解可由计算机程序指令实现流程图和/或方框图中的每一流程和/或方框、以及流程图和/或方框图中的流程和/或方框的结合。可提供这些计算机程序指令到通用计算机、专用计算机、嵌入式处理机或其他可编程数据处理设备的处理器以产生一个机器，使得通过计算机或其他可编程数据处理设备的处理器执行的指令产生用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的装置。
这些计算机程序指令也可存储在能引导计算机或其他可编程数据处理设备以特定方式工作的计算机可读存储器中，使得存储在该计算机可读存储器中的指令产生包括指令装置 的制造品，该指令装置实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能。
这些计算机程序指令也可装载到计算机或其他可编程数据处理设备上，使得在计算机或其他可编程设备上执行一系列操作步骤以产生计算机实现的处理，从而在计算机或其他可编程设备上执行的指令提供用于实现在流程图一个流程或多个流程和/或方框图一个方框或多个方框中指定的功能的步骤。
尽管已描述了本申请的优选实施例，但本领域内的技术人员一旦得知了基本创造性概念，则可对这些实施例作出另外的变更和修改。所以，所附权利要求意欲解释为包括优选实施例以及落入本申请范围的所有变更和修改。
显然，本领域的技术人员可以对本申请实施例进行各种改动和变型而不脱离本申请实施例的精神和范围。这样，倘若本申请实施例的这些修改和变型属于本申请权利要求及其等同技术的范围之内，则本申请也意图包含这些改动和变型在内。
附录一：第一序列的一个例子：
序列1：
Punc_Q＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,10 79,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,4096,4097,4098,4099,4100,4101,4102,4103,4104,4105,4106,4107,4108,4109,4110,4111,4112,4113,4114,4115,4116,4117,4118,4119,4120,4121,4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133,4134,4135,4136,4137,4138,4139,4140,4141,4142,4143,4144,4145,4146,4147,4148,4149,4150,4151,4152,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4164,4165,4166,4167,4168,4169,4170,4171,4172,4173,4174,4175,4176,4177,4178,4179,4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212,4213,4214,4215,4216,4217,4218,4219,4220,4221,4222,4223,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1 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序列2：
Punc_Q_half＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258, 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1824,1825,1826,1827,1828,1829,1830,1831,1832,1833,1834,1835,1836,1837,1838,1839,1840,1841,1842,1843,1844,1845,1846,1847,1848,1849,1850,1851,1852,1853,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,6144,6145,6146,6147,6148,6149,6150,6151,6152,6153,6154,6155,6156,6157,6158,6159,6160,6161,6162,6163,6164,6165,6166,6167,6168,6169,6170,6171,6172,6173,6174,6175,6176,6177,6178,6179,6180,6181,6182,6183,6184,6185,6186,6187,6188,6189,6190,6191,6192,6193,6194,6195,6196,6197,6198,6199,6200,6201,6202,6203,6204,6205,6206,6207,6208,6209,6210,6211,6212,6213,6214,6215,6216,6217,6218,6219,6220,6221,6222,6223,6224,6225,6226,6227,6228,6229,6230,6231,6232,6233,6234,6235,6236,6237,6238,6239,6240,6241,6242,6243,6244,6245,6246,6247,6248,6249,6250,6251,6252,6253,6254,6255,6256,6257,6258,6259,6260,6261,6262,6263,6264,6265,6266,6267,6268,6269,6270,6271,2688,2689,2690,2691,2692,2693,2694,2695,2696,2697,2698,2699,2700,2701,2702,2703,2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2716,2717,2718,2719,2720,2721,2722,2723,2724,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2771,2772,2773,2774,2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,2814,2815,4480,4481,4482,4483,4484,4485,4486,4487,4488,4489,4490,4491,4492,4493,4494,4495,4496,4497,4498,4499,4500,4501,4502,4503,4504,4505,4506,4507,4508,4509,4510,4511,4512,4513,4514,4515,4516,4517,4518,4519,4520,4521,4522,4523,4524,4525,4526,4527,4528,4529,4530,4531,4532,4533,4534,4535,4536,4537,4538,4539,4540,4541,4542,4543,4544,4545,4546,4547,4548,4549,4550,4551,4552,4553,4554,4555,4556,4557,4558,4559,4560,4561,4562,4563,4564,4565,4566,4567,4568,4569,4570,4571,4572,4573,4574,4575,4576,4577,4578,4579,4580,4581,4582,4583,4584,4585,4586,4587,4588,4589,4590,4591,4592,4593,4594,4595,4596,4597,4598,4599,4600,4601,4602,4603,4604,4605,4606,4607,2816,2817,2818,2819,2820,2821,2822,2823,2824,2825,2826,2827,2828,2829,2830,2831,2832,2833,2834,2835,2836,2837,2838,2839,2840,2841,2842,2843,2844,2845,2846,2847,2848,2849,2850,2851,2852,2853,2854,2855,2856,2857,2858,2859,2860,2861,2862,2863,2864,2865,2866,2867,2868,2869,2870,2871,2872,2873,2874,2875,2876,2877,2878,2879,2880,2881,2882,2883,2884,2885,2886,2887,2888,2889,2890,2891,2892,2893,2894,2895,2896,2897,2898,2899,2900,2901,2902,2903,2904,2905,2906,2907,2908,2909,2910,2911,2912,2913,2914,2915,2916,2917,2918,2919,2920,2921,2922,2923,2924,2925,2926,2927,2928,2929,2930,2931,2932,2933,2934,2935,2936,2937,2938,2939,2940,2941,2942,2943,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210,3211,3212,3213,3214,3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3228,3229,3230,3231,3232,3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263,3264,3265,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3277,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293, 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序列3：
Punc_Z＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,110 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86,7087,7088,7089,7090,7091,7092,7093,7094,7095,7096,7097,7098,7099,7100,7101,7102,7103,7104,7105,7106,7107,7108,7109,7110,7111,7112,7113,7114,7115,7116,7117,7118,7119,7120,7121,7122,7123,7124,7125,7126,7127,7128,7129,7130,7131,7132,7133,7134,7135,7136,7137,7138,7139,7140,7141,7142,7143,7144,7145,7146,7147,7148,7149,7150,7151,7152,7153,7154,7155,7156,7157,7158,7159,7160,7161,7162,7163,7164,7165,7166,7167,7680,7681,7682,7683,7684,7685,7686,7687,7688,7689,7690,7691,7692,7693,7694,7695,7696,7697,7698,7699,7700,7701,7702,7703,7704,7705,7706,7707,7708,7709,7710,7711,7712,7713,7714,7715,7716,7717,7718,7719,7720,7721,7722,7723,7724,7725,7726,7727,7728,7729,7730,7731,7732,7733,7734,7735,7736,7737,7738,7739,7740,7741,7742,7743,7744,7745,7746,7747,7748,7749,7750,7751,7752,7753,7754,7755,7756,7757,7758,7759,7760,7761,7762,7763,7764,7765,7766,7767,7768,7769,7770,7771,7772,7773,7774,7775,7776,7777,7778,7779,7780,7781,7782,7783,7784,7785,7786,7787,7788,7789,7790,7791,7792,7793,7794,7795,7796,7797,7798,7799,7800,7801,7802,7803,7804,7805,7806,7807,7296,7297,7298,7299,7300,7301,7302,7303,7304,7305,7306,7307,7308,7309,7310,7311,7312,7313,7314,7315,7316,7317,7318,7319,7320,7321,7322,7323,7324,7325,7326,7327,7328,7329,7330,7331,7332,7333,7334,7335,7336,7337,7338,7339,7340,7341,7342,7343,7344,7345,7346,7347,7348,7349,7350,7351,7352,7353,7354,7355,7356,7357,7358,7359,7360,7361,7362,7363,7364,7365,7366,7367,7368,7369,7370,7371,7372,7373,7374,7375,7376,7377,7378,7379,7380,7381,7382,7383,7384,7385,7386,7387,7388,7389,7390,7391,7392,7393,7394,7395,7396,7397,7398,7399,7400,7401,7402,7403,7404,7405,7406,7407,7408,7409,7410,7411,7412,7413,7414,7415,7416,7417,7418,7419,7420,7421,7422,7423,7808,7809,7810,7811,7812,7813,7814,7815,7816,7817,7818,7819,7820,7821,7822,7823,7824,7825,7826,7827,7828,7829,7830,7831,7832,7833,7834,7835,7836,7837,7838,7839,7840,7841,7842,7843,7844,7845,7846,7847,7848,7849,7850,7851,7852,7853,7854,7855,7856,7857,7858,7859,7860,7861,7862,7863,7864,7865,7866,7867,7868,7869,7870,7871,7872,7873,7874,7875,7876,7877,7878,7879,7880,7881,7882,7883,7884,7885,7886,7887,7888,7889,7890,7891,7892,7893,7894,7895,7896,7897,7898,7899,7900,7901,7902,7903,7904,7905,7906,7907,7908,7909,7910,7911,7912,7913,7914,7915,7916,7917,7918,7919,7920,7921,7922,7923,7924,7925,7926,7927,7928,7929,7930,7931,7932,7933,7934,7935,7936,7937,7938,7939,7940,7941,7942,7943,7944,7945,7946,7947,7948,7949,7950,7951,7952,7953,7954,7955,7956,7957,7958,7959,7960,7961,7962,7963,7964,7965,7966,7967,7968,7969,7970,7971,7972,7973,7974,7975,7976,7977,7978,7979,7980,7981,7982,7983,7984,7985,7986,7987,7988,7989,7990,7991,7992,7993,7994,7995,7996,7997,7998,7999,8000,8001,8002,8003,8004,8005,8006,8007,8008,8009,8010,8011,8012,8013,8014,8015,8016,8017,8018,8019,8020,8021,8022,8023,8024,8025,8026,8027,8028,8029,8030,8031,8032,8033,8034,8035,8036,8037,8038,8039,8040,8041,8042,8043,8044,8045,8046,8047,8048,8049,8050,8051,8052,8053,8054,8055,8056,8057,8058,8059,8060,8061,8062,8063,8064,8065,8066,8067,8068,8069,8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080,8081,8082,8083,8084,8085,8086,8087,8088,8089,8090,8091,8092,8093,8094,8095,8096,8097,8098,8099,8100,8101,8102,8103,8104,8105,8106,8107,8108,8109,8110,8111,8112,8113,8114,8115,8116,8117,8118,8119,8120,8121,8122,8123,8124,8125,8126,8127,8128,8129,8130,8131,8132,8133,8134,8135,8136,8137,8138,8139,8140,8141,8142,8143,8144,8145,8146,8147,8148,8149,8150,8151,8152,8153,8154,8155,8156,8157,8158,8159,8160,8161,8162,8163,8164,8165,8166,8167,8168,8169,8170,8171,8172,8173,8174,8175,8176,8177,8178,8179,8180,8181,8182,8183,8184,8185,8186,8187,8188,8189,8190,8191]。
当N=8192，序列1a即为序列1，序列2a即为序列2，序列3a即为序列3；
当N＝4096，
序列1b为：
Punc_Q＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478 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31,3432,3433,3434,3435,3436,3437,3438,3439,3440,3441,3442,3443,3444,3445,3446,3447,3448,3449,3450,3451,3452,3453,3454,3455,3584,3585,3586,3587,3588,3589,3590,3591,3592,3593,3594,3595,3596,3597,3598,3599,3600,3601,3602,3603,3604,3605,3606,3607,3608,3609,3610,3611,3612,3613,3614,3615,3616,3617,3618,3619,3620,3621,3622,3623,3624,3625,3626,3627,3628,3629,3630,3631,3632,3633,3634,3635,3636,3637,3638,3639,3640,3641,3642,3643,3644,3645,3646,3647,3648,3649,3650,3651,3652,3653,3654,3655,3656,3657,3658,3659,3660,3661,3662,3663,3664,3665,3666,3667,3668,3669,3670,3671,3672,3673,3674,3675,3676,3677,3678,3679,3680,3681,3682,3683,3684,3685,3686,3687,3688,3689,3690,3691,3692,3693,3694,3695,3696,3697,3698,3699,3700,3701,3702,3703,3704,3705,3706,3707,3708,3709,3710,3711,1920,1921,1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968,1969,1970,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2026,2027,2028,2029,2030,2031,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044,2045,2046,2047,2944,2945,2946,2947,2948,2949,2950,2951,2952,2953,2954,2955,2956,2957,2958,2959,2960,2961,2962,2963,2964,2965,2966,2967,2968,2969,2970,2971,2972,2973,2974,2975,2976,2977,2978,2979,2980,2981,2982,2983,2984,2985,2986,2987,2988,2989,2990,2991,2992,2993,2994,2995,2996,2997,2998,2999,3000,3001,3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044,3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3058,3059,3060,3061,3062,3063,3064,3065,3066,3067,3068,3069,3070,3071,3456,3457,3458,3459,3460,3461,3462,3463,3464,3465,3466,3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493,3494,3495,3496,3497,3498,3499,3500,3501,3502,3503,3504,3505,3506,3507,3508,3509,3510,3511,3512,3513,3514,3515,3516,3517,3518,3519,3520,3521,3522,3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3535,3536,3537,3538,3539,3540,3541,3542,3543,3544,3545,3546,3547,3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561,3562,3563,3564,3565,3566,3567,3568,3569,3570,3571,3572,3573,3574,3575,3576,3577,3578,3579,3580,3581,3582,3583,3712,3713,3714,3715,3716,3717,3718,3719,3720,3721,3722,3723,3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739,3740,3741,3742,3743,3744,3745,3746,3747,3748,3749,3750,3751,3752,3753,3754,3755,3756,3757,3758,3759,3760,3761,3762,3763,3764,3765,3766,3767,3768,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780,3781,3782,3783,3784,3785,3786,3787,3788,3789,3790,3791,3792,3793,3794,3795,3796,3797,3798,3799,3800,3801,3802,3803,3804,3805,3806,3807,3808,3809,3810,3811,3812,3813,3814,3815,3816,3817,3818,3819,3820,3821,3822,3823,3824,3825,3826,3827,3828,3829,3830,3831,3832,3833,3834,3835,3836,3837,3838,3839,3840,3841,3842,3843,3844,3845,3846,3847,3848,3849,3850,3851,3852,3853,3854,3855,3856,3857,3858,3859,3860,3861,3862,3863,3864,3865,3866,3867,3868,3869,3870,3871,3872,3873,3874,3875,3876,3877,3878,3879,3880,3881,3882,3883,3884,3885,3886,3887,3888,3889,3890,3891,3892,3893,3894,3895,3896,3897,3898,3899,3900,3901,3902,3903,3904,3905,3906,3907,3908,3909,3910,3911,3912,3913,3914,3915,39 16,3917,3918,3919,3920,3921,3922,3923,3924,3925,3926,3927,3928,3929,3930,3931,3932,3933,3934,3935,3936,3937,3938,3939,3940,3941,3942,3943,3944,3945,3946,3947,3948,3949,3950,3951,3952,3953,3954,3955,3956,3957,3958,3959,3960,3961,3962,3963,3964,3965,3966,3967,3968,3969,3970,3971,3972,3973,3974,3975,3976,3977,3978,3979,3980,3981,3982,3983,3984,3985,3986,3987,3988,3989,3990,3991,3992,3993,3994,3995,3996,3997,3998,3999,4000,4001,4002,4003,4004,4005,4006,4007,4008,4009,4010,4011,4012,4013,4014,4015,4016,4017,4018,4019,4020,4021,4022,4023,4024,4025,4026,4027,4028,4029,4030,4031,4032,4033,4034,4035,4036,4037,4038,4039,4040,4041,4042,4043,4044,4045,4046,4047,4048,4049,4050,4051,4052,4053,4054,4055,4056,4057,4058,4059,4060,4061,4062,4063,4064,4065,4066,4067,4068,4069,4070,4071,4072,4073,4074,4075,4076,4077,4078,4079,4080,4081,4082,4083,4084,4085,4086,4087,4088,4089,4090,4091,4092,4093,4094,4095]。
序列2b为：
Punc_Q_half＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2 077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,2176,2177,2178,2179,2180,2181,2182,2183,2184,2185,2186,2187,2188,2189,2190,2191,2192,2193,2194,2195,2196,2197,2198,2199,2200,2201,2202,2203,2204,2205,2206,2207,2208,2209,2210,2211,2212,2213,2214,2215,2216,2217,2218,2219,2220,2221,2222,2223,2224,2225,2226,2227,2228,2229,2230,2231,2232,2233,2234,2235,2236,2237,2238,2239,2240,2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251,2252,2253,2254,2255,2256,2257,2258,2259,2260,2261,2262,2263,2264,226 5,2266,2267,2268,2269,2270,2271,2272,2273,2274,2275,2276,2277,2278,2279,2280,2281,2282,2283,2284,2285,2286,2287,2288,2289,2290,2291,2292,2293,2294,2295,2296,2297,2298,2299,2300,2301,2302,2303,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2560,2561,2562,2563,2564,2565,2566,2567,2568,2569,2570,2571,2572,2573,2574,2575,2576,2577,2578,2579,2580,2581,2582,2583,2584,2585,2586,2587,2588,2589,2590,2591,2592,2593,2594,2595,2596,2597,2598,2599,2600,2601,2602,2603,2604,2605,2606,2607,2608,2609,2610,2611,2612,2613,2614,2615,2616,2617,2618,2619,2620,2621,2622,2623,2624,2625,2626,2627,2628,2629,2630,2631,2632,2633,2634,2635,2636,2637,2638,2639,2640,2641,2642,2643,2644,2645,2646,2647,2648,2649,2650,2651,2652,2653,2654,2655,2656,2657,2658,2659,2660,2661,2662,2663,2664,2665,2666,2667,2668,2669,2670,2671,2672,2673,2674,2675,2676,2677,2678,2679,2680,2681,2682,2683,2684,2685,2686,2687,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023]。
序列3b为：
Punc_Z＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,2 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当N＝2048，
序列1c为：
Punc_Q＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795 ,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,171 1,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761,1762,1763,1764,1765,1766,1767,1768,1769,1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,1792,1793,1794,1795,1796,1797,1798,1799,1800,1801,1802,1803,1804,1805,1806,1807,1808,1809,1810,1811,1812,1813,1814,1815,1816,1817,1818,1819,1820,1821,1822,1823,1824,1825,1826,1827,1828,1829,1830,1831,1832,1833,1834,1835,1836,1837,1838,1839,1840,1841,1842,1843,1844,1845,1846,1847,1848,1849,1850,1851,1852,1853,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,1920,1921,1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968,1969,1970,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2026,2027,2028,2029,2030,2031,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044,2045,2046,2047]。
序列2c为：
Punc_Q_half＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027, 1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895]。
序列3c为；
Punc_Z＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779 ,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350, 1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,2688,2689,2690,2691,2692,2693,2694,2695,2696,2697,2698,2699,2700,2701,2702,2703,2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2716,2717,2718,2719,2720,2721,2722,2723,2724,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2771,2772,2773,2774,2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,2814,2815,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761,1762,1763,1764,1765,1766,1767,1768,1769,1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,2944,2945,2946,2947,2948,2949,2950,2951,2952,2953,2954,2955,2956,2957,2958,2959,2960,2961,2962,2963,2964,2965,2966,2967,2968,2969,2970,2971,2972,2973,2974,2975,2976,2977,2978,2979,2980,2981,2982,2983,2984,2985,2986,2987,2988,2989,2990,2991,2992,2993,2994,2995,2996,2997,2998,2999,3000,3001,3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044,3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3058,3059,3060,3061,3062,3063,3064,3065,3066,3067,3068,3069,3070,3071,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210,3211,3212,3213,3214,3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3228,3229,3230,3231,3232,3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263,3264,3265,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3277,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301,3302,3303,3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317,3318,3319,3320,3321,3322,3323,3324,3325,3326,3327,4736,4737,4738,4739,4740,4741,4742,4743,4744,4745,4746,4747,4748,4749,4750,4751,4752,4753,4754,4755,4756,4757,4758,4759,4760,4761,4762,4763,4764,4765,4766,4767,4768,4769,4770,4771,4772,4773,4774,4775,4776,4777,4778,4779, 4780,4781,4782,4783,4784,4785,4786,4787,4788,4789,4790,4791,4792,4793,4794,4795,4796,4797,4798,4799,4800,4801,4802,4803,4804,4805,4806,4807,4808,4809,4810,4811,4812,4813,4814,4815,4816,4817,4818,4819,4820,4821,4822,4823,4824,4825,4826,4827,4828,4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844,4845,4846,4847,4848,4849,4850,4851,4852,4853,4854,4855,4856,4857,4858,4859,4860,4861,4862,4863]。
当N＝1024，
序列1d为：
Punc_Q＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,8 12,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023]。
序列2d为：
Punc_Q_half＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639]。
序列3d为；
Punc_Z＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,2 24,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431]。
当N＝512，
序列1e为：
Punc_Q＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511]。
序列2e为：
Punc_Q_half＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255]。
序列3e为：
Punc_Z＝[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,1 75,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895]。
附录二：第一序列的另一个例子：
序列1：
Shorten_Q＝[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,4832,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3 760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176,8,4104,2056,6152,1032,5128,3080,7176,520,4616,2568,6664,1544,5640,3592,7688,264,4360,2312,6408,1288,5384,3336,7432,776,4872,2824,6920,1800,5896,3848,7944,136,4232,2184,6280,1160,5256,3208,7304,648,4744,2696,6792,1672,5768,3720,7816,392,4488,2440,6536,1416,5512,3464,7560,904,5000,2952,7048,1928,6024,3976,8072,72,4168,2120,6216,1096,5192,3144,7240,584,4680,2632,6728,1608,5704,3656,7752,328,4424,2376,6472,1352,5448,3400,7496,840,4936,2888,6984,1864,5960,3912,8008,200,4296,2248,6344,1224,5320,3272,7368,712,4808,2760,6856,1736,5832,3784,7880,456,4552,2504,6600,1480,5576,3528,7624,968,5064,3016,7112,1992,6088,4040,8136,40,4136,2088,6184,1064,5160,3112,7208,552,4648,2600,6696,1576,5672,3624,7720,296,4392,2344,6440,1320,5416,3368,7464,808,4904,2856,6952,1832,5928,3880,7976,168,4264,2216,6312,1192,5288,3240,7336,680,4776,2728,6824,1704,5800,3752,7848,424,4520,2472,6568,1448,5544,3496,7592,936,5032,2984,7080,1960,6056,4008,8104,104,4200,2152,6248,1128,5224,3176,7272,616,4712,2664,6760,1640,5736,3688,7784,360,4456,2408,6504,1384,5480,3432,7528,872,4968,2920,7016,1896,5992,3944,8040,232,4328,2280,6376,1256,5352,3304,7400,744,4840,2792,6888,1768,5864,3816,7912,488,4584,2536,6632,1512,5608,3560,7656,1000,5096,3048,7144,2024,6120,4072,8168,24,4120,2072,6168,1048,5144,3096,7192,536,4632,2584,6680,1560,5656,3608,7704,280,4376,2328,6424,1304,5400,3352,7448,792,4888,2840,6936,1816,5912,3864,7960,152,4248,2200,6296,1176,5272,3224,7320,664,4760,2712,6808,1688,5784,3736,7832,408,4504,2456,6552,1432,5528,3480,7576,920,5016,2968,7064,1944,6040,3992,8088,88,4184,2136,6232,1112,5208,3160,7256,600,4696,2648,6744,1624,5720,3672,7768,344,4440,2392,6488,1368,5464,3416,7512,856,4952,2904,7000,1880,5976,3928,8024,216,4312,2264,6360,1240,5336,3288,7384,728,4824,2776,6872,1752,5848,3800,7896,472,4568,2520,6616,1496,5592,3544,7640,984,5080,3032,7128,2008,6104,4056,8152,56,4152,2104,6200,1080,5176,3128,7224,568,4664,2616,6712,1592,5688,3640,7736,312,4408,2360,6456,1336,5432,3384,7480,824,4920,2872,6968,1848,5944,3896,7992,184,4280,2232,6328,1208,5304,3256,7352,696,4792,2744,6840,1720,5816,3768,7864,440,4536,2488,6584,1464,5560,3512,7608,952,5048,3000,7096,1976,6072,4024,8120,120,4216,2168,6264,1144,5240,3192,7288,632,4728,2680,6776,1656,5752,3704,7800,376,4472,2424,6520,1400,5496,3448,7544,888,4984,2936,7032,1912,6008,3960,8056,248,4344,2296,6392,1272,5368,3320,7416,760,4856,2808,6904,1784,5880,3832,7928,504,4600,2552,6648,1528,5624,3576,7672,1016,5112,3064,7160,2040,6136,4088,8184,4,4100,2052,6148,1028,5124,3076,7172,516,4612,2564,6660,1540,5636,3588,7684,260,4356,2308,6404,1284,5380,3332,7428,772,4868,2820,6916,1796,5892,3844,7940,132,4228,2180,6276,1156,5252,3204,7300,644,4740,2692,6788,1668,5764,3716,7812,388,4484,2436,6532,1412,5508,3460,7556,900,4996,2948,7044,1924,6020,3972,8068,68,4164,2116,6212,1092,5188,3140,7236,580,4676,2628,6724,1604,5700,3652,7748,324,4420,2372,6468,1348,5444,3396,7492,836,4932,2884,6980,1860,5956,3908,8004,196,4292,2244,6340,1220,5316,3268,7364,708,4804,2756,6852,1732,5828,3780,7876,452,4548,2500,6596,1476,5572,3524,7620,964,5060,3012,7108,1988,6084,4036,8132,36,4132,2084,6180,1060,5156,3108,7204,548,4644,2596,6692,1572,5668,3620,7716,292,4388,2340,6436,1316,5412,3364,7460,804,4900,2852,6948,1828,5924,3876,7972,164,4260,2212,6308,1188,5284,3236,7332,676,4772,2724,6820,1700,5 796,3748,7844,420,4516,2468,6564,1444,5540,3492,7588,932,5028,2980,7076,1956,6052,4004,8100,100,4196,2148,6244,1124,5220,3172,7268,612,4708,2660,6756,1636,5732,3684,7780,356,4452,2404,6500,1380,5476,3428,7524,868,4964,2916,7012,1892,5988,3940,8036,228,4324,2276,6372,1252,5348,3300,7396,740,4836,2788,6884,1764,5860,3812,7908,484,4580,2532,6628,1508,5604,3556,7652,996,5092,3044,7140,2020,6116,4068,8164,20,4116,2068,6164,1044,5140,3092,7188,532,4628,2580,6676,1556,5652,3604,7700,276,4372,2324,6420,1300,5396,3348,7444,788,4884,2836,6932,1812,5908,3860,7956,148,4244,2196,6292,1172,5268,3220,7316,660,4756,2708,6804,1684,5780,3732,7828,404,4500,2452,6548,1428,5524,3476,7572,916,5012,2964,7060,1940,6036,3988,8084,84,4180,2132,6228,1108,5204,3156,7252,596,4692,2644,6740,1620,5716,3668,7764,340,4436,2388,6484,1364,5460,3412,7508,852,4948,2900,6996,1876,5972,3924,8020,212,4308,2260,6356,1236,5332,3284,7380,724,4820,2772,6868,1748,5844,3796,7892,468,4564,2516,6612,1492,5588,3540,7636,980,5076,3028,7124,2004,6100,4052,8148,52,4148,2100,6196,1076,5172,3124,7220,564,4660,2612,6708,1588,5684,3636,7732,308,4404,2356,6452,1332,5428,3380,7476,820,4916,2868,6964,1844,5940,3892,7988,180,4276,2228,6324,1204,5300,3252,7348,692,4788,2740,6836,1716,5812,3764,7860,436,4532,2484,6580,1460,5556,3508,7604,948,5044,2996,7092,1972,6068,4020,8116,116,4212,2164,6260,1140,5236,3188,7284,628,4724,2676,6772,1652,5748,3700,7796,372,4468,2420,6516,1396,5492,3444,7540,884,4980,2932,7028,1908,6004,3956,8052,244,4340,2292,6388,1268,5364,3316,7412,756,4852,2804,6900,1780,5876,3828,7924,500,4596,2548,6644,1524,5620,3572,7668,1012,5108,3060,7156,2036,6132,4084,8180,12,4108,2060,6156,1036,5132,3084,7180,524,4620,2572,6668,1548,5644,3596,7692,268,4364,2316,6412,1292,5388,3340,7436,780,4876,2828,6924,1804,5900,3852,7948,140,4236,2188,6284,1164,5260,3212,7308,652,4748,2700,6796,1676,5772,3724,7820,396,4492,2444,6540,1420,5516,3468,7564,908,5004,2956,7052,1932,6028,3980,8076,76,4172,2124,6220,1100,5196,3148,7244,588,4684,2636,6732,1612,5708,3660,7756,332,4428,2380,6476,1356,5452,3404,7500,844,4940,2892,6988,1868,5964,3916,8012,204,4300,2252,6348,1228,5324,3276,7372,716,4812,2764,6860,1740,5836,3788,7884,460,4556,2508,6604,1484,5580,3532,7628,972,5068,3020,7116,1996,6092,4044,8140,44,4140,2092,6188,1068,5164,3116,7212,556,4652,2604,6700,1580,5676,3628,7724,300,4396,2348,6444,1324,5420,3372,7468,812,4908,2860,6956,1836,5932,3884,7980,172,4268,2220,6316,1196,5292,3244,7340,684,4780,2732,6828,1708,5804,3756,7852,428,4524,2476,6572,1452,5548,3500,7596,940,5036,2988,7084,1964,6060,4012,8108,108,4204,2156,6252,1132,5228,3180,7276,620,4716,2668,6764,1644,5740,3692,7788,364,4460,2412,6508,1388,5484,3436,7532,876,4972,2924,7020,1900,5996,3948,8044,236,4332,2284,6380,1260,5356,3308,7404,748,4844,2796,6892,1772,5868,3820,7916,492,4588,2540,6636,1516,5612,3564,7660,1004,5100,3052,7148,2028,6124,4076,8172,28,4124,2076,6172,1052,5148,3100,7196,540,4636,2588,6684,1564,5660,3612,7708,284,4380,2332,6428,1308,5404,3356,7452,796,4892,2844,6940,1820,5916,3868,7964,156,4252,2204,6300,1180,5276,3228,7324,668,4764,2716,6812,1692,5788,3740,7836,412,4508,2460,6556,1436,5532,3484,7580,924,5020,2972,7068,1948,6044,3996,8092,92,4188,2140,6236,1116,5212,3164,7260,604,4700,2652,6748,1628,5724,3676,7772,348,4444,2396,6492,1372,5468,3420,7516,860,4956,2908,7004,1884,5980,3932,8028,220,4316,2268,6364,1244,5340,3292,7388,732,4828,2780,6876,1756,5852,3804,7900,476,4572,2524,6620,1500,5596,3548,7644,988,5084,3036,7132,2012,6108,4060,8156,60,4156,2108,6204,1084,5180,3132,7228,572,4668,2620,6716,1596,5692,3644,7740,316,4412,2364,6460,1340,5436,3388,7484,828,4924,2876,6972,1852,5948,3900,7996,188,4284,2236,6332,1212,5308,3260,7356,700,4796,2748,6844, 1724,5820,3772,7868,444,4540,2492,6588,1468,5564,3516,7612,956,5052,3004,7100,1980,6076,4028,8124,124,4220,2172,6268,1148,5244,3196,7292,636,4732,2684,6780,1660,5756,3708,7804,380,4476,2428,6524,1404,5500,3452,7548,892,4988,2940,7036,1916,6012,3964,8060,252,4348,2300,6396,1276,5372,3324,7420,764,4860,2812,6908,1788,5884,3836,7932,508,4604,2556,6652,1532,5628,3580,7676,1020,5116,3068,7164,2044,6140,4092,8188,2,4098,2050,6146,1026,5122,3074,7170,514,4610,2562,6658,1538,5634,3586,7682,258,4354,2306,6402,1282,5378,3330,7426,770,4866,2818,6914,1794,5890,3842,7938,130,4226,2178,6274,1154,5250,3202,7298,642,4738,2690,6786,1666,5762,3714,7810,386,4482,2434,6530,1410,5506,3458,7554,898,4994,2946,7042,1922,6018,3970,8066,66,4162,2114,6210,1090,5186,3138,7234,578,4674,2626,6722,1602,5698,3650,7746,322,4418,2370,6466,1346,5442,3394,7490,834,4930,2882,6978,1858,5954,3906,8002,194,4290,2242,6338,1218,5314,3266,7362,706,4802,2754,6850,1730,5826,3778,7874,450,4546,2498,6594,1474,5570,3522,7618,962,5058,3010,7106,1986,6082,4034,8130,34,4130,2082,6178,1058,5154,3106,7202,546,4642,2594,6690,1570,5666,3618,7714,290,4386,2338,6434,1314,5410,3362,7458,802,4898,2850,6946,1826,5922,3874,7970,162,4258,2210,6306,1186,5282,3234,7330,674,4770,2722,6818,1698,5794,3746,7842,418,4514,2466,6562,1442,5538,3490,7586,930,5026,2978,7074,1954,6050,4002,8098,98,4194,2146,6242,1122,5218,3170,7266,610,4706,2658,6754,1634,5730,3682,7778,354,4450,2402,6498,1378,5474,3426,7522,866,4962,2914,7010,1890,5986,3938,8034,226,4322,2274,6370,1250,5346,3298,7394,738,4834,2786,6882,1762,5858,3810,7906,482,4578,2530,6626,1506,5602,3554,7650,994,5090,3042,7138,2018,6114,4066,8162,18,4114,2066,6162,1042,5138,3090,7186,530,4626,2578,6674,1554,5650,3602,7698,274,4370,2322,6418,1298,5394,3346,7442,786,4882,2834,6930,1810,5906,3858,7954,146,4242,2194,6290,1170,5266,3218,7314,658,4754,2706,6802,1682,5778,3730,7826,402,4498,2450,6546,1426,5522,3474,7570,914,5010,2962,7058,1938,6034,3986,8082,82,4178,2130,6226,1106,5202,3154,7250,594,4690,2642,6738,1618,5714,3666,7762,338,4434,2386,6482,1362,5458,3410,7506,850,4946,2898,6994,1874,5970,3922,8018,210,4306,2258,6354,1234,5330,3282,7378,722,4818,2770,6866,1746,5842,3794,7890,466,4562,2514,6610,1490,5586,3538,7634,978,5074,3026,7122,2002,6098,4050,8146,50,4146,2098,6194,1074,5170,3122,7218,562,4658,2610,6706,1586,5682,3634,7730,306,4402,2354,6450,1330,5426,3378,7474,818,4914,2866,6962,1842,5938,3890,7986,178,4274,2226,6322,1202,5298,3250,7346,690,4786,2738,6834,1714,5810,3762,7858,434,4530,2482,6578,1458,5554,3506,7602,946,5042,2994,7090,1970,6066,4018,8114,114,4210,2162,6258,1138,5234,3186,7282,626,4722,2674,6770,1650,5746,3698,7794,370,4466,2418,6514,1394,5490,3442,7538,882,4978,2930,7026,1906,6002,3954,8050,242,4338,2290,6386,1266,5362,3314,7410,754,4850,2802,6898,1778,5874,3826,7922,498,4594,2546,6642,1522,5618,3570,7666,1010,5106,3058,7154,2034,6130,4082,8178,10,4106,2058,6154,1034,5130,3082,7178,522,4618,2570,6666,1546,5642,3594,7690,266,4362,2314,6410,1290,5386,3338,7434,778,4874,2826,6922,1802,5898,3850,7946,138,4234,2186,6282,1162,5258,3210,7306,650,4746,2698,6794,1674,5770,3722,7818,394,4490,2442,6538,1418,5514,3466,7562,906,5002,2954,7050,1930,6026,3978,8074,74,4170,2122,6218,1098,5194,3146,7242,586,4682,2634,6730,1610,5706,3658,7754,330,4426,2378,6474,1354,5450,3402,7498,842,4938,2890,6986,1866,5962,3914,8010,202,4298,2250,6346,1226,5322,3274,7370,714,4810,2762,6858,1738,5834,3786,7882,458,4554,2506,6602,1482,5578,3530,7626,970,5066,3018,7114,1994,6090,4042,8138,42,4138,2090,6186,1066,5162,3114,7210,554,4650,2602,6698,1578,5674,3626,7722,298,4394,2346,6442,1322,5418,3370,7466,810,4906,2858,6954,1834,5930,3882,7978,170,4266,2218,6314,1194,5290,3242,7338,682,4778,2730,6 826,1706,5802,3754,7850,426,4522,2474,6570,1450,5546,3498,7594,938,5034,2986,7082,1962,6058,4010,8106,106,4202,2154,6250,1130,5226,3178,7274,618,4714,2666,6762,1642,5738,3690,7786,362,4458,2410,6506,1386,5482,3434,7530,874,4970,2922,7018,1898,5994,3946,8042,234,4330,2282,6378,1258,5354,3306,7402,746,4842,2794,6890,1770,5866,3818,7914,490,4586,2538,6634,1514,5610,3562,7658,1002,5098,3050,7146,2026,6122,4074,8170,26,4122,2074,6170,1050,5146,3098,7194,538,4634,2586,6682,1562,5658,3610,7706,282,4378,2330,6426,1306,5402,3354,7450,794,4890,2842,6938,1818,5914,3866,7962,154,4250,2202,6298,1178,5274,3226,7322,666,4762,2714,6810,1690,5786,3738,7834,410,4506,2458,6554,1434,5530,3482,7578,922,5018,2970,7066,1946,6042,3994,8090,90,4186,2138,6234,1114,5210,3162,7258,602,4698,2650,6746,1626,5722,3674,7770,346,4442,2394,6490,1370,5466,3418,7514,858,4954,2906,7002,1882,5978,3930,8026,218,4314,2266,6362,1242,5338,3290,7386,730,4826,2778,6874,1754,5850,3802,7898,474,4570,2522,6618,1498,5594,3546,7642,986,5082,3034,7130,2010,6106,4058,8154,58,4154,2106,6202,1082,5178,3130,7226,570,4666,2618,6714,1594,5690,3642,7738,314,4410,2362,6458,1338,5434,3386,7482,826,4922,2874,6970,1850,5946,3898,7994,186,4282,2234,6330,1210,5306,3258,7354,698,4794,2746,6842,1722,5818,3770,7866,442,4538,2490,6586,1466,5562,3514,7610,954,5050,3002,7098,1978,6074,4026,8122,122,4218,2170,6266,1146,5242,3194,7290,634,4730,2682,6778,1658,5754,3706,7802,378,4474,2426,6522,1402,5498,3450,7546,890,4986,2938,7034,1914,6010,3962,8058,250,4346,2298,6394,1274,5370,3322,7418,762,4858,2810,6906,1786,5882,3834,7930,506,4602,2554,6650,1530,5626,3578,7674,1018,5114,3066,7162,2042,6138,4090,8186,6,4102,2054,6150,1030,5126,3078,7174,518,4614,2566,6662,1542,5638,3590,7686,262,4358,2310,6406,1286,5382,3334,7430,774,4870,2822,6918,1798,5894,3846,7942,134,4230,2182,6278,1158,5254,3206,7302,646,4742,2694,6790,1670,5766,3718,7814,390,4486,2438,6534,1414,5510,3462,7558,902,4998,2950,7046,1926,6022,3974,8070,70,4166,2118,6214,1094,5190,3142,7238,582,4678,2630,6726,1606,5702,3654,7750,326,4422,2374,6470,1350,5446,3398,7494,838,4934,2886,6982,1862,5958,3910,8006,198,4294,2246,6342,1222,5318,3270,7366,710,4806,2758,6854,1734,5830,3782,7878,454,4550,2502,6598,1478,5574,3526,7622,966,5062,3014,7110,1990,6086,4038,8134,38,4134,2086,6182,1062,5158,3110,7206,550,4646,2598,6694,1574,5670,3622,7718,294,4390,2342,6438,1318,5414,3366,7462,806,4902,2854,6950,1830,5926,3878,7974,166,4262,2214,6310,1190,5286,3238,7334,678,4774,2726,6822,1702,5798,3750,7846,422,4518,2470,6566,1446,5542,3494,7590,934,5030,2982,7078,1958,6054,4006,8102,102,4198,2150,6246,1126,5222,3174,7270,614,4710,2662,6758,1638,5734,3686,7782,358,4454,2406,6502,1382,5478,3430,7526,870,4966,2918,7014,1894,5990,3942,8038,230,4326,2278,6374,1254,5350,3302,7398,742,4838,2790,6886,1766,5862,3814,7910,486,4582,2534,6630,1510,5606,3558,7654,998,5094,3046,7142,2022,6118,4070,8166,22,4118,2070,6166,1046,5142,3094,7190,534,4630,2582,6678,1558,5654,3606,7702,278,4374,2326,6422,1302,5398,3350,7446,790,4886,2838,6934,1814,5910,3862,7958,150,4246,2198,6294,1174,5270,3222,7318,662,4758,2710,6806,1686,5782,3734,7830,406,4502,2454,6550,1430,5526,3478,7574,918,5014,2966,7062,1942,6038,3990,8086,86,4182,2134,6230,1110,5206,3158,7254,598,4694,2646,6742,1622,5718,3670,7766,342,4438,2390,6486,1366,5462,3414,7510,854,4950,2902,6998,1878,5974,3926,8022,214,4310,2262,6358,1238,5334,3286,7382,726,4822,2774,6870,1750,5846,3798,7894,470,4566,2518,6614,1494,5590,3542,7638,982,5078,3030,7126,2006,6102,4054,8150,54,4150,2102,6198,1078,5174,3126,7222,566,4662,2614,6710,1590,5686,3638,7734,310,4406,2358,6454,1334,5430,3382,7478,822,4918,2870,6966,1846,5942,3894,7990,182,4278,2230,6326,1206,5302,3254,7350,694,4790,2 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697,4793,2745,6841,1721,5817,3769,7865,441,4537,2489,6585,1465,5561,3513,7609,953,5049,3001,7097,1977,6073,4025,8121,121,4217,2169,6265,1145,5241,3193,7289,633,4729,2681,6777,1657,5753,3705,7801,377,4473,2425,6521,1401,5497,3449,7545,889,4985,2937,7033,1913,6009,3961,8057,249,4345,2297,6393,1273,5369,3321,7417,761,4857,2809,6905,1785,5881,3833,7929,505,4601,2553,6649,1529,5625,3577,7673,1017,5113,3065,7161,2041,6137,4089,8185,5,4101,2053,6149,1029,5125,3077,7173,517,4613,2565,6661,1541,5637,3589,7685,261,4357,2309,6405,1285,5381,3333,7429,773,4869,2821,6917,1797,5893,3845,7941,133,4229,2181,6277,1157,5253,3205,7301,645,4741,2693,6789,1669,5765,3717,7813,389,4485,2437,6533,1413,5509,3461,7557,901,4997,2949,7045,1925,6021,3973,8069,69,4165,2117,6213,1093,5189,3141,7237,581,4677,2629,6725,1605,5701,3653,7749,325,4421,2373,6469,1349,5445,3397,7493,837,4933,2885,6981,1861,5957,3909,8005,197,4293,2245,6341,1221,5317,3269,7365,709,4805,2757,6853,1733,5829,3781,7877,453,4549,2501,6597,1477,5573,3525,7621,965,5061,3013,7109,1989,6085,4037,8133,37,4133,2085,6181,1061,5157,3109,7205,549,4645,2597,6693,1573,5669,3621,7717,293,4389,2341,6437,1317,5413,3365,7461,805,4901,2853,6949,1829,5925,3877,7973,165,4261,2213,6309,1189,5285,3237,7333,677,4773,2725,6821,1701,5797,3749,7845,421,4517,2469,6565,1445,5541,3493,7589,933,5029,2981,7077,1957,6053,4005,8101,101,4197,2149,6245,1125,5221,3173,7269,613,4709,2661,6757,1637,5733,3685,7781,357,4453,2405,6501,1381,5477,3429,7525,869,4965,2917,7013,1893,5989,3941,8037,229,4325,2277,6373,1253,5349,3301,7397,741,4837,2789,6885,1765,5861,3813,7909,485,4581,2533,6629,1509,5605,3557,7653,997,5093,3045,7141,2021,6117,4069,8165,21,4117,2069,6165,1045,5141,3093,7189,533,4629,2581,6677,1557,5653,3605,7701,277,4373,2325,6421,1301,5397,3349,7445,789,4885,2837,6933,1813,5909,3861,7957,149,4245,2197,6293,1173,5269,3221,7317,661,4757,2709,6805,1685,5781,3733,7829,405,4501,2453,6549,1429,5525,3477,7573,917,5013,2965,7061,1941,6037,3989,8085,85,4181,2133,6229,1109,5205,3157,7253,597,4693,2645,6741,1621,5717,3669,7765,341,4437,2389,6485,1365,5461,3413,7509,853,4949,2901,6997,1877,5973,3925,8021,213,4309,2261,6357,1237,5333,3285,7381,725,4821,2773,6869,1749,5845,3797,7893,469,4565,2517,6613,1493,5589,3541,7637,981,5077,3029,7125,2005,6101,4053,8149,53,4149,2101,6197,1077,5173,3125,7221,565,4661,2613,6709,1589,5685,3637,7733,309,4405,2357,6453,1333,5429,3381,7477,821,4917,2869,6965,1845,5941,3893,7989,181,4277,2229,6325,1205,5301,3253,7349,693,4789,2741,6837,1717,5813,3765,7861,437,4533,2485,6581,1461,5557,3509,7605,949,5045,2997,7093,1973,6069,4021,8117,117,4213,2165,6261,1141,5237,3189,7285,629,4725,2677,6773,1653,5749,3701,7797,373,4469,2421,6517,1397,5493,3445,7541,885,4981,2933,7029,1909,6005,3957,8053,245,4341,2293,6389,1269,5365,3317,7413,757,4853,2805,6901,1781,5877,3829,7925,501,4597,2549,6645,1525,5621,3573,7669,1013,5109,3061,7157,2037,6133,4085,8181,13,4109,2061,6157,1037,5133,3085,7181,525,4621,2573,6669,1549,5645,3597,7693,269,4365,2317,6413,1293,5389,3341,7437,781,4877,2829,6925,1805,5901,3853,7949,141,4237,2189,6285,1165,5261,3213,7309,653,4749,2701,6797,1677,5773,3725,7821,397,4493,2445,6541,1421,5517,3469,7565,909,5005,2957,7053,1933,6029,3981,8077,77,4173,2125,6221,1101,5197,3149,7245,589,4685,2637,6733,1613,5709,3661,7757,333,4429,2381,6477,1357,5453,3405,7501,845,4941,2893,6989,1869,5965,3917,8013,205,4301,2253,6349,1229,5325,3277,7373,717,4813,2765,6861,1741,5837,3789,7885,461,4557,2509,6605,1485,5581,3533,7629,973,5069,3021,7117,1997,6093,4045,8141,45,4141,2093,6189,1069,5165,3117,7213,557,4653,2605,6701,1581,5677,3629,7725,301,4397,2349,6445,1325,5421,3373,7469,813,4909,2861,6957,1837,5933,3885,7981,173,4269,2221,6317,1197,5293,3245, 7341,685,4781,2733,6829,1709,5805,3757,7853,429,4525,2477,6573,1453,5549,3501,7597,941,5037,2989,7085,1965,6061,4013,8109,109,4205,2157,6253,1133,5229,3181,7277,621,4717,2669,6765,1645,5741,3693,7789,365,4461,2413,6509,1389,5485,3437,7533,877,4973,2925,7021,1901,5997,3949,8045,237,4333,2285,6381,1261,5357,3309,7405,749,4845,2797,6893,1773,5869,3821,7917,493,4589,2541,6637,1517,5613,3565,7661,1005,5101,3053,7149,2029,6125,4077,8173,29,4125,2077,6173,1053,5149,3101,7197,541,4637,2589,6685,1565,5661,3613,7709,285,4381,2333,6429,1309,5405,3357,7453,797,4893,2845,6941,1821,5917,3869,7965,157,4253,2205,6301,1181,5277,3229,7325,669,4765,2717,6813,1693,5789,3741,7837,413,4509,2461,6557,1437,5533,3485,7581,925,5021,2973,7069,1949,6045,3997,8093,93,4189,2141,6237,1117,5213,3165,7261,605,4701,2653,6749,1629,5725,3677,7773,349,4445,2397,6493,1373,5469,3421,7517,861,4957,2909,7005,1885,5981,3933,8029,221,4317,2269,6365,1245,5341,3293,7389,733,4829,2781,6877,1757,5853,3805,7901,477,4573,2525,6621,1501,5597,3549,7645,989,5085,3037,7133,2013,6109,4061,8157,61,4157,2109,6205,1085,5181,3133,7229,573,4669,2621,6717,1597,5693,3645,7741,317,4413,2365,6461,1341,5437,3389,7485,829,4925,2877,6973,1853,5949,3901,7997,189,4285,2237,6333,1213,5309,3261,7357,701,4797,2749,6845,1725,5821,3773,7869,445,4541,2493,6589,1469,5565,3517,7613,957,5053,3005,7101,1981,6077,4029,8125,125,4221,2173,6269,1149,5245,3197,7293,637,4733,2685,6781,1661,5757,3709,7805,381,4477,2429,6525,1405,5501,3453,7549,893,4989,2941,7037,1917,6013,3965,8061,253,4349,2301,6397,1277,5373,3325,7421,765,4861,2813,6909,1789,5885,3837,7933,509,4605,2557,6653,1533,5629,3581,7677,1021,5117,3069,7165,2045,6141,4093,8189,3,4099,2051,6147,1027,5123,3075,7171,515,4611,2563,6659,1539,5635,3587,7683,259,4355,2307,6403,1283,5379,3331,7427,771,4867,2819,6915,1795,5891,3843,7939,131,4227,2179,6275,1155,5251,3203,7299,643,4739,2691,6787,1667,5763,3715,7811,387,4483,2435,6531,1411,5507,3459,7555,899,4995,2947,7043,1923,6019,3971,8067,67,4163,2115,6211,1091,5187,3139,7235,579,4675,2627,6723,1603,5699,3651,7747,323,4419,2371,6467,1347,5443,3395,7491,835,4931,2883,6979,1859,5955,3907,8003,195,4291,2243,6339,1219,5315,3267,7363,707,4803,2755,6851,1731,5827,3779,7875,451,4547,2499,6595,1475,5571,3523,7619,963,5059,3011,7107,1987,6083,4035,8131,35,4131,2083,6179,1059,5155,3107,7203,547,4643,2595,6691,1571,5667,3619,7715,291,4387,2339,6435,1315,5411,3363,7459,803,4899,2851,6947,1827,5923,3875,7971,163,4259,2211,6307,1187,5283,3235,7331,675,4771,2723,6819,1699,5795,3747,7843,419,4515,2467,6563,1443,5539,3491,7587,931,5027,2979,7075,1955,6051,4003,8099,99,4195,2147,6243,1123,5219,3171,7267,611,4707,2659,6755,1635,5731,3683,7779,355,4451,2403,6499,1379,5475,3427,7523,867,4963,2915,7011,1891,5987,3939,8035,227,4323,2275,6371,1251,5347,3299,7395,739,4835,2787,6883,1763,5859,3811,7907,483,4579,2531,6627,1507,5603,3555,7651,995,5091,3043,7139,2019,6115,4067,8163,19,4115,2067,6163,1043,5139,3091,7187,531,4627,2579,6675,1555,5651,3603,7699,275,4371,2323,6419,1299,5395,3347,7443,787,4883,2835,6931,1811,5907,3859,7955,147,4243,2195,6291,1171,5267,3219,7315,659,4755,2707,6803,1683,5779,3731,7827,403,4499,2451,6547,1427,5523,3475,7571,915,5011,2963,7059,1939,6035,3987,8083,83,4179,2131,6227,1107,5203,3155,7251,595,4691,2643,6739,1619,5715,3667,7763,339,4435,2387,6483,1363,5459,3411,7507,851,4947,2899,6995,1875,5971,3923,8019,211,4307,2259,6355,1235,5331,3283,7379,723,4819,2771,6867,1747,5843,3795,7891,467,4563,2515,6611,1491,5587,3539,7635,979,5075,3027,7123,2003,6099,4051,8147,51,4147,2099,6195,1075,5171,3123,7219,563,4659,2611,6707,1587,5683,3635,7731,307,4403,2355,6451,1331,5427,3379,7475,819,4915,2867,6963,1843,5939,3891,7987,179,4275,2227,6323,1203,5299,3 251,7347,691,4787,2739,6835,1715,5811,3763,7859,435,4531,2483,6579,1459,5555,3507,7603,947,5043,2995,7091,1971,6067,4019,8115,115,4211,2163,6259,1139,5235,3187,7283,627,4723,2675,6771,1651,5747,3699,7795,371,4467,2419,6515,1395,5491,3443,7539,883,4979,2931,7027,1907,6003,3955,8051,243,4339,2291,6387,1267,5363,3315,7411,755,4851,2803,6899,1779,5875,3827,7923,499,4595,2547,6643,1523,5619,3571,7667,1011,5107,3059,7155,2035,6131,4083,8179,11,4107,2059,6155,1035,5131,3083,7179,523,4619,2571,6667,1547,5643,3595,7691,267,4363,2315,6411,1291,5387,3339,7435,779,4875,2827,6923,1803,5899,3851,7947,139,4235,2187,6283,1163,5259,3211,7307,651,4747,2699,6795,1675,5771,3723,7819,395,4491,2443,6539,1419,5515,3467,7563,907,5003,2955,7051,1931,6027,3979,8075,75,4171,2123,6219,1099,5195,3147,7243,587,4683,2635,6731,1611,5707,3659,7755,331,4427,2379,6475,1355,5451,3403,7499,843,4939,2891,6987,1867,5963,3915,8011,203,4299,2251,6347,1227,5323,3275,7371,715,4811,2763,6859,1739,5835,3787,7883,459,4555,2507,6603,1483,5579,3531,7627,971,5067,3019,7115,1995,6091,4043,8139,43,4139,2091,6187,1067,5163,3115,7211,555,4651,2603,6699,1579,5675,3627,7723,299,4395,2347,6443,1323,5419,3371,7467,811,4907,2859,6955,1835,5931,3883,7979,171,4267,2219,6315,1195,5291,3243,7339,683,4779,2731,6827,1707,5803,3755,7851,427,4523,2475,6571,1451,5547,3499,7595,939,5035,2987,7083,1963,6059,4011,8107,107,4203,2155,6251,1131,5227,3179,7275,619,4715,2667,6763,1643,5739,3691,7787,363,4459,2411,6507,1387,5483,3435,7531,875,4971,2923,7019,1899,5995,3947,8043,235,4331,2283,6379,1259,5355,3307,7403,747,4843,2795,6891,1771,5867,3819,7915,491,4587,2539,6635,1515,5611,3563,7659,1003,5099,3051,7147,2027,6123,4075,8171,27,4123,2075,6171,1051,5147,3099,7195,539,4635,2587,6683,1563,5659,3611,7707,283,4379,2331,6427,1307,5403,3355,7451,795,4891,2843,6939,1819,5915,3867,7963,155,4251,2203,6299,1179,5275,3227,7323,667,4763,2715,6811,1691,5787,3739,7835,411,4507,2459,6555,1435,5531,3483,7579,923,5019,2971,7067,1947,6043,3995,8091,91,4187,2139,6235,1115,5211,3163,7259,603,4699,2651,6747,1627,5723,3675,7771,347,4443,2395,6491,1371,5467,3419,7515,859,4955,2907,7003,1883,5979,3931,8027,219,4315,2267,6363,1243,5339,3291,7387,731,4827,2779,6875,1755,5851,3803,7899,475,4571,2523,6619,1499,5595,3547,7643,987,5083,3035,7131,2011,6107,4059,8155,59,4155,2107,6203,1083,5179,3131,7227,571,4667,2619,6715,1595,5691,3643,7739,315,4411,2363,6459,1339,5435,3387,7483,827,4923,2875,6971,1851,5947,3899,7995,187,4283,2235,6331,1211,5307,3259,7355,699,4795,2747,6843,1723,5819,3771,7867,443,4539,2491,6587,1467,5563,3515,7611,955,5051,3003,7099,1979,6075,4027,8123,123,4219,2171,6267,1147,5243,3195,7291,635,4731,2683,6779,1659,5755,3707,7803,379,4475,2427,6523,1403,5499,3451,7547,891,4987,2939,7035,1915,6011,3963,8059,251,4347,2299,6395,1275,5371,3323,7419,763,4859,2811,6907,1787,5883,3835,7931,507,4603,2555,6651,1531,5627,3579,7675,1019,5115,3067,7163,2043,6139,4091,8187,7,4103,2055,6151,1031,5127,3079,7175,519,4615,2567,6663,1543,5639,3591,7687,263,4359,2311,6407,1287,5383,3335,7431,775,4871,2823,6919,1799,5895,3847,7943,135,4231,2183,6279,1159,5255,3207,7303,647,4743,2695,6791,1671,5767,3719,7815,391,4487,2439,6535,1415,5511,3463,7559,903,4999,2951,7047,1927,6023,3975,8071,71,4167,2119,6215,1095,5191,3143,7239,583,4679,2631,6727,1607,5703,3655,7751,327,4423,2375,6471,1351,5447,3399,7495,839,4935,2887,6983,1863,5959,3911,8007,199,4295,2247,6343,1223,5319,3271,7367,711,4807,2759,6855,1735,5831,3783,7879,455,4551,2503,6599,1479,5575,3527,7623,967,5063,3015,7111,1991,6087,4039,8135,39,4135,2087,6183,1063,5159,3111,7207,551,4647,2599,6695,1575,5671,3623,7719,295,4391,2343,6439,1319,5415,3367,7463,807,4903,2855,6951,1831,5927,3879,7975,167,4263,2215,6311,1191, 5287,3239,7335,679,4775,2727,6823,1703,5799,3751,7847,423,4519,2471,6567,1447,5543,3495,7591,935,5031,2983,7079,1959,6055,4007,8103,103,4199,2151,6247,1127,5223,3175,7271,615,4711,2663,6759,1639,5735,3687,7783,359,4455,2407,6503,1383,5479,3431,7527,871,4967,2919,7015,1895,5991,3943,8039,231,4327,2279,6375,1255,5351,3303,7399,743,4839,2791,6887,1767,5863,3815,7911,487,4583,2535,6631,1511,5607,3559,7655,999,5095,3047,7143,2023,6119,4071,8167,23,4119,2071,6167,1047,5143,3095,7191,535,4631,2583,6679,1559,5655,3607,7703,279,4375,2327,6423,1303,5399,3351,7447,791,4887,2839,6935,1815,5911,3863,7959,151,4247,2199,6295,1175,5271,3223,7319,663,4759,2711,6807,1687,5783,3735,7831,407,4503,2455,6551,1431,5527,3479,7575,919,5015,2967,7063,1943,6039,3991,8087,87,4183,2135,6231,1111,5207,3159,7255,599,4695,2647,6743,1623,5719,3671,7767,343,4439,2391,6487,1367,5463,3415,7511,855,4951,2903,6999,1879,5975,3927,8023,215,4311,2263,6359,1239,5335,3287,7383,727,4823,2775,6871,1751,5847,3799,7895,471,4567,2519,6615,1495,5591,3543,7639,983,5079,3031,7127,2007,6103,4055,8151,55,4151,2103,6199,1079,5175,3127,7223,567,4663,2615,6711,1591,5687,3639,7735,311,4407,2359,6455,1335,5431,3383,7479,823,4919,2871,6967,1847,5943,3895,7991,183,4279,2231,6327,1207,5303,3255,7351,695,4791,2743,6839,1719,5815,3767,7863,439,4535,2487,6583,1463,5559,3511,7607,951,5047,2999,7095,1975,6071,4023,8119,119,4215,2167,6263,1143,5239,3191,7287,631,4727,2679,6775,1655,5751,3703,7799,375,4471,2423,6519,1399,5495,3447,7543,887,4983,2935,7031,1911,6007,3959,8055,247,4343,2295,6391,1271,5367,3319,7415,759,4855,2807,6903,1783,5879,3831,7927,503,4599,2551,6647,1527,5623,3575,7671,1015,5111,3063,7159,2039,6135,4087,8183,15,4111,2063,6159,1039,5135,3087,7183,527,4623,2575,6671,1551,5647,3599,7695,271,4367,2319,6415,1295,5391,3343,7439,783,4879,2831,6927,1807,5903,3855,7951,143,4239,2191,6287,1167,5263,3215,7311,655,4751,2703,6799,1679,5775,3727,7823,399,4495,2447,6543,1423,5519,3471,7567,911,5007,2959,7055,1935,6031,3983,8079,79,4175,2127,6223,1103,5199,3151,7247,591,4687,2639,6735,1615,5711,3663,7759,335,4431,2383,6479,1359,5455,3407,7503,847,4943,2895,6991,1871,5967,3919,8015,207,4303,2255,6351,1231,5327,3279,7375,719,4815,2767,6863,1743,5839,3791,7887,463,4559,2511,6607,1487,5583,3535,7631,975,5071,3023,7119,1999,6095,4047,8143,47,4143,2095,6191,1071,5167,3119,7215,559,4655,2607,6703,1583,5679,3631,7727,303,4399,2351,6447,1327,5423,3375,7471,815,4911,2863,6959,1839,5935,3887,7983,175,4271,2223,6319,1199,5295,3247,7343,687,4783,2735,6831,1711,5807,3759,7855,431,4527,2479,6575,1455,5551,3503,7599,943,5039,2991,7087,1967,6063,4015,8111,111,4207,2159,6255,1135,5231,3183,7279,623,4719,2671,6767,1647,5743,3695,7791,367,4463,2415,6511,1391,5487,3439,7535,879,4975,2927,7023,1903,5999,3951,8047,239,4335,2287,6383,1263,5359,3311,7407,751,4847,2799,6895,1775,5871,3823,7919,495,4591,2543,6639,1519,5615,3567,7663,1007,5103,3055,7151,2031,6127,4079,8175,31,4127,2079,6175,1055,5151,3103,7199,543,4639,2591,6687,1567,5663,3615,7711,287,4383,2335,6431,1311,5407,3359,7455,799,4895,2847,6943,1823,5919,3871,7967,159,4255,2207,6303,1183,5279,3231,7327,671,4767,2719,6815,1695,5791,3743,7839,415,4511,2463,6559,1439,5535,3487,7583,927,5023,2975,7071,1951,6047,3999,8095,95,4191,2143,6239,1119,5215,3167,7263,607,4703,2655,6751,1631,5727,3679,7775,351,4447,2399,6495,1375,5471,3423,7519,863,4959,2911,7007,1887,5983,3935,8031,223,4319,2271,6367,1247,5343,3295,7391,735,4831,2783,6879,1759,5855,3807,7903,479,4575,2527,6623,1503,5599,3551,7647,991,5087,3039,7135,2015,6111,4063,8159,63,4159,2111,6207,1087,5183,3135,7231,575,4671,2623,6719,1599,5695,3647,7743,319,4415,2367,6463,1343,5439,3391,7487,831,4927,2879,6975,1855,5951,3903,7999,191,4287,2239,633 5,1215,5311,3263,7359,703,4799,2751,6847,1727,5823,3775,7871,447,4543,2495,6591,1471,5567,3519,7615,959,5055,3007,7103,1983,6079,4031,8127,127,4223,2175,6271,1151,5247,3199,7295,639,4735,2687,6783,1663,5759,3711,7807,383,4479,2431,6527,1407,5503,3455,7551,895,4991,2943,7039,1919,6015,3967,8063,255,4351,2303,6399,1279,5375,3327,7423,767,4863,2815,6911,1791,5887,3839,7935,511,4607,2559,6655,1535,5631,3583,7679,1023,5119,3071,7167,2047,6143,4095,8191]。
序列2：
Shorten_Q_half＝[1,4097,2049,6145,1025,5121,3073,7169,513,4609,2561,6657,1537,5633,3585,7681,257,4353,2305,6401,1281,5377,3329,7425,769,4865,2817,6913,1793,5889,3841,7937,129,4225,2177,6273,1153,5249,3201,7297,641,4737,2689,6785,1665,5761,3713,7809,385,4481,2433,6529,1409,5505,3457,7553,897,4993,2945,7041,1921,6017,3969,8065,65,4161,2113,6209,1089,5185,3137,7233,577,4673,2625,6721,1601,5697,3649,7745,321,4417,2369,6465,1345,5441,3393,7489,833,4929,2881,6977,1857,5953,3905,8001,193,4289,2241,6337,1217,5313,3265,7361,705,4801,2753,6849,1729,5825,3777,7873,449,4545,2497,6593,1473,5569,3521,7617,961,5057,3009,7105,1985,6081,4033,8129,33,4129,2081,6177,1057,5153,3105,7201,545,4641,2593,6689,1569,5665,3617,7713,289,4385,2337,6433,1313,5409,3361,7457,801,4897,2849,6945,1825,5921,3873,7969,161,4257,2209,6305,1185,5281,3233,7329,673,4769,2721,6817,1697,5793,3745,7841,417,4513,2465,6561,1441,5537,3489,7585,929,5025,2977,7073,1953,6049,4001,8097,97,4193,2145,6241,1121,5217,3169,7265,609,4705,2657,6753,1633,5729,3681,7777,353,4449,2401,6497,1377,5473,3425,7521,865,4961,2913,7009,1889,5985,3937,8033,225,4321,2273,6369,1249,5345,3297,7393,737,4833,2785,6881,1761,5857,3809,7905,481,4577,2529,6625,1505,5601,3553,7649,993,5089,3041,7137,2017,6113,4065,8161,17,4113,2065,6161,1041,5137,3089,7185,529,4625,2577,6673,1553,5649,3601,7697,273,4369,2321,6417,1297,5393,3345,7441,785,4881,2833,6929,1809,5905,3857,7953,145,4241,2193,6289,1169,5265,3217,7313,657,4753,2705,6801,1681,5777,3729,7825,401,4497,2449,6545,1425,5521,3473,7569,913,5009,2961,7057,1937,6033,3985,8081,81,4177,2129,6225,1105,5201,3153,7249,593,4689,2641,6737,1617,5713,3665,7761,337,4433,2385,6481,1361,5457,3409,7505,849,4945,2897,6993,1873,5969,3921,8017,209,4305,2257,6353,1233,5329,3281,7377,721,4817,2769,6865,1745,5841,3793,7889,465,4561,2513,6609,1489,5585,3537,7633,977,5073,3025,7121,2001,6097,4049,8145,49,4145,2097,6193,1073,5169,3121,7217,561,4657,2609,6705,1585,5681,3633,7729,305,4401,2353,6449,1329,5425,3377,7473,817,4913,2865,6961,1841,5937,3889,7985,177,4273,2225,6321,1201,5297,3249,7345,689,4785,2737,6833,1713,5809,3761,7857,433,4529,2481,6577,1457,5553,3505,7601,945,5041,2993,7089,1969,6065,4017,8113,113,4209,2161,6257,1137,5233,3185,7281,625,4721,2673,6769,1649,5745,3697,7793,369,4465,2417,6513,1393,5489,3441,7537,881,4977,2929,7025,1905,6001,3953,8049,241,4337,2289,6385,1265,5361,3313,7409,753,4849,2801,6897,1777,5873,3825,7921,497,4593,2545,6641,1521,5617,3569,7665,1009,5105,3057,7153,2033,6129,4081,8177,9,4105,2057,6153,1033,5129,3081,7177,521,4617,2569,6665,1545,5641,3593,7689,265,4361,2313,6409,1289,5385,3337,7433,777,4873,2825,6921,1801,5897,3849,7945,137,4233,2185,6281,1161,5257,3209,7305,649,4745,2697,6793,1673,5769,3721,7817,393,4489,2441,6537,1417,5513,3465,7561,905,5001,2953,7049,1929,6025,3977,8073,73,4169,2121,6217,1097,5193,3145,7241,585,4681,2633,6729,1609,5705,3657,7753,329,4425,2377,6473,1353,5449,3401,7497,841,4937,2889,6985,1865,5961,3913,8009,201,4297,2249,6345,1225,5321,3273,7369,713,4809,2761,6857,1737,5833,3785,7881,457,4553,2505,6601,1481,5577,3529,7625,969,5065,3017,7113,1993,6089,4041,8137,41,4137,2089,6185,1065,5161, 3113,7209,553,4649,2601,6697,1577,5673,3625,7721,297,4393,2345,6441,1321,5417,3369,7465,809,4905,2857,6953,1833,5929,3881,7977,169,4265,2217,6313,1193,5289,3241,7337,681,4777,2729,6825,1705,5801,3753,7849,425,4521,2473,6569,1449,5545,3497,7593,937,5033,2985,7081,1961,6057,4009,8105,105,4201,2153,6249,1129,5225,3177,7273,617,4713,2665,6761,1641,5737,3689,7785,361,4457,2409,6505,1385,5481,3433,7529,873,4969,2921,7017,1897,5993,3945,8041,233,4329,2281,6377,1257,5353,3305,7401,745,4841,2793,6889,1769,5865,3817,7913,489,4585,2537,6633,1513,5609,3561,7657,1001,5097,3049,7145,2025,6121,4073,8169,25,4121,2073,6169,1049,5145,3097,7193,537,4633,2585,6681,1561,5657,3609,7705,281,4377,2329,6425,1305,5401,3353,7449,793,4889,2841,6937,1817,5913,3865,7961,153,4249,2201,6297,1177,5273,3225,7321,665,4761,2713,6809,1689,5785,3737,7833,409,4505,2457,6553,1433,5529,3481,7577,921,5017,2969,7065,1945,6041,3993,8089,89,4185,2137,6233,1113,5209,3161,7257,601,4697,2649,6745,1625,5721,3673,7769,345,4441,2393,6489,1369,5465,3417,7513,857,4953,2905,7001,1881,5977,3929,8025,217,4313,2265,6361,1241,5337,3289,7385,729,4825,2777,6873,1753,5849,3801,7897,473,4569,2521,6617,1497,5593,3545,7641,985,5081,3033,7129,2009,6105,4057,8153,57,4153,2105,6201,1081,5177,3129,7225,569,4665,2617,6713,1593,5689,3641,7737,313,4409,2361,6457,1337,5433,3385,7481,825,4921,2873,6969,1849,5945,3897,7993,185,4281,2233,6329,1209,5305,3257,7353,697,4793,2745,6841,1721,5817,3769,7865,441,4537,2489,6585,1465,5561,3513,7609,953,5049,3001,7097,1977,6073,4025,8121,121,4217,2169,6265,1145,5241,3193,7289,633,4729,2681,6777,1657,5753,3705,7801,377,4473,2425,6521,1401,5497,3449,7545,889,4985,2937,7033,1913,6009,3961,8057,249,4345,2297,6393,1273,5369,3321,7417,761,4857,2809,6905,1785,5881,3833,7929,505,4601,2553,6649,1529,5625,3577,7673,1017,5113,3065,7161,2041,6137,4089,8185,5,4101,2053,6149,1029,5125,3077,7173,517,4613,2565,6661,1541,5637,3589,7685,261,4357,2309,6405,1285,5381,3333,7429,773,4869,2821,6917,1797,5893,3845,7941,133,4229,2181,6277,1157,5253,3205,7301,645,4741,2693,6789,1669,5765,3717,7813,389,4485,2437,6533,1413,5509,3461,7557,901,4997,2949,7045,1925,6021,3973,8069,69,4165,2117,6213,1093,5189,3141,7237,581,4677,2629,6725,1605,5701,3653,7749,325,4421,2373,6469,1349,5445,3397,7493,837,4933,2885,6981,1861,5957,3909,8005,197,4293,2245,6341,1221,5317,3269,7365,709,4805,2757,6853,1733,5829,3781,7877,453,4549,2501,6597,1477,5573,3525,7621,965,5061,3013,7109,1989,6085,4037,8133,37,4133,2085,6181,1061,5157,3109,7205,549,4645,2597,6693,1573,5669,3621,7717,293,4389,2341,6437,1317,5413,3365,7461,805,4901,2853,6949,1829,5925,3877,7973,165,4261,2213,6309,1189,5285,3237,7333,677,4773,2725,6821,1701,5797,3749,7845,421,4517,2469,6565,1445,5541,3493,7589,933,5029,2981,7077,1957,6053,4005,8101,101,4197,2149,6245,1125,5221,3173,7269,613,4709,2661,6757,1637,5733,3685,7781,357,4453,2405,6501,1381,5477,3429,7525,869,4965,2917,7013,1893,5989,3941,8037,229,4325,2277,6373,1253,5349,3301,7397,741,4837,2789,6885,1765,5861,3813,7909,485,4581,2533,6629,1509,5605,3557,7653,997,5093,3045,7141,2021,6117,4069,8165,21,4117,2069,6165,1045,5141,3093,7189,533,4629,2581,6677,1557,5653,3605,7701,277,4373,2325,6421,1301,5397,3349,7445,789,4885,2837,6933,1813,5909,3861,7957,149,4245,2197,6293,1173,5269,3221,7317,661,4757,2709,6805,1685,5781,3733,7829,405,4501,2453,6549,1429,5525,3477,7573,917,5013,2965,7061,1941,6037,3989,8085,85,4181,2133,6229,1109,5205,3157,7253,597,4693,2645,6741,1621,5717,3669,7765,341,4437,2389,6485,1365,5461,3413,7509,853,4949,2901,6997,1877,5973,3925,8021,213,4309,2261,6357,1237,5333,3285,7381,725,4821,2773,6869,1749,5845,3797,7893,469,4565,2517,6613,1493,5589,3541,7637,981,5077,3029,7125,2005,6101,4053,8149,53,4149,2101,6197,1077, 5173,3125,7221,565,4661,2613,6709,1589,5685,3637,7733,309,4405,2357,6453,1333,5429,3381,7477,821,4917,2869,6965,1845,5941,3893,7989,181,4277,2229,6325,1205,5301,3253,7349,693,4789,2741,6837,1717,5813,3765,7861,437,4533,2485,6581,1461,5557,3509,7605,949,5045,2997,7093,1973,6069,4021,8117,117,4213,2165,6261,1141,5237,3189,7285,629,4725,2677,6773,1653,5749,3701,7797,373,4469,2421,6517,1397,5493,3445,7541,885,4981,2933,7029,1909,6005,3957,8053,245,4341,2293,6389,1269,5365,3317,7413,757,4853,2805,6901,1781,5877,3829,7925,501,4597,2549,6645,1525,5621,3573,7669,1013,5109,3061,7157,2037,6133,4085,8181,13,4109,2061,6157,1037,5133,3085,7181,525,4621,2573,6669,1549,5645,3597,7693,269,4365,2317,6413,1293,5389,3341,7437,781,4877,2829,6925,1805,5901,3853,7949,141,4237,2189,6285,1165,5261,3213,7309,653,4749,2701,6797,1677,5773,3725,7821,397,4493,2445,6541,1421,5517,3469,7565,909,5005,2957,7053,1933,6029,3981,8077,77,4173,2125,6221,1101,5197,3149,7245,589,4685,2637,6733,1613,5709,3661,7757,333,4429,2381,6477,1357,5453,3405,7501,845,4941,2893,6989,1869,5965,3917,8013,205,4301,2253,6349,1229,5325,3277,7373,717,4813,2765,6861,1741,5837,3789,7885,461,4557,2509,6605,1485,5581,3533,7629,973,5069,3021,7117,1997,6093,4045,8141,45,4141,2093,6189,1069,5165,3117,7213,557,4653,2605,6701,1581,5677,3629,7725,301,4397,2349,6445,1325,5421,3373,7469,813,4909,2861,6957,1837,5933,3885,7981,173,4269,2221,6317,1197,5293,3245,7341,685,4781,2733,6829,1709,5805,3757,7853,429,4525,2477,6573,1453,5549,3501,7597,941,5037,2989,7085,1965,6061,4013,8109,109,4205,2157,6253,1133,5229,3181,7277,621,4717,2669,6765,1645,5741,3693,7789,365,4461,2413,6509,1389,5485,3437,7533,877,4973,2925,7021,1901,5997,3949,8045,237,4333,2285,6381,1261,5357,3309,7405,749,4845,2797,6893,1773,5869,3821,7917,493,4589,2541,6637,1517,5613,3565,7661,1005,5101,3053,7149,2029,6125,4077,8173,29,4125,2077,6173,1053,5149,3101,7197,541,4637,2589,6685,1565,5661,3613,7709,285,4381,2333,6429,1309,5405,3357,7453,797,4893,2845,6941,1821,5917,3869,7965,157,4253,2205,6301,1181,5277,3229,7325,669,4765,2717,6813,1693,5789,3741,7837,413,4509,2461,6557,1437,5533,3485,7581,925,5021,2973,7069,1949,6045,3997,8093,93,4189,2141,6237,1117,5213,3165,7261,605,4701,2653,6749,1629,5725,3677,7773,349,4445,2397,6493,1373,5469,3421,7517,861,4957,2909,7005,1885,5981,3933,8029,221,4317,2269,6365,1245,5341,3293,7389,733,4829,2781,6877,1757,5853,3805,7901,477,4573,2525,6621,1501,5597,3549,7645,989,5085,3037,7133,2013,6109,4061,8157,61,4157,2109,6205,1085,5181,3133,7229,573,4669,2621,6717,1597,5693,3645,7741,317,4413,2365,6461,1341,5437,3389,7485,829,4925,2877,6973,1853,5949,3901,7997,189,4285,2237,6333,1213,5309,3261,7357,701,4797,2749,6845,1725,5821,3773,7869,445,4541,2493,6589,1469,5565,3517,7613,957,5053,3005,7101,1981,6077,4029,8125,125,4221,2173,6269,1149,5245,3197,7293,637,4733,2685,6781,1661,5757,3709,7805,381,4477,2429,6525,1405,5501,3453,7549,893,4989,2941,7037,1917,6013,3965,8061,253,4349,2301,6397,1277,5373,3325,7421,765,4861,2813,6909,1789,5885,3837,7933,509,4605,2557,6653,1533,5629,3581,7677,1021,5117,3069,7165,2045,6141,4093,8189,3,4099,2051,6147,1027,5123,3075,7171,515,4611,2563,6659,1539,5635,3587,7683,259,4355,2307,6403,1283,5379,3331,7427,771,4867,2819,6915,1795,5891,3843,7939,131,4227,2179,6275,1155,5251,3203,7299,643,4739,2691,6787,1667,5763,3715,7811,387,4483,2435,6531,1411,5507,3459,7555,899,4995,2947,7043,1923,6019,3971,8067,67,4163,2115,6211,1091,5187,3139,7235,579,4675,2627,6723,1603,5699,3651,7747,323,4419,2371,6467,1347,5443,3395,7491,835,4931,2883,6979,1859,5955,3907,8003,195,4291,2243,6339,1219,5315,3267,7363,707,4803,2755,6851,1731,5827,3779,7875,451,4547,2499,6595,1475,5571,3523,7619,963,5059,3011,7107,1987,6083,4035,8131,35,4131,2083,617 9,1059,5155,3107,7203,547,4643,2595,6691,1571,5667,3619,7715,291,4387,2339,6435,1315,5411,3363,7459,803,4899,2851,6947,1827,5923,3875,7971,163,4259,2211,6307,1187,5283,3235,7331,675,4771,2723,6819,1699,5795,3747,7843,419,4515,2467,6563,1443,5539,3491,7587,931,5027,2979,7075,1955,6051,4003,8099,99,4195,2147,6243,1123,5219,3171,7267,611,4707,2659,6755,1635,5731,3683,7779,355,4451,2403,6499,1379,5475,3427,7523,867,4963,2915,7011,1891,5987,3939,8035,227,4323,2275,6371,1251,5347,3299,7395,739,4835,2787,6883,1763,5859,3811,7907,483,4579,2531,6627,1507,5603,3555,7651,995,5091,3043,7139,2019,6115,4067,8163,19,4115,2067,6163,1043,5139,3091,7187,531,4627,2579,6675,1555,5651,3603,7699,275,4371,2323,6419,1299,5395,3347,7443,787,4883,2835,6931,1811,5907,3859,7955,147,4243,2195,6291,1171,5267,3219,7315,659,4755,2707,6803,1683,5779,3731,7827,403,4499,2451,6547,1427,5523,3475,7571,915,5011,2963,7059,1939,6035,3987,8083,83,4179,2131,6227,1107,5203,3155,7251,595,4691,2643,6739,1619,5715,3667,7763,339,4435,2387,6483,1363,5459,3411,7507,851,4947,2899,6995,1875,5971,3923,8019,211,4307,2259,6355,1235,5331,3283,7379,723,4819,2771,6867,1747,5843,3795,7891,467,4563,2515,6611,1491,5587,3539,7635,979,5075,3027,7123,2003,6099,4051,8147,51,4147,2099,6195,1075,5171,3123,7219,563,4659,2611,6707,1587,5683,3635,7731,307,4403,2355,6451,1331,5427,3379,7475,819,4915,2867,6963,1843,5939,3891,7987,179,4275,2227,6323,1203,5299,3251,7347,691,4787,2739,6835,1715,5811,3763,7859,435,4531,2483,6579,1459,5555,3507,7603,947,5043,2995,7091,1971,6067,4019,8115,115,4211,2163,6259,1139,5235,3187,7283,627,4723,2675,6771,1651,5747,3699,7795,371,4467,2419,6515,1395,5491,3443,7539,883,4979,2931,7027,1907,6003,3955,8051,243,4339,2291,6387,1267,5363,3315,7411,755,4851,2803,6899,1779,5875,3827,7923,499,4595,2547,6643,1523,5619,3571,7667,1011,5107,3059,7155,2035,6131,4083,8179,11,4107,2059,6155,1035,5131,3083,7179,523,4619,2571,6667,1547,5643,3595,7691,267,4363,2315,6411,1291,5387,3339,7435,779,4875,2827,6923,1803,5899,3851,7947,139,4235,2187,6283,1163,5259,3211,7307,651,4747,2699,6795,1675,5771,3723,7819,395,4491,2443,6539,1419,5515,3467,7563,907,5003,2955,7051,1931,6027,3979,8075,75,4171,2123,6219,1099,5195,3147,7243,587,4683,2635,6731,1611,5707,3659,7755,331,4427,2379,6475,1355,5451,3403,7499,843,4939,2891,6987,1867,5963,3915,8011,203,4299,2251,6347,1227,5323,3275,7371,715,4811,2763,6859,1739,5835,3787,7883,459,4555,2507,6603,1483,5579,3531,7627,971,5067,3019,7115,1995,6091,4043,8139,43,4139,2091,6187,1067,5163,3115,7211,555,4651,2603,6699,1579,5675,3627,7723,299,4395,2347,6443,1323,5419,3371,7467,811,4907,2859,6955,1835,5931,3883,7979,171,4267,2219,6315,1195,5291,3243,7339,683,4779,2731,6827,1707,5803,3755,7851,427,4523,2475,6571,1451,5547,3499,7595,939,5035,2987,7083,1963,6059,4011,8107,107,4203,2155,6251,1131,5227,3179,7275,619,4715,2667,6763,1643,5739,3691,7787,363,4459,2411,6507,1387,5483,3435,7531,875,4971,2923,7019,1899,5995,3947,8043,235,4331,2283,6379,1259,5355,3307,7403,747,4843,2795,6891,1771,5867,3819,7915,491,4587,2539,6635,1515,5611,3563,7659,1003,5099,3051,7147,2027,6123,4075,8171,27,4123,2075,6171,1051,5147,3099,7195,539,4635,2587,6683,1563,5659,3611,7707,283,4379,2331,6427,1307,5403,3355,7451,795,4891,2843,6939,1819,5915,3867,7963,155,4251,2203,6299,1179,5275,3227,7323,667,4763,2715,6811,1691,5787,3739,7835,411,4507,2459,6555,1435,5531,3483,7579,923,5019,2971,7067,1947,6043,3995,8091,91,4187,2139,6235,1115,5211,3163,7259,603,4699,2651,6747,1627,5723,3675,7771,347,4443,2395,6491,1371,5467,3419,7515,859,4955,2907,7003,1883,5979,3931,8027,219,4315,2267,6363,1243,5339,3291,7387,731,4827,2779,6875,1755,5851,3803,7899,475,4571,2523,6619,1499,5595,3547,7643,987,5083,3035,7131,2011,6107,4059,8155,59,4155,210 7,6203,1083,5179,3131,7227,571,4667,2619,6715,1595,5691,3643,7739,315,4411,2363,6459,1339,5435,3387,7483,827,4923,2875,6971,1851,5947,3899,7995,187,4283,2235,6331,1211,5307,3259,7355,699,4795,2747,6843,1723,5819,3771,7867,443,4539,2491,6587,1467,5563,3515,7611,955,5051,3003,7099,1979,6075,4027,8123,123,4219,2171,6267,1147,5243,3195,7291,635,4731,2683,6779,1659,5755,3707,7803,379,4475,2427,6523,1403,5499,3451,7547,891,4987,2939,7035,1915,6011,3963,8059,251,4347,2299,6395,1275,5371,3323,7419,763,4859,2811,6907,1787,5883,3835,7931,507,4603,2555,6651,1531,5627,3579,7675,1019,5115,3067,7163,2043,6139,4091,8187,7,4103,2055,6151,1031,5127,3079,7175,519,4615,2567,6663,1543,5639,3591,7687,263,4359,2311,6407,1287,5383,3335,7431,775,4871,2823,6919,1799,5895,3847,7943,135,4231,2183,6279,1159,5255,3207,7303,647,4743,2695,6791,1671,5767,3719,7815,391,4487,2439,6535,1415,5511,3463,7559,903,4999,2951,7047,1927,6023,3975,8071,71,4167,2119,6215,1095,5191,3143,7239,583,4679,2631,6727,1607,5703,3655,7751,327,4423,2375,6471,1351,5447,3399,7495,839,4935,2887,6983,1863,5959,3911,8007,199,4295,2247,6343,1223,5319,3271,7367,711,4807,2759,6855,1735,5831,3783,7879,455,4551,2503,6599,1479,5575,3527,7623,967,5063,3015,7111,1991,6087,4039,8135,39,4135,2087,6183,1063,5159,3111,7207,551,4647,2599,6695,1575,5671,3623,7719,295,4391,2343,6439,1319,5415,3367,7463,807,4903,2855,6951,1831,5927,3879,7975,167,4263,2215,6311,1191,5287,3239,7335,679,4775,2727,6823,1703,5799,3751,7847,423,4519,2471,6567,1447,5543,3495,7591,935,5031,2983,7079,1959,6055,4007,8103,103,4199,2151,6247,1127,5223,3175,7271,615,4711,2663,6759,1639,5735,3687,7783,359,4455,2407,6503,1383,5479,3431,7527,871,4967,2919,7015,1895,5991,3943,8039,231,4327,2279,6375,1255,5351,3303,7399,743,4839,2791,6887,1767,5863,3815,7911,487,4583,2535,6631,1511,5607,3559,7655,999,5095,3047,7143,2023,6119,4071,8167,23,4119,2071,6167,1047,5143,3095,7191,535,4631,2583,6679,1559,5655,3607,7703,279,4375,2327,6423,1303,5399,3351,7447,791,4887,2839,6935,1815,5911,3863,7959,151,4247,2199,6295,1175,5271,3223,7319,663,4759,2711,6807,1687,5783,3735,7831,407,4503,2455,6551,1431,5527,3479,7575,919,5015,2967,7063,1943,6039,3991,8087,87,4183,2135,6231,1111,5207,3159,7255,599,4695,2647,6743,1623,5719,3671,7767,343,4439,2391,6487,1367,5463,3415,7511,855,4951,2903,6999,1879,5975,3927,8023,215,4311,2263,6359,1239,5335,3287,7383,727,4823,2775,6871,1751,5847,3799,7895,471,4567,2519,6615,1495,5591,3543,7639,983,5079,3031,7127,2007,6103,4055,8151,55,4151,2103,6199,1079,5175,3127,7223,567,4663,2615,6711,1591,5687,3639,7735,311,4407,2359,6455,1335,5431,3383,7479,823,4919,2871,6967,1847,5943,3895,7991,183,4279,2231,6327,1207,5303,3255,7351,695,4791,2743,6839,1719,5815,3767,7863,439,4535,2487,6583,1463,5559,3511,7607,951,5047,2999,7095,1975,6071,4023,8119,119,4215,2167,6263,1143,5239,3191,7287,631,4727,2679,6775,1655,5751,3703,7799,375,4471,2423,6519,1399,5495,3447,7543,887,4983,2935,7031,1911,6007,3959,8055,247,4343,2295,6391,1271,5367,3319,7415,759,4855,2807,6903,1783,5879,3831,7927,503,4599,2551,6647,1527,5623,3575,7671,1015,5111,3063,7159,2039,6135,4087,8183,15,4111,2063,6159,1039,5135,3087,7183,527,4623,2575,6671,1551,5647,3599,7695,271,4367,2319,6415,1295,5391,3343,7439,783,4879,2831,6927,1807,5903,3855,7951,143,4239,2191,6287,1167,5263,3215,7311,655,4751,2703,6799,1679,5775,3727,7823,399,4495,2447,6543,1423,5519,3471,7567,911,5007,2959,7055,1935,6031,3983,8079,79,4175,2127,6223,1103,5199,3151,7247,591,4687,2639,6735,1615,5711,3663,7759,335,4431,2383,6479,1359,5455,3407,7503,847,4943,2895,6991,1871,5967,3919,8015,207,4303,2255,6351,1231,5327,3279,7375,719,4815,2767,6863,1743,5839,3791,7887,463,4559,2511,6607,1487,5583,3535,7631,975,5071,3023,7119,1999,6095,4047,8143,47,414 3,2095,6191,1071,5167,3119,7215,559,4655,2607,6703,1583,5679,3631,7727,303,4399,2351,6447,1327,5423,3375,7471,815,4911,2863,6959,1839,5935,3887,7983,175,4271,2223,6319,1199,5295,3247,7343,687,4783,2735,6831,1711,5807,3759,7855,431,4527,2479,6575,1455,5551,3503,7599,943,5039,2991,7087,1967,6063,4015,8111,111,4207,2159,6255,1135,5231,3183,7279,623,4719,2671,6767,1647,5743,3695,7791,367,4463,2415,6511,1391,5487,3439,7535,879,4975,2927,7023,1903,5999,3951,8047,239,4335,2287,6383,1263,5359,3311,7407,751,4847,2799,6895,1775,5871,3823,7919,495,4591,2543,6639,1519,5615,3567,7663,1007,5103,3055,7151,2031,6127,4079,8175,31,4127,2079,6175,1055,5151,3103,7199,543,4639,2591,6687,1567,5663,3615,7711,287,4383,2335,6431,1311,5407,3359,7455,799,4895,2847,6943,1823,5919,3871,7967,159,4255,2207,6303,1183,5279,3231,7327,671,4767,2719,6815,1695,5791,3743,7839,415,4511,2463,6559,1439,5535,3487,7583,927,5023,2975,7071,1951,6047,3999,8095,95,4191,2143,6239,1119,5215,3167,7263,607,4703,2655,6751,1631,5727,3679,7775,351,4447,2399,6495,1375,5471,3423,7519,863,4959,2911,7007,1887,5983,3935,8031,223,4319,2271,6367,1247,5343,3295,7391,735,4831,2783,6879,1759,5855,3807,7903,479,4575,2527,6623,1503,5599,3551,7647,991,5087,3039,7135,2015,6111,4063,8159,63,4159,2111,6207,1087,5183,3135,7231,575,4671,2623,6719,1599,5695,3647,7743,319,4415,2367,6463,1343,5439,3391,7487,831,4927,2879,6975,1855,5951,3903,7999,191,4287,2239,6335,1215,5311,3263,7359,703,4799,2751,6847,1727,5823,3775,7871,447,4543,2495,6591,1471,5567,3519,7615,959,5055,3007,7103,1983,6079,4031,8127,127,4223,2175,6271,1151,5247,3199,7295,639,4735,2687,6783,1663,5759,3711,7807,383,4479,2431,6527,1407,5503,3455,7551,895,4991,2943,7039,1919,6015,3967,8063,255,4351,2303,6399,1279,5375,3327,7423,767,4863,2815,6911,1791,5887,3839,7935,511,4607,2559,6655,1535,5631,3583,7679,1023,5119,3071,7167,2047,6143,4095,8191]。
序列3：
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08,8004,196,4292,2244,6340,1220,5316,3268,7364,708,4804,2756,6852,1732,5828,3780,7876,452,4548,2500,6596,1476,5572,3524,7620,964,5060,3012,7108,1988,6084,4036,8132,36,4132,2084,6180,1060,5156,3108,7204,548,4644,2596,6692,1572,5668,3620,7716,292,4388,2340,6436,1316,5412,3364,7460,804,4900,2852,6948,1828,5924,3876,7972,164,4260,2212,6308,1188,5284,3236,7332,676,4772,2724,6820,1700,5796,3748,7844,420,4516,2468,6564,1444,5540,3492,7588,932,5028,2980,7076,1956,6052,4004,8100,100,4196,2148,6244,1124,5220,3172,7268,612,4708,2660,6756,1636,5732,3684,7780,356,4452,2404,6500,1380,5476,3428,7524,868,4964,2916,7012,1892,5988,3940,8036,228,4324,2276,6372,1252,5348,3300,7396,740,4836,2788,6884,1764,5860,3812,7908,484,4580,2532,6628,1508,5604,3556,7652,996,5092,3044,7140,2020,6116,4068,8164,20,4116,2068,6164,1044,5140,3092,7188,532,4628,2580,6676,1556,5652,3604,7700,276,4372,2324,6420,1300,5396,3348,7444,788,4884,2836,6932,1812,5908,3860,7956,148,4244,2196,6292,1172,5268,3220,7316,660,4756,2708,6804,1684,5780,3732,7828,404,4500,2452,6548,1428,5524,3476,7572,916,5012,2964,7060,1940,6036,3988,8084,84,4180,2132,6228,1108,5204,3156,7252,596,4692,2644,6740,1620,5716,3668,7764,340,4436,2388,6484,1364,5460,3412,7508,852,4948,2900,6996,1876,5972,3924,8020,212,4308,2260,6356,1236,5332,3284,7380,724,4820,2772,6868,1748,5844,3796,7892,468,4564,2516,6612,1492,5588,3540,7636,980,5076,3028,7124,2004,6100,4052,8148,52,4148,2100,6196,1076,5172,3124,7220,564,4660,2612,6708,1588,5684,3636,7732,308,4404,2356,6452,1332,5428,3380,7476,820,4916,2868,6964,1844,5940,3892,7988,180,4276,2228,6324,1204,5300,3252,7348,692,4788,2740,6836,1716,5812,3764,7860,436,4532,2484,6580,1460,5556,3508,7604,948,5044,2996,7092,1972,6068,4020,8116,116,4212,2164,6260,1140,5236,3188,7284,628,4724,2676,6772,1652,5748,3700,7796,372,4468,2420,6516,1396,5492,3444,7540,884,4980,2932,7028,1908,6004,3956,8052,244,4340,2292,6388,1268,5364,3316,7412,756,4852,2804,6900,1780,5876,3828,7924,500,4596,2548,6644,1524,5620,3572,7668,1012,5108,3060,7156,2036,6132,4084,8180,12,4108,2060,6156,1036,5132,3084,7180,524,4620,2572,6668,1548,5644,3596,7692,268,4364,2316,6412,1292,5388,3340,7436,780,4876,2828,6924,1804,5900,3852,7948,140,4236,2188,6284,1164,5260,3212,7308,652,4748,2700,6796,1676,5772,3724,7820,396,4492,2444,6540,1420,5516,3468,7564,908,5004,2956,7052,1932,6028,3980,8076,76,4172,2124,6220,1100,5196,3148,7244,588,4684,2636,6732,1612,5708,3660,7756,332,4428,2380,6476,1356,5452,3404,7500,844,4940,2892,6988,1868,5964,3916,8012,204,4300,2252,6348,1228,5324,3276,7372,716,4812,2764,6860,1740,5836,3788,7884,460,4556,2508,6604,1484,5580,3532,7628,972,5068,3020,7116,1996,6092,4044,8140,44,4140,2092,6188,1068,5164,3116,7212,556,4652,2604,6700,1580,5676,3628,7724,300,4396,2348,6444,1324,5420,3372,7468,812,4908,2860,6956,1836,5932,3884,7980,172,4268,2220,6316,1196,5292,3244,7340,684,4780,2732,6828,1708,5804,3756,7852,428,4524,2476,6572,1452,5548,3500,7596,940,5036,2988,7084,1964,6060,4012,8108,108,4204,2156,6252,1132,5228,3180,7276,620,4716,2668,6764,1644,5740,3692,7788,364,4460,2412,6508,1388,5484,3436,7532,876,4972,2924,7020,1900,5996,3948,8044,236,4332,2284,6380,1260,5356,3308,7404,748,4844,2796,6892,1772,5868,3820,7916,492,4588,2540,6636,1516,5612,3564,7660,1004,5100,3052,7148,2028,6124,4076,8172,28,4124,2076,6172,1052,5148,3100,7196,540,4636,2588,6684,1564,5660,3612,7708,284,4380,2332,6428,1308,5404,3356,7452,796,4892,2844,6940,1820,5916,3868,7964,156,4252,2204,6300,1180,5276,3228,7324,668,4764,2716,6812,1692,5788,3740,7836,412,4508,2460,6556,1436,5532,3484,7580,924,5020,2972,7068,1948,6044,3996,8092,92,4188,2140,6236,1116,5212,3164,7260,604,4700,2652,6748,1628,5724,3676,7772,348,4444,2396,6492,1372,5468,3420,7516,860,4956,2908,7004,1884,5 980,3932,8028,220,4316,2268,6364,1244,5340,3292,7388,732,4828,2780,6876,1756,5852,3804,7900,476,4572,2524,6620,1500,5596,3548,7644,988,5084,3036,7132,2012,6108,4060,8156,60,4156,2108,6204,1084,5180,3132,7228,572,4668,2620,6716,1596,5692,3644,7740,316,4412,2364,6460,1340,5436,3388,7484,828,4924,2876,6972,1852,5948,3900,7996,188,4284,2236,6332,1212,5308,3260,7356,700,4796,2748,6844,1724,5820,3772,7868,444,4540,2492,6588,1468,5564,3516,7612,956,5052,3004,7100,1980,6076,4028,8124,124,4220,2172,6268,1148,5244,3196,7292,636,4732,2684,6780,1660,5756,3708,7804,380,4476,2428,6524,1404,5500,3452,7548,892,4988,2940,7036,1916,6012,3964,8060,252,4348,2300,6396,1276,5372,3324,7420,764,4860,2812,6908,1788,5884,3836,7932,508,4604,2556,6652,1532,5628,3580,7676,1020,5116,3068,7164,2044,6140,4092,8188,2,4098,2050,6146,1026,5122,3074,7170,514,4610,2562,6658,1538,5634,3586,7682,258,4354,2306,6402,1282,5378,3330,7426,770,4866,2818,6914,1794,5890,3842,7938,130,4226,2178,6274,1154,5250,3202,7298,642,4738,2690,6786,1666,5762,3714,7810,386,4482,2434,6530,1410,5506,3458,7554,898,4994,2946,7042,1922,6018,3970,8066,66,4162,2114,6210,1090,5186,3138,7234,578,4674,2626,6722,1602,5698,3650,7746,322,4418,2370,6466,1346,5442,3394,7490,834,4930,2882,6978,1858,5954,3906,8002,194,4290,2242,6338,1218,5314,3266,7362,706,4802,2754,6850,1730,5826,3778,7874,450,4546,2498,6594,1474,5570,3522,7618,962,5058,3010,7106,1986,6082,4034,8130,34,4130,2082,6178,1058,5154,3106,7202,546,4642,2594,6690,1570,5666,3618,7714,290,4386,2338,6434,1314,5410,3362,7458,802,4898,2850,6946,1826,5922,3874,7970,162,4258,2210,6306,1186,5282,3234,7330,674,4770,2722,6818,1698,5794,3746,7842,418,4514,2466,6562,1442,5538,3490,7586,930,5026,2978,7074,1954,6050,4002,8098,98,4194,2146,6242,1122,5218,3170,7266,610,4706,2658,6754,1634,5730,3682,7778,354,4450,2402,6498,1378,5474,3426,7522,866,4962,2914,7010,1890,5986,3938,8034,226,4322,2274,6370,1250,5346,3298,7394,738,4834,2786,6882,1762,5858,3810,7906,482,4578,2530,6626,1506,5602,3554,7650,994,5090,3042,7138,2018,6114,4066,8162,18,4114,2066,6162,1042,5138,3090,7186,530,4626,2578,6674,1554,5650,3602,7698,274,4370,2322,6418,1298,5394,3346,7442,786,4882,2834,6930,1810,5906,3858,7954,146,4242,2194,6290,1170,5266,3218,7314,658,4754,2706,6802,1682,5778,3730,7826,402,4498,2450,6546,1426,5522,3474,7570,914,5010,2962,7058,1938,6034,3986,8082,82,4178,2130,6226,1106,5202,3154,7250,594,4690,2642,6738,1618,5714,3666,7762,338,4434,2386,6482,1362,5458,3410,7506,850,4946,2898,6994,1874,5970,3922,8018,210,4306,2258,6354,1234,5330,3282,7378,722,4818,2770,6866,1746,5842,3794,7890,466,4562,2514,6610,1490,5586,3538,7634,978,5074,3026,7122,2002,6098,4050,8146,50,4146,2098,6194,1074,5170,3122,7218,562,4658,2610,6706,1586,5682,3634,7730,306,4402,2354,6450,1330,5426,3378,7474,818,4914,2866,6962,1842,5938,3890,7986,178,4274,2226,6322,1202,5298,3250,7346,690,4786,2738,6834,1714,5810,3762,7858,434,4530,2482,6578,1458,5554,3506,7602,946,5042,2994,7090,1970,6066,4018,8114,114,4210,2162,6258,1138,5234,3186,7282,626,4722,2674,6770,1650,5746,3698,7794,370,4466,2418,6514,1394,5490,3442,7538,882,4978,2930,7026,1906,6002,3954,8050,242,4338,2290,6386,1266,5362,3314,7410,754,4850,2802,6898,1778,5874,3826,7922,498,4594,2546,6642,1522,5618,3570,7666,1010,5106,3058,7154,2034,6130,4082,8178,10,4106,2058,6154,1034,5130,3082,7178,522,4618,2570,6666,1546,5642,3594,7690,266,4362,2314,6410,1290,5386,3338,7434,778,4874,2826,6922,1802,5898,3850,7946,138,4234,2186,6282,1162,5258,3210,7306,650,4746,2698,6794,1674,5770,3722,7818,394,4490,2442,6538,1418,5514,3466,7562,906,5002,2954,7050,1930,6026,3978,8074,74,4170,2122,6218,1098,5194,3146,7242,586,4682,2634,6730,1610,5706,3658,7754,330,4426,2378,6474,1354,5450,3402,7498,842,4938,2890,6986,18 66,5962,3914,8010,202,4298,2250,6346,1226,5322,3274,7370,714,4810,2762,6858,1738,5834,3786,7882,458,4554,2506,6602,1482,5578,3530,7626,970,5066,3018,7114,1994,6090,4042,8138,42,4138,2090,6186,1066,5162,3114,7210,554,4650,2602,6698,1578,5674,3626,7722,298,4394,2346,6442,1322,5418,3370,7466,810,4906,2858,6954,1834,5930,3882,7978,170,4266,2218,6314,1194,5290,3242,7338,682,4778,2730,6826,1706,5802,3754,7850,426,4522,2474,6570,1450,5546,3498,7594,938,5034,2986,7082,1962,6058,4010,8106,106,4202,2154,6250,1130,5226,3178,7274,618,4714,2666,6762,1642,5738,3690,7786,362,4458,2410,6506,1386,5482,3434,7530,874,4970,2922,7018,1898,5994,3946,8042,234,4330,2282,6378,1258,5354,3306,7402,746,4842,2794,6890,1770,5866,3818,7914,490,4586,2538,6634,1514,5610,3562,7658,1002,5098,3050,7146,2026,6122,4074,8170,26,4122,2074,6170,1050,5146,3098,7194,538,4634,2586,6682,1562,5658,3610,7706,282,4378,2330,6426,1306,5402,3354,7450,794,4890,2842,6938,1818,5914,3866,7962,154,4250,2202,6298,1178,5274,3226,7322,666,4762,2714,6810,1690,5786,3738,7834,410,4506,2458,6554,1434,5530,3482,7578,922,5018,2970,7066,1946,6042,3994,8090,90,4186,2138,6234,1114,5210,3162,7258,602,4698,2650,6746,1626,5722,3674,7770,346,4442,2394,6490,1370,5466,3418,7514,858,4954,2906,7002,1882,5978,3930,8026,218,4314,2266,6362,1242,5338,3290,7386,730,4826,2778,6874,1754,5850,3802,7898,474,4570,2522,6618,1498,5594,3546,7642,986,5082,3034,7130,2010,6106,4058,8154,58,4154,2106,6202,1082,5178,3130,7226,570,4666,2618,6714,1594,5690,3642,7738,314,4410,2362,6458,1338,5434,3386,7482,826,4922,2874,6970,1850,5946,3898,7994,186,4282,2234,6330,1210,5306,3258,7354,698,4794,2746,6842,1722,5818,3770,7866,442,4538,2490,6586,1466,5562,3514,7610,954,5050,3002,7098,1978,6074,4026,8122,122,4218,2170,6266,1146,5242,3194,7290,634,4730,2682,6778,1658,5754,3706,7802,378,4474,2426,6522,1402,5498,3450,7546,890,4986,2938,7034,1914,6010,3962,8058,250,4346,2298,6394,1274,5370,3322,7418,762,4858,2810,6906,1786,5882,3834,7930,506,4602,2554,6650,1530,5626,3578,7674,1018,5114,3066,7162,2042,6138,4090,8186,6,4102,2054,6150,1030,5126,3078,7174,518,4614,2566,6662,1542,5638,3590,7686,262,4358,2310,6406,1286,5382,3334,7430,774,4870,2822,6918,1798,5894,3846,7942,134,4230,2182,6278,1158,5254,3206,7302,646,4742,2694,6790,1670,5766,3718,7814,390,4486,2438,6534,1414,5510,3462,7558,902,4998,2950,7046,1926,6022,3974,8070,70,4166,2118,6214,1094,5190,3142,7238,582,4678,2630,6726,1606,5702,3654,7750,326,4422,2374,6470,1350,5446,3398,7494,838,4934,2886,6982,1862,5958,3910,8006,198,4294,2246,6342,1222,5318,3270,7366,710,4806,2758,6854,1734,5830,3782,7878,454,4550,2502,6598,1478,5574,3526,7622,966,5062,3014,7110,1990,6086,4038,8134,38,4134,2086,6182,1062,5158,3110,7206,550,4646,2598,6694,1574,5670,3622,7718,294,4390,2342,6438,1318,5414,3366,7462,806,4902,2854,6950,1830,5926,3878,7974,166,4262,2214,6310,1190,5286,3238,7334,678,4774,2726,6822,1702,5798,3750,7846,422,4518,2470,6566,1446,5542,3494,7590,934,5030,2982,7078,1958,6054,4006,8102,102,4198,2150,6246,1126,5222,3174,7270,614,4710,2662,6758,1638,5734,3686,7782,358,4454,2406,6502,1382,5478,3430,7526,870,4966,2918,7014,1894,5990,3942,8038,230,4326,2278,6374,1254,5350,3302,7398,742,4838,2790,6886,1766,5862,3814,7910,486,4582,2534,6630,1510,5606,3558,7654,998,5094,3046,7142,2022,6118,4070,8166,22,4118,2070,6166,1046,5142,3094,7190,534,4630,2582,6678,1558,5654,3606,7702,278,4374,2326,6422,1302,5398,3350,7446,790,4886,2838,6934,1814,5910,3862,7958,150,4246,2198,6294,1174,5270,3222,7318,662,4758,2710,6806,1686,5782,3734,7830,406,4502,2454,6550,1430,5526,3478,7574,918,5014,2966,7062,1942,6038,3990,8086,86,4182,2134,6230,1110,5206,3158,7254,598,4694,2646,6742,1622,5718,3670,7766,342,4438,2390,6486,1366,5462,3414,7510,854,4950,2902,69 98,1878,5974,3926,8022,214,4310,2262,6358,1238,5334,3286,7382,726,4822,2774,6870,1750,5846,3798,7894,470,4566,2518,6614,1494,5590,3542,7638,982,5078,3030,7126,2006,6102,4054,8150,54,4150,2102,6198,1078,5174,3126,7222,566,4662,2614,6710,1590,5686,3638,7734,310,4406,2358,6454,1334,5430,3382,7478,822,4918,2870,6966,1846,5942,3894,7990,182,4278,2230,6326,1206,5302,3254,7350,694,4790,2742,6838,1718,5814,3766,7862,438,4534,2486,6582,1462,5558,3510,7606,950,5046,2998,7094,1974,6070,4022,8118,118,4214,2166,6262,1142,5238,3190,7286,630,4726,2678,6774,1654,5750,3702,7798,374,4470,2422,6518,1398,5494,3446,7542,886,4982,2934,7030,1910,6006,3958,8054,246,4342,2294,6390,1270,5366,3318,7414,758,4854,2806,6902,1782,5878,3830,7926,502,4598,2550,6646,1526,5622,3574,7670,1014,5110,3062,7158,2038,6134,4086,8182,14,4110,2062,6158,1038,5134,3086,7182,526,4622,2574,6670,1550,5646,3598,7694,270,4366,2318,6414,1294,5390,3342,7438,782,4878,2830,6926,1806,5902,3854,7950,142,4238,2190,6286,1166,5262,3214,7310,654,4750,2702,6798,1678,5774,3726,7822,398,4494,2446,6542,1422,5518,3470,7566,910,5006,2958,7054,1934,6030,3982,8078,78,4174,2126,6222,1102,5198,3150,7246,590,4686,2638,6734,1614,5710,3662,7758,334,4430,2382,6478,1358,5454,3406,7502,846,4942,2894,6990,1870,5966,3918,8014,206,4302,2254,6350,1230,5326,3278,7374,718,4814,2766,6862,1742,5838,3790,7886,462,4558,2510,6606,1486,5582,3534,7630,974,5070,3022,7118,1998,6094,4046,8142,46,4142,2094,6190,1070,5166,3118,7214,558,4654,2606,6702,1582,5678,3630,7726,302,4398,2350,6446,1326,5422,3374,7470,814,4910,2862,6958,1838,5934,3886,7982,174,4270,2222,6318,1198,5294,3246,7342,686,4782,2734,6830,1710,5806,3758,7854,430,4526,2478,6574,1454,5550,3502,7598,942,5038,2990,7086,1966,6062,4014,8110,110,4206,2158,6254,1134,5230,3182,7278,622,4718,2670,6766,1646,5742,3694,7790,366,4462,2414,6510,1390,5486,3438,7534,878,4974,2926,7022,1902,5998,3950,8046,238,4334,2286,6382,1262,5358,3310,7406,750,4846,2798,6894,1774,5870,3822,7918,494,4590,2542,6638,1518,5614,3566,7662,1006,5102,3054,7150,2030,6126,4078,8174,30,4126,2078,6174,1054,5150,3102,7198,542,4638,2590,6686,1566,5662,3614,7710,286,4382,2334,6430,1310,5406,3358,7454,798,4894,2846,6942,1822,5918,3870,7966,158,4254,2206,6302,1182,5278,3230,7326,670,4766,2718,6814,1694,5790,3742,7838,414,4510,2462,6558,1438,5534,3486,7582,926,5022,2974,7070,1950,6046,3998,8094,94,4190,2142,6238,1118,5214,3166,7262,606,4702,2654,6750,1630,5726,3678,7774,350,4446,2398,6494,1374,5470,3422,7518,862,4958,2910,7006,1886,5982,3934,8030,222,4318,2270,6366,1246,5342,3294,7390,734,4830,2782,6878,1758,5854,3806,7902,478,4574,2526,6622,1502,5598,3550,7646,990,5086,3038,7134,2014,6110,4062,8158,62,4158,2110,6206,1086,5182,3134,7230,574,4670,2622,6718,1598,5694,3646,7742,318,4414,2366,6462,1342,5438,3390,7486,830,4926,2878,6974,1854,5950,3902,7998,190,4286,2238,6334,1214,5310,3262,7358,702,4798,2750,6846,1726,5822,3774,7870,446,4542,2494,6590,1470,5566,3518,7614,958,5054,3006,7102,1982,6078,4030,8126,126,4222,2174,6270,1150,5246,3198,7294,638,4734,2686,6782,1662,5758,3710,7806,382,4478,2430,6526,1406,5502,3454,7550,894,4990,2942,7038,1918,6014,3966,8062,254,4350,2302,6398,1278,5374,3326,7422,766,4862,2814,6910,1790,5886,3838,7934,510,4606,2558,6654,1534,5630,3582,7678,1022,5118,3070,7166,2046,6142,4094,8190,1,4097,2049,6145,1025,5121,3073,7169,513,4609,2561,6657,1537,5633,3585,7681,257,4353,2305,6401,1281,5377,3329,7425,769,4865,2817,6913,1793,5889,3841,7937,129,4225,2177,6273,1153,5249,3201,7297,641,4737,2689,6785,1665,5761,3713,7809,385,4481,2433,6529,1409,5505,3457,7553,897,4993,2945,7041,1921,6017,3969,8065,65,4161,2113,6209,1089,5185,3137,7233,577,4673,2625,6721,1601,5697,3649,7745,321,4417,2369,6465,1345,5441,3393,7489,833,4929,2 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53,2905,7001,1881,5977,3929,8025,217,4313,2265,6361,1241,5337,3289,7385,729,4825,2777,6873,1753,5849,3801,7897,473,4569,2521,6617,1497,5593,3545,7641,985,5081,3033,7129,2009,6105,4057,8153,57,4153,2105,6201,1081,5177,3129,7225,569,4665,2617,6713,1593,5689,3641,7737,313,4409,2361,6457,1337,5433,3385,7481,825,4921,2873,6969,1849,5945,3897,7993,185,4281,2233,6329,1209,5305,3257,7353,697,4793,2745,6841,1721,5817,3769,7865,441,4537,2489,6585,1465,5561,3513,7609,953,5049,3001,7097,1977,6073,4025,8121,121,4217,2169,6265,1145,5241,3193,7289,633,4729,2681,6777,1657,5753,3705,7801,377,4473,2425,6521,1401,5497,3449,7545,889,4985,2937,7033,1913,6009,3961,8057,249,4345,2297,6393,1273,5369,3321,7417,761,4857,2809,6905,1785,5881,3833,7929,505,4601,2553,6649,1529,5625,3577,7673,1017,5113,3065,7161,2041,6137,4089,8185,5,4101,2053,6149,1029,5125,3077,7173,517,4613,2565,6661,1541,5637,3589,7685,261,4357,2309,6405,1285,5381,3333,7429,773,4869,2821,6917,1797,5893,3845,7941,133,4229,2181,6277,1157,5253,3205,7301,645,4741,2693,6789,1669,5765,3717,7813,389,4485,2437,6533,1413,5509,3461,7557,901,4997,2949,7045,1925,6021,3973,8069,69,4165,2117,6213,1093,5189,3141,7237,581,4677,2629,6725,1605,5701,3653,7749,325,4421,2373,6469,1349,5445,3397,7493,837,4933,2885,6981,1861,5957,3909,8005,197,4293,2245,6341,1221,5317,3269,7365,709,4805,2757,6853,1733,5829,3781,7877,453,4549,2501,6597,1477,5573,3525,7621,965,5061,3013,7109,1989,6085,4037,8133,37,4133,2085,6181,1061,5157,3109,7205,549,4645,2597,6693,1573,5669,3621,7717,293,4389,2341,6437,1317,5413,3365,7461,805,4901,2853,6949,1829,5925,3877,7973,165,4261,2213,6309,1189,5285,3237,7333,677,4773,2725,6821,1701,5797,3749,7845,421,4517,2469,6565,1445,5541,3493,7589,933,5029,2981,7077,1957,6053,4005,8101,101,4197,2149,6245,1125,5221,3173,7269,613,4709,2661,6757,1637,5733,3685,7781,357,4453,2405,6501,1381,5477,3429,7525,869,4965,2917,7013,1893,5989,3941,8037,229,4325,2277,6373,1253,5349,3301,7397,741,4837,2789,6885,1765,5861,3813,7909,485,4581,2533,6629,1509,5605,3557,7653,997,5093,3045,7141,2021,6117,4069,8165,21,4117,2069,6165,1045,5141,3093,7189,533,4629,2581,6677,1557,5653,3605,7701,277,4373,2325,6421,1301,5397,3349,7445,789,4885,2837,6933,1813,5909,3861,7957,149,4245,2197,6293,1173,5269,3221,7317,661,4757,2709,6805,1685,5781,3733,7829,405,4501,2453,6549,1429,5525,3477,7573,917,5013,2965,7061,1941,6037,3989,8085,85,4181,2133,6229,1109,5205,3157,7253,597,4693,2645,6741,1621,5717,3669,7765,341,4437,2389,6485,1365,5461,3413,7509,853,4949,2901,6997,1877,5973,3925,8021,213,4309,2261,6357,1237,5333,3285,7381,725,4821,2773,6869,1749,5845,3797,7893,469,4565,2517,6613,1493,5589,3541,7637,981,5077,3029,7125,2005,6101,4053,8149,53,4149,2101,6197,1077,5173,3125,7221,565,4661,2613,6709,1589,5685,3637,7733,309,4405,2357,6453,1333,5429,3381,7477,821,4917,2869,6965,1845,5941,3893,7989,181,4277,2229,6325,1205,5301,3253,7349,693,4789,2741,6837,1717,5813,3765,7861,437,4533,2485,6581,1461,5557,3509,7605,949,5045,2997,7093,1973,6069,4021,8117,117,4213,2165,6261,1141,5237,3189,7285,629,4725,2677,6773,1653,5749,3701,7797,373,4469,2421,6517,1397,5493,3445,7541,885,4981,2933,7029,1909,6005,3957,8053,245,4341,2293,6389,1269,5365,3317,7413,757,4853,2805,6901,1781,5877,3829,7925,501,4597,2549,6645,1525,5621,3573,7669,1013,5109,3061,7157,2037,6133,4085,8181,13,4109,2061,6157,1037,5133,3085,7181,525,4621,2573,6669,1549,5645,3597,7693,269,4365,2317,6413,1293,5389,3341,7437,781,4877,2829,6925,1805,5901,3853,7949,141,4237,2189,6285,1165,5261,3213,7309,653,4749,2701,6797,1677,5773,3725,7821,397,4493,2445,6541,1421,5517,3469,7565,909,5005,2957,7053,1933,6029,3981,8077,77,4173,2125,6221,1101,5197,3149,7245,589,4685,2637,6733,1613,5709,3661,7757,333,4429,2381,6477,1357,5453,3405,7501,8 45,4941,2893,6989,1869,5965,3917,8013,205,4301,2253,6349,1229,5325,3277,7373,717,4813,2765,6861,1741,5837,3789,7885,461,4557,2509,6605,1485,5581,3533,7629,973,5069,3021,7117,1997,6093,4045,8141,45,4141,2093,6189,1069,5165,3117,7213,557,4653,2605,6701,1581,5677,3629,7725,301,4397,2349,6445,1325,5421,3373,7469,813,4909,2861,6957,1837,5933,3885,7981,173,4269,2221,6317,1197,5293,3245,7341,685,4781,2733,6829,1709,5805,3757,7853,429,4525,2477,6573,1453,5549,3501,7597,941,5037,2989,7085,1965,6061,4013,8109,109,4205,2157,6253,1133,5229,3181,7277,621,4717,2669,6765,1645,5741,3693,7789,365,4461,2413,6509,1389,5485,3437,7533,877,4973,2925,7021,1901,5997,3949,8045,237,4333,2285,6381,1261,5357,3309,7405,749,4845,2797,6893,1773,5869,3821,7917,493,4589,2541,6637,1517,5613,3565,7661,1005,5101,3053,7149,2029,6125,4077,8173,29,4125,2077,6173,1053,5149,3101,7197,541,4637,2589,6685,1565,5661,3613,7709,285,4381,2333,6429,1309,5405,3357,7453,797,4893,2845,6941,1821,5917,3869,7965,157,4253,2205,6301,1181,5277,3229,7325,669,4765,2717,6813,1693,5789,3741,7837,413,4509,2461,6557,1437,5533,3485,7581,925,5021,2973,7069,1949,6045,3997,8093,93,4189,2141,6237,1117,5213,3165,7261,605,4701,2653,6749,1629,5725,3677,7773,349,4445,2397,6493,1373,5469,3421,7517,861,4957,2909,7005,1885,5981,3933,8029,221,4317,2269,6365,1245,5341,3293,7389,733,4829,2781,6877,1757,5853,3805,7901,477,4573,2525,6621,1501,5597,3549,7645,989,5085,3037,7133,2013,6109,4061,8157,61,4157,2109,6205,1085,5181,3133,7229,573,4669,2621,6717,1597,5693,3645,7741,317,4413,2365,6461,1341,5437,3389,7485,829,4925,2877,6973,1853,5949,3901,7997,189,4285,2237,6333,1213,5309,3261,7357,701,4797,2749,6845,1725,5821,3773,7869,445,4541,2493,6589,1469,5565,3517,7613,957,5053,3005,7101,1981,6077,4029,8125,125,4221,2173,6269,1149,5245,3197,7293,637,4733,2685,6781,1661,5757,3709,7805,381,4477,2429,6525,1405,5501,3453,7549,893,4989,2941,7037,1917,6013,3965,8061,253,4349,2301,6397,1277,5373,3325,7421,765,4861,2813,6909,1789,5885,3837,7933,509,4605,2557,6653,1533,5629,3581,7677,1021,5117,3069,7165,2045,6141,4093,8189,3,4099,2051,6147,1027,5123,3075,7171,515,4611,2563,6659,1539,5635,3587,7683,259,4355,2307,6403,1283,5379,3331,7427,771,4867,2819,6915,1795,5891,3843,7939,131,4227,2179,6275,1155,5251,3203,7299,643,4739,2691,6787,1667,5763,3715,7811,387,4483,2435,6531,1411,5507,3459,7555,899,4995,2947,7043,1923,6019,3971,8067,67,4163,2115,6211,1091,5187,3139,7235,579,4675,2627,6723,1603,5699,3651,7747,323,4419,2371,6467,1347,5443,3395,7491,835,4931,2883,6979,1859,5955,3907,8003,195,4291,2243,6339,1219,5315,3267,7363,707,4803,2755,6851,1731,5827,3779,7875,451,4547,2499,6595,1475,5571,3523,7619,963,5059,3011,7107,1987,6083,4035,8131,35,4131,2083,6179,1059,5155,3107,7203,547,4643,2595,6691,1571,5667,3619,7715,291,4387,2339,6435,1315,5411,3363,7459,803,4899,2851,6947,1827,5923,3875,7971,163,4259,2211,6307,1187,5283,3235,7331,675,4771,2723,6819,1699,5795,3747,7843,419,4515,2467,6563,1443,5539,3491,7587,931,5027,2979,7075,1955,6051,4003,8099,99,4195,2147,6243,1123,5219,3171,7267,611,4707,2659,6755,1635,5731,3683,7779,355,4451,2403,6499,1379,5475,3427,7523,867,4963,2915,7011,1891,5987,3939,8035,227,4323,2275,6371,1251,5347,3299,7395,739,4835,2787,6883,1763,5859,3811,7907,483,4579,2531,6627,1507,5603,3555,7651,995,5091,3043,7139,2019,6115,4067,8163,19,4115,2067,6163,1043,5139,3091,7187,531,4627,2579,6675,1555,5651,3603,7699,275,4371,2323,6419,1299,5395,3347,7443,787,4883,2835,6931,1811,5907,3859,7955,147,4243,2195,6291,1171,5267,3219,7315,659,4755,2707,6803,1683,5779,3731,7827,403,4499,2451,6547,1427,5523,3475,7571,915,5011,2963,7059,1939,6035,3987,8083,83,4179,2131,6227,1107,5203,3155,7251,595,4691,2643,6739,1619,5715,3667,7763,339,4435,2387,6483,1363,5459,3411,75 07,851,4947,2899,6995,1875,5971,3923,8019,211,4307,2259,6355,1235,5331,3283,7379,723,4819,2771,6867,1747,5843,3795,7891,467,4563,2515,6611,1491,5587,3539,7635,979,5075,3027,7123,2003,6099,4051,8147,51,4147,2099,6195,1075,5171,3123,7219,563,4659,2611,6707,1587,5683,3635,7731,307,4403,2355,6451,1331,5427,3379,7475,819,4915,2867,6963,1843,5939,3891,7987,179,4275,2227,6323,1203,5299,3251,7347,691,4787,2739,6835,1715,5811,3763,7859,435,4531,2483,6579,1459,5555,3507,7603,947,5043,2995,7091,1971,6067,4019,8115,115,4211,2163,6259,1139,5235,3187,7283,627,4723,2675,6771,1651,5747,3699,7795,371,4467,2419,6515,1395,5491,3443,7539,883,4979,2931,7027,1907,6003,3955,8051,243,4339,2291,6387,1267,5363,3315,7411,755,4851,2803,6899,1779,5875,3827,7923,499,4595,2547,6643,1523,5619,3571,7667,1011,5107,3059,7155,2035,6131,4083,8179,11,4107,2059,6155,1035,5131,3083,7179,523,4619,2571,6667,1547,5643,3595,7691,267,4363,2315,6411,1291,5387,3339,7435,779,4875,2827,6923,1803,5899,3851,7947,139,4235,2187,6283,1163,5259,3211,7307,651,4747,2699,6795,1675,5771,3723,7819,395,4491,2443,6539,1419,5515,3467,7563,907,5003,2955,7051,1931,6027,3979,8075,75,4171,2123,6219,1099,5195,3147,7243,587,4683,2635,6731,1611,5707,3659,7755,331,4427,2379,6475,1355,5451,3403,7499,843,4939,2891,6987,1867,5963,3915,8011,203,4299,2251,6347,1227,5323,3275,7371,715,4811,2763,6859,1739,5835,3787,7883,459,4555,2507,6603,1483,5579,3531,7627,971,5067,3019,7115,1995,6091,4043,8139,43,4139,2091,6187,1067,5163,3115,7211,555,4651,2603,6699,1579,5675,3627,7723,299,4395,2347,6443,1323,5419,3371,7467,811,4907,2859,6955,1835,5931,3883,7979,171,4267,2219,6315,1195,5291,3243,7339,683,4779,2731,6827,1707,5803,3755,7851,427,4523,2475,6571,1451,5547,3499,7595,939,5035,2987,7083,1963,6059,4011,8107,107,4203,2155,6251,1131,5227,3179,7275,619,4715,2667,6763,1643,5739,3691,7787,363,4459,2411,6507,1387,5483,3435,7531,875,4971,2923,7019,1899,5995,3947,8043,235,4331,2283,6379,1259,5355,3307,7403,747,4843,2795,6891,1771,5867,3819,7915,491,4587,2539,6635,1515,5611,3563,7659,1003,5099,3051,7147,2027,6123,4075,8171,27,4123,2075,6171,1051,5147,3099,7195,539,4635,2587,6683,1563,5659,3611,7707,283,4379,2331,6427,1307,5403,3355,7451,795,4891,2843,6939,1819,5915,3867,7963,155,4251,2203,6299,1179,5275,3227,7323,667,4763,2715,6811,1691,5787,3739,7835,411,4507,2459,6555,1435,5531,3483,7579,923,5019,2971,7067,1947,6043,3995,8091,91,4187,2139,6235,1115,5211,3163,7259,603,4699,2651,6747,1627,5723,3675,7771,347,4443,2395,6491,1371,5467,3419,7515,859,4955,2907,7003,1883,5979,3931,8027,219,4315,2267,6363,1243,5339,3291,7387,731,4827,2779,6875,1755,5851,3803,7899,475,4571,2523,6619,1499,5595,3547,7643,987,5083,3035,7131,2011,6107,4059,8155,59,4155,2107,6203,1083,5179,3131,7227,571,4667,2619,6715,1595,5691,3643,7739,315,4411,2363,6459,1339,5435,3387,7483,827,4923,2875,6971,1851,5947,3899,7995,187,4283,2235,6331,1211,5307,3259,7355,699,4795,2747,6843,1723,5819,3771,7867,443,4539,2491,6587,1467,5563,3515,7611,955,5051,3003,7099,1979,6075,4027,8123,123,4219,2171,6267,1147,5243,3195,7291,635,4731,2683,6779,1659,5755,3707,7803,379,4475,2427,6523,1403,5499,3451,7547,891,4987,2939,7035,1915,6011,3963,8059,251,4347,2299,6395,1275,5371,3323,7419,763,4859,2811,6907,1787,5883,3835,7931,507,4603,2555,6651,1531,5627,3579,7675,1019,5115,3067,7163,2043,6139,4091,8187,7,4103,2055,6151,1031,5127,3079,7175,519,4615,2567,6663,1543,5639,3591,7687,263,4359,2311,6407,1287,5383,3335,7431,775,4871,2823,6919,1799,5895,3847,7943,135,4231,2183,6279,1159,5255,3207,7303,647,4743,2695,6791,1671,5767,3719,7815,391,4487,2439,6535,1415,5511,3463,7559,903,4999,2951,7047,1927,6023,3975,8071,71,4167,2119,6215,1095,5191,3143,7239,583,4679,2631,6727,1607,5703,3655,7751,327,4423,2375,6471,1351,5447,3 399,7495,839,4935,2887,6983,1863,5959,3911,8007,199,4295,2247,6343,1223,5319,3271,7367,711,4807,2759,6855,1735,5831,3783,7879,455,4551,2503,6599,1479,5575,3527,7623,967,5063,3015,7111,1991,6087,4039,8135,39,4135,2087,6183,1063,5159,3111,7207,551,4647,2599,6695,1575,5671,3623,7719,295,4391,2343,6439,1319,5415,3367,7463,807,4903,2855,6951,1831,5927,3879,7975,167,4263,2215,6311,1191,5287,3239,7335,679,4775,2727,6823,1703,5799,3751,7847,423,4519,2471,6567,1447,5543,3495,7591,935,5031,2983,7079,1959,6055,4007,8103,103,4199,2151,6247,1127,5223,3175,7271,615,4711,2663,6759,1639,5735,3687,7783,359,4455,2407,6503,1383,5479,3431,7527,871,4967,2919,7015,1895,5991,3943,8039,231,4327,2279,6375,1255,5351,3303,7399,743,4839,2791,6887,1767,5863,3815,7911,487,4583,2535,6631,1511,5607,3559,7655,999,5095,3047,7143,2023,6119,4071,8167,23,4119,2071,6167,1047,5143,3095,7191,535,4631,2583,6679,1559,5655,3607,7703,279,4375,2327,6423,1303,5399,3351,7447,791,4887,2839,6935,1815,5911,3863,7959,151,4247,2199,6295,1175,5271,3223,7319,663,4759,2711,6807,1687,5783,3735,7831,407,4503,2455,6551,1431,5527,3479,7575,919,5015,2967,7063,1943,6039,3991,8087,87,4183,2135,6231,1111,5207,3159,7255,599,4695,2647,6743,1623,5719,3671,7767,343,4439,2391,6487,1367,5463,3415,7511,855,4951,2903,6999,1879,5975,3927,8023,215,4311,2263,6359,1239,5335,3287,7383,727,4823,2775,6871,1751,5847,3799,7895,471,4567,2519,6615,1495,5591,3543,7639,983,5079,3031,7127,2007,6103,4055,8151,55,4151,2103,6199,1079,5175,3127,7223,567,4663,2615,6711,1591,5687,3639,7735,311,4407,2359,6455,1335,5431,3383,7479,823,4919,2871,6967,1847,5943,3895,7991,183,4279,2231,6327,1207,5303,3255,7351,695,4791,2743,6839,1719,5815,3767,7863,439,4535,2487,6583,1463,5559,3511,7607,951,5047,2999,7095,1975,6071,4023,8119,119,4215,2167,6263,1143,5239,3191,7287,631,4727,2679,6775,1655,5751,3703,7799,375,4471,2423,6519,1399,5495,3447,7543,887,4983,2935,7031,1911,6007,3959,8055,247,4343,2295,6391,1271,5367,3319,7415,759,4855,2807,6903,1783,5879,3831,7927,503,4599,2551,6647,1527,5623,3575,7671,1015,5111,3063,7159,2039,6135,4087,8183,15,4111,2063,6159,1039,5135,3087,7183,527,4623,2575,6671,1551,5647,3599,7695,271,4367,2319,6415,1295,5391,3343,7439,783,4879,2831,6927,1807,5903,3855,7951,143,4239,2191,6287,1167,5263,3215,7311,655,4751,2703,6799,1679,5775,3727,7823,399,4495,2447,6543,1423,5519,3471,7567,911,5007,2959,7055,1935,6031,3983,8079,79,4175,2127,6223,1103,5199,3151,7247,591,4687,2639,6735,1615,5711,3663,7759,335,4431,2383,6479,1359,5455,3407,7503,847,4943,2895,6991,1871,5967,3919,8015,207,4303,2255,6351,1231,5327,3279,7375,719,4815,2767,6863,1743,5839,3791,7887,463,4559,2511,6607,1487,5583,3535,7631,975,5071,3023,7119,1999,6095,4047,8143,47,4143,2095,6191,1071,5167,3119,7215,559,4655,2607,6703,1583,5679,3631,7727,303,4399,2351,6447,1327,5423,3375,7471,815,4911,2863,6959,1839,5935,3887,7983,175,4271,2223,6319,1199,5295,3247,7343,687,4783,2735,6831,1711,5807,3759,7855,431,4527,2479,6575,1455,5551,3503,7599,943,5039,2991,7087,1967,6063,4015,8111,111,4207,2159,6255,1135,5231,3183,7279,623,4719,2671,6767,1647,5743,3695,7791,367,4463,2415,6511,1391,5487,3439,7535,879,4975,2927,7023,1903,5999,3951,8047,239,4335,2287,6383,1263,5359,3311,7407,751,4847,2799,6895,1775,5871,3823,7919,495,4591,2543,6639,1519,5615,3567,7663,1007,5103,3055,7151,2031,6127,4079,8175,31,4127,2079,6175,1055,5151,3103,7199,543,4639,2591,6687,1567,5663,3615,7711,287,4383,2335,6431,1311,5407,3359,7455,799,4895,2847,6943,1823,5919,3871,7967,159,4255,2207,6303,1183,5279,3231,7327,671,4767,2719,6815,1695,5791,3743,7839,415,4511,2463,6559,1439,5535,3487,7583,927,5023,2975,7071,1951,6047,3999,8095,95,4191,2143,6239,1119,5215,3167,7263,607,4703,2655,6751,1631,5727,3679,7775,351,4447,2399,6495,1375, 5471,3423,7519,863,4959,2911,7007,1887,5983,3935,8031,223,4319,2271,6367,1247,5343,3295,7391,735,4831,2783,6879,1759,5855,3807,7903,479,4575,2527,6623,1503,5599,3551,7647,991,5087,3039,7135,2015,6111,4063,8159,63,4159,2111,6207,1087,5183,3135,7231,575,4671,2623,6719,1599,5695,3647,7743,319,4415,2367,6463,1343,5439,3391,7487,831,4927,2879,6975,1855,5951,3903,7999,191,4287,2239,6335,1215,5311,3263,7359,703,4799,2751,6847,1727,5823,3775,7871,447,4543,2495,6591,1471,5567,3519,7615,959,5055,3007,7103,1983,6079,4031,8127,127,4223,2175,6271,1151,5247,3199,7295,639,4735,2687,6783,1663,5759,3711,7807,383,4479,2431,6527,1407,5503,3455,7551,895,4991,2943,7039,1919,6015,3967,8063,255,4351,2303,6399,1279,5375,3327,7423,767,4863,2815,6911,1791,5887,3839,7935,511,4607,2559,6655,1535,5631,3583,7679,1023,5119,3071,7167,2047,6143,4095,8191]。
当N＝8192，序列1a即为序列1，序列2a即为序列2，序列3a即为序列3；
当N＝4096，
序列1b为：
Shorten_Q＝[0,2048,1024,3072,512,2560,1536,3584,256,2304,1280,3328,768,2816,1792,3840,128,2176,1152,3200,640,2688,1664,3712,384,2432,1408,3456,896,2944,1920,3968,64,2112,1088,3136,576,2624,1600,3648,320,2368,1344,3392,832,2880,1856,3904,192,2240,1216,3264,704,2752,1728,3776,448,2496,1472,3520,960,3008,1984,4032,32,2080,1056,3104,544,2592,1568,3616,288,2336,1312,3360,800,2848,1824,3872,160,2208,1184,3232,672,2720,1696,3744,416,2464,1440,3488,928,2976,1952,4000,96,2144,1120,3168,608,2656,1632,3680,352,2400,1376,3424,864,2912,1888,3936,224,2272,1248,3296,736,2784,1760,3808,480,2528,1504,3552,992,3040,2016,4064,16,2064,1040,3088,528,2576,1552,3600,272,2320,1296,3344,784,2832,1808,3856,144,2192,1168,3216,656,2704,1680,3728,400,2448,1424,3472,912,2960,1936,3984,80,2128,1104,3152,592,2640,1616,3664,336,2384,1360,3408,848,2896,1872,3920,208,2256,1232,3280,720,2768,1744,3792,464,2512,1488,3536,976,3024,2000,4048,48,2096,1072,3120,560,2608,1584,3632,304,2352,1328,3376,816,2864,1840,3888,176,2224,1200,3248,688,2736,1712,3760,432,2480,1456,3504,944,2992,1968,4016,112,2160,1136,3184,624,2672,1648,3696,368,2416,1392,3440,880,2928,1904,3952,240,2288,1264,3312,752,2800,1776,3824,496,2544,1520,3568,1008,3056,2032,4080,8,2056,1032,3080,520,2568,1544,3592,264,2312,1288,3336,776,2824,1800,3848,136,2184,1160,3208,648,2696,1672,3720,392,2440,1416,3464,904,2952,1928,3976,72,2120,1096,3144,584,2632,1608,3656,328,2376,1352,3400,840,2888,1864,3912,200,2248,1224,3272,712,2760,1736,3784,456,2504,1480,3528,968,3016,1992,4040,40,2088,1064,3112,552,2600,1576,3624,296,2344,1320,3368,808,2856,1832,3880,168,2216,1192,3240,680,2728,1704,3752,424,2472,1448,3496,936,2984,1960,4008,104,2152,1128,3176,616,2664,1640,3688,360,2408,1384,3432,872,2920,1896,3944,232,2280,1256,3304,744,2792,1768,3816,488,2536,1512,3560,1000,3048,2024,4072,24,2072,1048,3096,536,2584,1560,3608,280,2328,1304,3352,792,2840,1816,3864,152,2200,1176,3224,664,2712,1688,3736,408,2456,1432,3480,920,2968,1944,3992,88,2136,1112,3160,600,2648,1624,3672,344,2392,1368,3416,856,2904,1880,3928,216,2264,1240,3288,728,2776,1752,3800,472,2520,1496,3544,984,3032,2008,4056,56,2104,1080,3128,568,2616,1592,3640,312,2360,1336,3384,824,2872,1848,3896,184,2232,1208,3256,696,2744,1720,3768,440,2488,1464,3512,952,3000,1976,4024,120,2168,1144,3192,632,2680,1656,3704,376,2424,1400,3448,888,2936,1912,3960,248,2296,1272,3320,760,2808,1784,3832,504,2552,1528,3576,1016,3064,2040,4088,4,2052,1028,3076,516,2564,1540,3588,260,2308,1284,3332,772,2820,1796,3844,132,2180,1156,3204,644,2692,1668,3716,388,2436,1412,3460,900,2948,1924,3972,68,211 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5,2413,1389,3437,877,2925,1901,3949,237,2285,1261,3309,749,2797,1773,3821,493,2541,1517,3565,1005,3053,2029,4077,29,2077,1053,3101,541,2589,1565,3613,285,2333,1309,3357,797,2845,1821,3869,157,2205,1181,3229,669,2717,1693,3741,413,2461,1437,3485,925,2973,1949,3997,93,2141,1117,3165,605,2653,1629,3677,349,2397,1373,3421,861,2909,1885,3933,221,2269,1245,3293,733,2781,1757,3805,477,2525,1501,3549,989,3037,2013,4061,61,2109,1085,3133,573,2621,1597,3645,317,2365,1341,3389,829,2877,1853,3901,189,2237,1213,3261,701,2749,1725,3773,445,2493,1469,3517,957,3005,1981,4029,125,2173,1149,3197,637,2685,1661,3709,381,2429,1405,3453,893,2941,1917,3965,253,2301,1277,3325,765,2813,1789,3837,509,2557,1533,3581,1021,3069,2045,4093,3,2051,1027,3075,515,2563,1539,3587,259,2307,1283,3331,771,2819,1795,3843,131,2179,1155,3203,643,2691,1667,3715,387,2435,1411,3459,899,2947,1923,3971,67,2115,1091,3139,579,2627,1603,3651,323,2371,1347,3395,835,2883,1859,3907,195,2243,1219,3267,707,2755,1731,3779,451,2499,1475,3523,963,3011,1987,4035,35,2083,1059,3107,547,2595,1571,3619,291,2339,1315,3363,803,2851,1827,3875,163,2211,1187,3235,675,2723,1699,3747,419,2467,1443,3491,931,2979,1955,4003,99,2147,1123,3171,611,2659,1635,3683,355,2403,1379,3427,867,2915,1891,3939,227,2275,1251,3299,739,2787,1763,3811,483,2531,1507,3555,995,3043,2019,4067,19,2067,1043,3091,531,2579,1555,3603,275,2323,1299,3347,787,2835,1811,3859,147,2195,1171,3219,659,2707,1683,3731,403,2451,1427,3475,915,2963,1939,3987,83,2131,1107,3155,595,2643,1619,3667,339,2387,1363,3411,851,2899,1875,3923,211,2259,1235,3283,723,2771,1747,3795,467,2515,1491,3539,979,3027,2003,4051,51,2099,1075,3123,563,2611,1587,3635,307,2355,1331,3379,819,2867,1843,3891,179,2227,1203,3251,691,2739,1715,3763,435,2483,1459,3507,947,2995,1971,4019,115,2163,1139,3187,627,2675,1651,3699,371,2419,1395,3443,883,2931,1907,3955,243,2291,1267,3315,755,2803,1779,3827,499,2547,1523,3571,1011,3059,2035,4083,11,2059,1035,3083,523,2571,1547,3595,267,2315,1291,3339,779,2827,1803,3851,139,2187,1163,3211,651,2699,1675,3723,395,2443,1419,3467,907,2955,1931,3979,75,2123,1099,3147,587,2635,1611,3659,331,2379,1355,3403,843,2891,1867,3915,203,2251,1227,3275,715,2763,1739,3787,459,2507,1483,3531,971,3019,1995,4043,43,2091,1067,3115,555,2603,1579,3627,299,2347,1323,3371,811,2859,1835,3883,171,2219,1195,3243,683,2731,1707,3755,427,2475,1451,3499,939,2987,1963,4011,107,2155,1131,3179,619,2667,1643,3691,363,2411,1387,3435,875,2923,1899,3947,235,2283,1259,3307,747,2795,1771,3819,491,2539,1515,3563,1003,3051,2027,4075,27,2075,1051,3099,539,2587,1563,3611,283,2331,1307,3355,795,2843,1819,3867,155,2203,1179,3227,667,2715,1691,3739,411,2459,1435,3483,923,2971,1947,3995,91,2139,1115,3163,603,2651,1627,3675,347,2395,1371,3419,859,2907,1883,3931,219,2267,1243,3291,731,2779,1755,3803,475,2523,1499,3547,987,3035,2011,4059,59,2107,1083,3131,571,2619,1595,3643,315,2363,1339,3387,827,2875,1851,3899,187,2235,1211,3259,699,2747,1723,3771,443,2491,1467,3515,955,3003,1979,4027,123,2171,1147,3195,635,2683,1659,3707,379,2427,1403,3451,891,2939,1915,3963,251,2299,1275,3323,763,2811,1787,3835,507,2555,1531,3579,1019,3067,2043,4091,7,2055,1031,3079,519,2567,1543,3591,263,2311,1287,3335,775,2823,1799,3847,135,2183,1159,3207,647,2695,1671,3719,391,2439,1415,3463,903,2951,1927,3975,71,2119,1095,3143,583,2631,1607,3655,327,2375,1351,3399,839,2887,1863,3911,199,2247,1223,3271,711,2759,1735,3783,455,2503,1479,3527,967,3015,1991,4039,39,2087,1063,3111,551,2599,1575,3623,295,2343,1319,3367,807,2855,1831,3879,167,2215,1191,3239,679,2727,1703,3751,423,2471,1447,3495,935,2983,1959,4007,103,2151,1127,3175,615,2663,1639,3687,359,2407,1383,3431,871,2919,1895,3943,231,2279,1255,3303,743,2791,1767,3815,487,2535,1511,3559,999,3047,2023,4071, 23,2071,1047,3095,535,2583,1559,3607,279,2327,1303,3351,791,2839,1815,3863,151,2199,1175,3223,663,2711,1687,3735,407,2455,1431,3479,919,2967,1943,3991,87,2135,1111,3159,599,2647,1623,3671,343,2391,1367,3415,855,2903,1879,3927,215,2263,1239,3287,727,2775,1751,3799,471,2519,1495,3543,983,3031,2007,4055,55,2103,1079,3127,567,2615,1591,3639,311,2359,1335,3383,823,2871,1847,3895,183,2231,1207,3255,695,2743,1719,3767,439,2487,1463,3511,951,2999,1975,4023,119,2167,1143,3191,631,2679,1655,3703,375,2423,1399,3447,887,2935,1911,3959,247,2295,1271,3319,759,2807,1783,3831,503,2551,1527,3575,1015,3063,2039,4087,15,2063,1039,3087,527,2575,1551,3599,271,2319,1295,3343,783,2831,1807,3855,143,2191,1167,3215,655,2703,1679,3727,399,2447,1423,3471,911,2959,1935,3983,79,2127,1103,3151,591,2639,1615,3663,335,2383,1359,3407,847,2895,1871,3919,207,2255,1231,3279,719,2767,1743,3791,463,2511,1487,3535,975,3023,1999,4047,47,2095,1071,3119,559,2607,1583,3631,303,2351,1327,3375,815,2863,1839,3887,175,2223,1199,3247,687,2735,1711,3759,431,2479,1455,3503,943,2991,1967,4015,111,2159,1135,3183,623,2671,1647,3695,367,2415,1391,3439,879,2927,1903,3951,239,2287,1263,3311,751,2799,1775,3823,495,2543,1519,3567,1007,3055,2031,4079,31,2079,1055,3103,543,2591,1567,3615,287,2335,1311,3359,799,2847,1823,3871,159,2207,1183,3231,671,2719,1695,3743,415,2463,1439,3487,927,2975,1951,3999,95,2143,1119,3167,607,2655,1631,3679,351,2399,1375,3423,863,2911,1887,3935,223,2271,1247,3295,735,2783,1759,3807,479,2527,1503,3551,991,3039,2015,4063,63,2111,1087,3135,575,2623,1599,3647,319,2367,1343,3391,831,2879,1855,3903,191,2239,1215,3263,703,2751,1727,3775,447,2495,1471,3519,959,3007,1983,4031,127,2175,1151,3199,639,2687,1663,3711,383,2431,1407,3455,895,2943,1919,3967,255,2303,1279,3327,767,2815,1791,3839,511,2559,1535,3583,1023,3071,2047,4095]。
序列2b为：
Shorten_Q_half＝[1,2049,1025,3073,513,2561,1537,3585,257,2305,1281,3329,769,2817,1793,3841,129,2177,1153,3201,641,2689,1665,3713,385,2433,1409,3457,897,2945,1921,3969,65,2113,1089,3137,577,2625,1601,3649,321,2369,1345,3393,833,2881,1857,3905,193,2241,1217,3265,705,2753,1729,3777,449,2497,1473,3521,961,3009,1985,4033,33,2081,1057,3105,545,2593,1569,3617,289,2337,1313,3361,801,2849,1825,3873,161,2209,1185,3233,673,2721,1697,3745,417,2465,1441,3489,929,2977,1953,4001,97,2145,1121,3169,609,2657,1633,3681,353,2401,1377,3425,865,2913,1889,3937,225,2273,1249,3297,737,2785,1761,3809,481,2529,1505,3553,993,3041,2017,4065,17,2065,1041,3089,529,2577,1553,3601,273,2321,1297,3345,785,2833,1809,3857,145,2193,1169,3217,657,2705,1681,3729,401,2449,1425,3473,913,2961,1937,3985,81,2129,1105,3153,593,2641,1617,3665,337,2385,1361,3409,849,2897,1873,3921,209,2257,1233,3281,721,2769,1745,3793,465,2513,1489,3537,977,3025,2001,4049,49,2097,1073,3121,561,2609,1585,3633,305,2353,1329,3377,817,2865,1841,3889,177,2225,1201,3249,689,2737,1713,3761,433,2481,1457,3505,945,2993,1969,4017,113,2161,1137,3185,625,2673,1649,3697,369,2417,1393,3441,881,2929,1905,3953,241,2289,1265,3313,753,2801,1777,3825,497,2545,1521,3569,1009,3057,2033,4081,9,2057,1033,3081,521,2569,1545,3593,265,2313,1289,3337,777,2825,1801,3849,137,2185,1161,3209,649,2697,1673,3721,393,2441,1417,3465,905,2953,1929,3977,73,2121,1097,3145,585,2633,1609,3657,329,2377,1353,3401,841,2889,1865,3913,201,2249,1225,3273,713,2761,1737,3785,457,2505,1481,3529,969,3017,1993,4041,41,2089,1065,3113,553,2601,1577,3625,297,2345,1321,3369,809,2857,1833,3881,169,2217,1193,3241,681,2729,1705,3753,425,2473,1449,3497,937,2985,1961,4009,105,2153,1129,3177,617,2665,1641,3689,361,2409,1385,3433,873,2921,1897,3945,233,2281,1257,3305,745,2793,1769,3817,489,2537,1513,3561,1001,3049 ,2025,4073,25,2073,1049,3097,537,2585,1561,3609,281,2329,1305,3353,793,2841,1817,3865,153,2201,1177,3225,665,2713,1689,3737,409,2457,1433,3481,921,2969,1945,3993,89,2137,1113,3161,601,2649,1625,3673,345,2393,1369,3417,857,2905,1881,3929,217,2265,1241,3289,729,2777,1753,3801,473,2521,1497,3545,985,3033,2009,4057,57,2105,1081,3129,569,2617,1593,3641,313,2361,1337,3385,825,2873,1849,3897,185,2233,1209,3257,697,2745,1721,3769,441,2489,1465,3513,953,3001,1977,4025,121,2169,1145,3193,633,2681,1657,3705,377,2425,1401,3449,889,2937,1913,3961,249,2297,1273,3321,761,2809,1785,3833,505,2553,1529,3577,1017,3065,2041,4089,5,2053,1029,3077,517,2565,1541,3589,261,2309,1285,3333,773,2821,1797,3845,133,2181,1157,3205,645,2693,1669,3717,389,2437,1413,3461,901,2949,1925,3973,69,2117,1093,3141,581,2629,1605,3653,325,2373,1349,3397,837,2885,1861,3909,197,2245,1221,3269,709,2757,1733,3781,453,2501,1477,3525,965,3013,1989,4037,37,2085,1061,3109,549,2597,1573,3621,293,2341,1317,3365,805,2853,1829,3877,165,2213,1189,3237,677,2725,1701,3749,421,2469,1445,3493,933,2981,1957,4005,101,2149,1125,3173,613,2661,1637,3685,357,2405,1381,3429,869,2917,1893,3941,229,2277,1253,3301,741,2789,1765,3813,485,2533,1509,3557,997,3045,2021,4069,21,2069,1045,3093,533,2581,1557,3605,277,2325,1301,3349,789,2837,1813,3861,149,2197,1173,3221,661,2709,1685,3733,405,2453,1429,3477,917,2965,1941,3989,85,2133,1109,3157,597,2645,1621,3669,341,2389,1365,3413,853,2901,1877,3925,213,2261,1237,3285,725,2773,1749,3797,469,2517,1493,3541,981,3029,2005,4053,53,2101,1077,3125,565,2613,1589,3637,309,2357,1333,3381,821,2869,1845,3893,181,2229,1205,3253,693,2741,1717,3765,437,2485,1461,3509,949,2997,1973,4021,117,2165,1141,3189,629,2677,1653,3701,373,2421,1397,3445,885,2933,1909,3957,245,2293,1269,3317,757,2805,1781,3829,501,2549,1525,3573,1013,3061,2037,4085,13,2061,1037,3085,525,2573,1549,3597,269,2317,1293,3341,781,2829,1805,3853,141,2189,1165,3213,653,2701,1677,3725,397,2445,1421,3469,909,2957,1933,3981,77,2125,1101,3149,589,2637,1613,3661,333,2381,1357,3405,845,2893,1869,3917,205,2253,1229,3277,717,2765,1741,3789,461,2509,1485,3533,973,3021,1997,4045,45,2093,1069,3117,557,2605,1581,3629,301,2349,1325,3373,813,2861,1837,3885,173,2221,1197,3245,685,2733,1709,3757,429,2477,1453,3501,941,2989,1965,4013,109,2157,1133,3181,621,2669,1645,3693,365,2413,1389,3437,877,2925,1901,3949,237,2285,1261,3309,749,2797,1773,3821,493,2541,1517,3565,1005,3053,2029,4077,29,2077,1053,3101,541,2589,1565,3613,285,2333,1309,3357,797,2845,1821,3869,157,2205,1181,3229,669,2717,1693,3741,413,2461,1437,3485,925,2973,1949,3997,93,2141,1117,3165,605,2653,1629,3677,349,2397,1373,3421,861,2909,1885,3933,221,2269,1245,3293,733,2781,1757,3805,477,2525,1501,3549,989,3037,2013,4061,61,2109,1085,3133,573,2621,1597,3645,317,2365,1341,3389,829,2877,1853,3901,189,2237,1213,3261,701,2749,1725,3773,445,2493,1469,3517,957,3005,1981,4029,125,2173,1149,3197,637,2685,1661,3709,381,2429,1405,3453,893,2941,1917,3965,253,2301,1277,3325,765,2813,1789,3837,509,2557,1533,3581,1021,3069,2045,4093,3,2051,1027,3075,515,2563,1539,3587,259,2307,1283,3331,771,2819,1795,3843,131,2179,1155,3203,643,2691,1667,3715,387,2435,1411,3459,899,2947,1923,3971,67,2115,1091,3139,579,2627,1603,3651,323,2371,1347,3395,835,2883,1859,3907,195,2243,1219,3267,707,2755,1731,3779,451,2499,1475,3523,963,3011,1987,4035,35,2083,1059,3107,547,2595,1571,3619,291,2339,1315,3363,803,2851,1827,3875,163,2211,1187,3235,675,2723,1699,3747,419,2467,1443,3491,931,2979,1955,4003,99,2147,1123,3171,611,2659,1635,3683,355,2403,1379,3427,867,2915,1891,3939,227,2275,1251,3299,739,2787,1763,3811,483,2531,1507,3555,995,3043,2019,4067,19,2067,1043,3091,531,2579,1555,3603,275,2323,1299,3347,787,2835,1811,3859,147,2195,1171,3219,659,2707 ,1683,3731,403,2451,1427,3475,915,2963,1939,3987,83,2131,1107,3155,595,2643,1619,3667,339,2387,1363,3411,851,2899,1875,3923,211,2259,1235,3283,723,2771,1747,3795,467,2515,1491,3539,979,3027,2003,4051,51,2099,1075,3123,563,2611,1587,3635,307,2355,1331,3379,819,2867,1843,3891,179,2227,1203,3251,691,2739,1715,3763,435,2483,1459,3507,947,2995,1971,4019,115,2163,1139,3187,627,2675,1651,3699,371,2419,1395,3443,883,2931,1907,3955,243,2291,1267,3315,755,2803,1779,3827,499,2547,1523,3571,1011,3059,2035,4083,11,2059,1035,3083,523,2571,1547,3595,267,2315,1291,3339,779,2827,1803,3851,139,2187,1163,3211,651,2699,1675,3723,395,2443,1419,3467,907,2955,1931,3979,75,2123,1099,3147,587,2635,1611,3659,331,2379,1355,3403,843,2891,1867,3915,203,2251,1227,3275,715,2763,1739,3787,459,2507,1483,3531,971,3019,1995,4043,43,2091,1067,3115,555,2603,1579,3627,299,2347,1323,3371,811,2859,1835,3883,171,2219,1195,3243,683,2731,1707,3755,427,2475,1451,3499,939,2987,1963,4011,107,2155,1131,3179,619,2667,1643,3691,363,2411,1387,3435,875,2923,1899,3947,235,2283,1259,3307,747,2795,1771,3819,491,2539,1515,3563,1003,3051,2027,4075,27,2075,1051,3099,539,2587,1563,3611,283,2331,1307,3355,795,2843,1819,3867,155,2203,1179,3227,667,2715,1691,3739,411,2459,1435,3483,923,2971,1947,3995,91,2139,1115,3163,603,2651,1627,3675,347,2395,1371,3419,859,2907,1883,3931,219,2267,1243,3291,731,2779,1755,3803,475,2523,1499,3547,987,3035,2011,4059,59,2107,1083,3131,571,2619,1595,3643,315,2363,1339,3387,827,2875,1851,3899,187,2235,1211,3259,699,2747,1723,3771,443,2491,1467,3515,955,3003,1979,4027,123,2171,1147,3195,635,2683,1659,3707,379,2427,1403,3451,891,2939,1915,3963,251,2299,1275,3323,763,2811,1787,3835,507,2555,1531,3579,1019,3067,2043,4091,7,2055,1031,3079,519,2567,1543,3591,263,2311,1287,3335,775,2823,1799,3847,135,2183,1159,3207,647,2695,1671,3719,391,2439,1415,3463,903,2951,1927,3975,71,2119,1095,3143,583,2631,1607,3655,327,2375,1351,3399,839,2887,1863,3911,199,2247,1223,3271,711,2759,1735,3783,455,2503,1479,3527,967,3015,1991,4039,39,2087,1063,3111,551,2599,1575,3623,295,2343,1319,3367,807,2855,1831,3879,167,2215,1191,3239,679,2727,1703,3751,423,2471,1447,3495,935,2983,1959,4007,103,2151,1127,3175,615,2663,1639,3687,359,2407,1383,3431,871,2919,1895,3943,231,2279,1255,3303,743,2791,1767,3815,487,2535,1511,3559,999,3047,2023,4071,23,2071,1047,3095,535,2583,1559,3607,279,2327,1303,3351,791,2839,1815,3863,151,2199,1175,3223,663,2711,1687,3735,407,2455,1431,3479,919,2967,1943,3991,87,2135,1111,3159,599,2647,1623,3671,343,2391,1367,3415,855,2903,1879,3927,215,2263,1239,3287,727,2775,1751,3799,471,2519,1495,3543,983,3031,2007,4055,55,2103,1079,3127,567,2615,1591,3639,311,2359,1335,3383,823,2871,1847,3895,183,2231,1207,3255,695,2743,1719,3767,439,2487,1463,3511,951,2999,1975,4023,119,2167,1143,3191,631,2679,1655,3703,375,2423,1399,3447,887,2935,1911,3959,247,2295,1271,3319,759,2807,1783,3831,503,2551,1527,3575,1015,3063,2039,4087,15,2063,1039,3087,527,2575,1551,3599,271,2319,1295,3343,783,2831,1807,3855,143,2191,1167,3215,655,2703,1679,3727,399,2447,1423,3471,911,2959,1935,3983,79,2127,1103,3151,591,2639,1615,3663,335,2383,1359,3407,847,2895,1871,3919,207,2255,1231,3279,719,2767,1743,3791,463,2511,1487,3535,975,3023,1999,4047,47,2095,1071,3119,559,2607,1583,3631,303,2351,1327,3375,815,2863,1839,3887,175,2223,1199,3247,687,2735,1711,3759,431,2479,1455,3503,943,2991,1967,4015,111,2159,1135,3183,623,2671,1647,3695,367,2415,1391,3439,879,2927,1903,3951,239,2287,1263,3311,751,2799,1775,3823,495,2543,1519,3567,1007,3055,2031,4079,31,2079,1055,3103,543,2591,1567,3615,287,2335,1311,3359,799,2847,1823,3871,159,2207,1183,3231,671,2719,1695,3743,415,2463,1439,3487,927,2975,1951,3999,95,2143,1119,3167,607,2655,1631,3679,351,2399,1375,3423,863,2 911,1887,3935,223,2271,1247,3295,735,2783,1759,3807,479,2527,1503,3551,991,3039,2015,4063,63,2111,1087,3135,575,2623,1599,3647,319,2367,1343,3391,831,2879,1855,3903,191,2239,1215,3263,703,2751,1727,3775,447,2495,1471,3519,959,3007,1983,4031,127,2175,1151,3199,639,2687,1663,3711,383,2431,1407,3455,895,2943,1919,3967,255,2303,1279,3327,767,2815,1791,3839,511,2559,1535,3583,1023,3071,2047,4095]。
序列3b为：
Shorten_Z＝[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,4832,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176,8,4104,2056,6152,1032,5128,3080,7176,520,4616,2568,6664,1544,5640,3592,7688,264,4360,2312,6408,1288,5384,3336,7432,776,4872,2824,6920,1800,5896,3848,7944,136,4232,2184,6280,1160,5256,3208,7304,648,4744,2696,6792,1672,5768,3720,7816,392,4488,2440,6536,1416,5512,3464,7560,904,5000,2952,7048,1928,6024,3976,8072,72,4168,2120,6216,1096,5192,3144,7240,584,4680,2632,6728,1608,5704,3656,7752,328,4424,2376,6472,1352,5448,3400,7496,840,4936,2888,6984,1864,5960,3912,8008,200,4296,2248,6344,1224,5320,3272,7368,712,4808,2760,6856,1736,5832,3784,7880,456,4552,2504,6600,1480,5576,3528,7624,968,5064,3016,7112,1992,6088,4040,8136,40,4136,2088,6184,1064,5160,311 2,7208,552,4648,2600,6696,1576,5672,3624,7720,296,4392,2344,6440,1320,5416,3368,7464,808,4904,2856,6952,1832,5928,3880,7976,168,4264,2216,6312,1192,5288,3240,7336,680,4776,2728,6824,1704,5800,3752,7848,424,4520,2472,6568,1448,5544,3496,7592,936,5032,2984,7080,1960,6056,4008,8104,104,4200,2152,6248,1128,5224,3176,7272,616,4712,2664,6760,1640,5736,3688,7784,360,4456,2408,6504,1384,5480,3432,7528,872,4968,2920,7016,1896,5992,3944,8040,232,4328,2280,6376,1256,5352,3304,7400,744,4840,2792,6888,1768,5864,3816,7912,488,4584,2536,6632,1512,5608,3560,7656,1000,5096,3048,7144,2024,6120,4072,8168,24,4120,2072,6168,1048,5144,3096,7192,536,4632,2584,6680,1560,5656,3608,7704,280,4376,2328,6424,1304,5400,3352,7448,792,4888,2840,6936,1816,5912,3864,7960,152,4248,2200,6296,1176,5272,3224,7320,664,4760,2712,6808,1688,5784,3736,7832,408,4504,2456,6552,1432,5528,3480,7576,920,5016,2968,7064,1944,6040,3992,8088,88,4184,2136,6232,1112,5208,3160,7256,600,4696,2648,6744,1624,5720,3672,7768,344,4440,2392,6488,1368,5464,3416,7512,856,4952,2904,7000,1880,5976,3928,8024,216,4312,2264,6360,1240,5336,3288,7384,728,4824,2776,6872,1752,5848,3800,7896,472,4568,2520,6616,1496,5592,3544,7640,984,5080,3032,7128,2008,6104,4056,8152,56,4152,2104,6200,1080,5176,3128,7224,568,4664,2616,6712,1592,5688,3640,7736,312,4408,2360,6456,1336,5432,3384,7480,824,4920,2872,6968,1848,5944,3896,7992,184,4280,2232,6328,1208,5304,3256,7352,696,4792,2744,6840,1720,5816,3768,7864,440,4536,2488,6584,1464,5560,3512,7608,952,5048,3000,7096,1976,6072,4024,8120,120,4216,2168,6264,1144,5240,3192,7288,632,4728,2680,6776,1656,5752,3704,7800,376,4472,2424,6520,1400,5496,3448,7544,888,4984,2936,7032,1912,6008,3960,8056,248,4344,2296,6392,1272,5368,3320,7416,760,4856,2808,6904,1784,5880,3832,7928,504,4600,2552,6648,1528,5624,3576,7672,1016,5112,3064,7160,2040,6136,4088,8184,4,4100,2052,6148,1028,5124,3076,7172,516,4612,2564,6660,1540,5636,3588,7684,260,4356,2308,6404,1284,5380,3332,7428,772,4868,2820,6916,1796,5892,3844,7940,132,4228,2180,6276,1156,5252,3204,7300,644,4740,2692,6788,1668,5764,3716,7812,388,4484,2436,6532,1412,5508,3460,7556,900,4996,2948,7044,1924,6020,3972,8068,68,4164,2116,6212,1092,5188,3140,7236,580,4676,2628,6724,1604,5700,3652,7748,324,4420,2372,6468,1348,5444,3396,7492,836,4932,2884,6980,1860,5956,3908,8004,196,4292,2244,6340,1220,5316,3268,7364,708,4804,2756,6852,1732,5828,3780,7876,452,4548,2500,6596,1476,5572,3524,7620,964,5060,3012,7108,1988,6084,4036,8132,36,4132,2084,6180,1060,5156,3108,7204,548,4644,2596,6692,1572,5668,3620,7716,292,4388,2340,6436,1316,5412,3364,7460,804,4900,2852,6948,1828,5924,3876,7972,164,4260,2212,6308,1188,5284,3236,7332,676,4772,2724,6820,1700,5796,3748,7844,420,4516,2468,6564,1444,5540,3492,7588,932,5028,2980,7076,1956,6052,4004,8100,100,4196,2148,6244,1124,5220,3172,7268,612,4708,2660,6756,1636,5732,3684,7780,356,4452,2404,6500,1380,5476,3428,7524,868,4964,2916,7012,1892,5988,3940,8036,228,4324,2276,6372,1252,5348,3300,7396,740,4836,2788,6884,1764,5860,3812,7908,484,4580,2532,6628,1508,5604,3556,7652,996,5092,3044,7140,2020,6116,4068,8164,20,4116,2068,6164,1044,5140,3092,7188,532,4628,2580,6676,1556,5652,3604,7700,276,4372,2324,6420,1300,5396,3348,7444,788,4884,2836,6932,1812,5908,3860,7956,148,4244,2196,6292,1172,5268,3220,7316,660,4756,2708,6804,1684,5780,3732,7828,404,4500,2452,6548,1428,5524,3476,7572,916,5012,2964,7060,1940,6036,3988,8084,84,4180,2132,6228,1108,5204,3156,7252,596,4692,2644,6740,1620,5716,3668,7764,340,4436,2388,6484,1364,5460,3412,7508,852,4948,2900,6996,1876,5972,3924,8020,212,4308,2260,6356,1236,5332,3284,7380,724,4820,2772,6868,1748,5844,3796,7892,468,4564,2516,6612,1492,5588,3540,7636,980,5076,3028,7124,2004,6100,4052,8148,52,4148,2100,6196,1076,517 2,3124,7220,564,4660,2612,6708,1588,5684,3636,7732,308,4404,2356,6452,1332,5428,3380,7476,820,4916,2868,6964,1844,5940,3892,7988,180,4276,2228,6324,1204,5300,3252,7348,692,4788,2740,6836,1716,5812,3764,7860,436,4532,2484,6580,1460,5556,3508,7604,948,5044,2996,7092,1972,6068,4020,8116,116,4212,2164,6260,1140,5236,3188,7284,628,4724,2676,6772,1652,5748,3700,7796,372,4468,2420,6516,1396,5492,3444,7540,884,4980,2932,7028,1908,6004,3956,8052,244,4340,2292,6388,1268,5364,3316,7412,756,4852,2804,6900,1780,5876,3828,7924,500,4596,2548,6644,1524,5620,3572,7668,1012,5108,3060,7156,2036,6132,4084,8180,12,4108,2060,6156,1036,5132,3084,7180,524,4620,2572,6668,1548,5644,3596,7692,268,4364,2316,6412,1292,5388,3340,7436,780,4876,2828,6924,1804,5900,3852,7948,140,4236,2188,6284,1164,5260,3212,7308,652,4748,2700,6796,1676,5772,3724,7820,396,4492,2444,6540,1420,5516,3468,7564,908,5004,2956,7052,1932,6028,3980,8076,76,4172,2124,6220,1100,5196,3148,7244,588,4684,2636,6732,1612,5708,3660,7756,332,4428,2380,6476,1356,5452,3404,7500,844,4940,2892,6988,1868,5964,3916,8012,204,4300,2252,6348,1228,5324,3276,7372,716,4812,2764,6860,1740,5836,3788,7884,460,4556,2508,6604,1484,5580,3532,7628,972,5068,3020,7116,1996,6092,4044,8140,44,4140,2092,6188,1068,5164,3116,7212,556,4652,2604,6700,1580,5676,3628,7724,300,4396,2348,6444,1324,5420,3372,7468,812,4908,2860,6956,1836,5932,3884,7980,172,4268,2220,6316,1196,5292,3244,7340,684,4780,2732,6828,1708,5804,3756,7852,428,4524,2476,6572,1452,5548,3500,7596,940,5036,2988,7084,1964,6060,4012,8108,108,4204,2156,6252,1132,5228,3180,7276,620,4716,2668,6764,1644,5740,3692,7788,364,4460,2412,6508,1388,5484,3436,7532,876,4972,2924,7020,1900,5996,3948,8044,236,4332,2284,6380,1260,5356,3308,7404,748,4844,2796,6892,1772,5868,3820,7916,492,4588,2540,6636,1516,5612,3564,7660,1004,5100,3052,7148,2028,6124,4076,8172,28,4124,2076,6172,1052,5148,3100,7196,540,4636,2588,6684,1564,5660,3612,7708,284,4380,2332,6428,1308,5404,3356,7452,796,4892,2844,6940,1820,5916,3868,7964,156,4252,2204,6300,1180,5276,3228,7324,668,4764,2716,6812,1692,5788,3740,7836,412,4508,2460,6556,1436,5532,3484,7580,924,5020,2972,7068,1948,6044,3996,8092,92,4188,2140,6236,1116,5212,3164,7260,604,4700,2652,6748,1628,5724,3676,7772,348,4444,2396,6492,1372,5468,3420,7516,860,4956,2908,7004,1884,5980,3932,8028,220,4316,2268,6364,1244,5340,3292,7388,732,4828,2780,6876,1756,5852,3804,7900,476,4572,2524,6620,1500,5596,3548,7644,988,5084,3036,7132,2012,6108,4060,8156,60,4156,2108,6204,1084,5180,3132,7228,572,4668,2620,6716,1596,5692,3644,7740,316,4412,2364,6460,1340,5436,3388,7484,828,4924,2876,6972,1852,5948,3900,7996,188,4284,2236,6332,1212,5308,3260,7356,700,4796,2748,6844,1724,5820,3772,7868,444,4540,2492,6588,1468,5564,3516,7612,956,5052,3004,7100,1980,6076,4028,8124,124,4220,2172,6268,1148,5244,3196,7292,636,4732,2684,6780,1660,5756,3708,7804,380,4476,2428,6524,1404,5500,3452,7548,892,4988,2940,7036,1916,6012,3964,8060,252,4348,2300,6396,1276,5372,3324,7420,764,4860,2812,6908,1788,5884,3836,7932,508,4604,2556,6652,1532,5628,3580,7676,1020,5116,3068,7164,2044,6140,4092,8188,2,4098,2050,6146,1026,5122,3074,7170,514,4610,2562,6658,1538,5634,3586,7682,258,4354,2306,6402,1282,5378,3330,7426,770,4866,2818,6914,1794,5890,3842,7938,130,4226,2178,6274,1154,5250,3202,7298,642,4738,2690,6786,1666,5762,3714,7810,386,4482,2434,6530,1410,5506,3458,7554,898,4994,2946,7042,1922,6018,3970,8066,66,4162,2114,6210,1090,5186,3138,7234,578,4674,2626,6722,1602,5698,3650,7746,322,4418,2370,6466,1346,5442,3394,7490,834,4930,2882,6978,1858,5954,3906,8002,194,4290,2242,6338,1218,5314,3266,7362,706,4802,2754,6850,1730,5826,3778,7874,450,4546,2498,6594,1474,5570,3522,7618,962,5058,3010,7106,1986,6082,4034,8130,34,4130,2082,6178,10 58,5154,3106,7202,546,4642,2594,6690,1570,5666,3618,7714,290,4386,2338,6434,1314,5410,3362,7458,802,4898,2850,6946,1826,5922,3874,7970,162,4258,2210,6306,1186,5282,3234,7330,674,4770,2722,6818,1698,5794,3746,7842,418,4514,2466,6562,1442,5538,3490,7586,930,5026,2978,7074,1954,6050,4002,8098,98,4194,2146,6242,1122,5218,3170,7266,610,4706,2658,6754,1634,5730,3682,7778,354,4450,2402,6498,1378,5474,3426,7522,866,4962,2914,7010,1890,5986,3938,8034,226,4322,2274,6370,1250,5346,3298,7394,738,4834,2786,6882,1762,5858,3810,7906,482,4578,2530,6626,1506,5602,3554,7650,994,5090,3042,7138,2018,6114,4066,8162,18,4114,2066,6162,1042,5138,3090,7186,530,4626,2578,6674,1554,5650,3602,7698,274,4370,2322,6418,1298,5394,3346,7442,786,4882,2834,6930,1810,5906,3858,7954,146,4242,2194,6290,1170,5266,3218,7314,658,4754,2706,6802,1682,5778,3730,7826,402,4498,2450,6546,1426,5522,3474,7570,914,5010,2962,7058,1938,6034,3986,8082,82,4178,2130,6226,1106,5202,3154,7250,594,4690,2642,6738,1618,5714,3666,7762,338,4434,2386,6482,1362,5458,3410,7506,850,4946,2898,6994,1874,5970,3922,8018,210,4306,2258,6354,1234,5330,3282,7378,722,4818,2770,6866,1746,5842,3794,7890,466,4562,2514,6610,1490,5586,3538,7634,978,5074,3026,7122,2002,6098,4050,8146,50,4146,2098,6194,1074,5170,3122,7218,562,4658,2610,6706,1586,5682,3634,7730,306,4402,2354,6450,1330,5426,3378,7474,818,4914,2866,6962,1842,5938,3890,7986,178,4274,2226,6322,1202,5298,3250,7346,690,4786,2738,6834,1714,5810,3762,7858,434,4530,2482,6578,1458,5554,3506,7602,946,5042,2994,7090,1970,6066,4018,8114,114,4210,2162,6258,1138,5234,3186,7282,626,4722,2674,6770,1650,5746,3698,7794,370,4466,2418,6514,1394,5490,3442,7538,882,4978,2930,7026,1906,6002,3954,8050,242,4338,2290,6386,1266,5362,3314,7410,754,4850,2802,6898,1778,5874,3826,7922,498,4594,2546,6642,1522,5618,3570,7666,1010,5106,3058,7154,2034,6130,4082,8178,10,4106,2058,6154,1034,5130,3082,7178,522,4618,2570,6666,1546,5642,3594,7690,266,4362,2314,6410,1290,5386,3338,7434,778,4874,2826,6922,1802,5898,3850,7946,138,4234,2186,6282,1162,5258,3210,7306,650,4746,2698,6794,1674,5770,3722,7818,394,4490,2442,6538,1418,5514,3466,7562,906,5002,2954,7050,1930,6026,3978,8074,74,4170,2122,6218,1098,5194,3146,7242,586,4682,2634,6730,1610,5706,3658,7754,330,4426,2378,6474,1354,5450,3402,7498,842,4938,2890,6986,1866,5962,3914,8010,202,4298,2250,6346,1226,5322,3274,7370,714,4810,2762,6858,1738,5834,3786,7882,458,4554,2506,6602,1482,5578,3530,7626,970,5066,3018,7114,1994,6090,4042,8138,42,4138,2090,6186,1066,5162,3114,7210,554,4650,2602,6698,1578,5674,3626,7722,298,4394,2346,6442,1322,5418,3370,7466,810,4906,2858,6954,1834,5930,3882,7978,170,4266,2218,6314,1194,5290,3242,7338,682,4778,2730,6826,1706,5802,3754,7850,426,4522,2474,6570,1450,5546,3498,7594,938,5034,2986,7082,1962,6058,4010,8106,106,4202,2154,6250,1130,5226,3178,7274,618,4714,2666,6762,1642,5738,3690,7786,362,4458,2410,6506,1386,5482,3434,7530,874,4970,2922,7018,1898,5994,3946,8042,234,4330,2282,6378,1258,5354,3306,7402,746,4842,2794,6890,1770,5866,3818,7914,490,4586,2538,6634,1514,5610,3562,7658,1002,5098,3050,7146,2026,6122,4074,8170,26,4122,2074,6170,1050,5146,3098,7194,538,4634,2586,6682,1562,5658,3610,7706,282,4378,2330,6426,1306,5402,3354,7450,794,4890,2842,6938,1818,5914,3866,7962,154,4250,2202,6298,1178,5274,3226,7322,666,4762,2714,6810,1690,5786,3738,7834,410,4506,2458,6554,1434,5530,3482,7578,922,5018,2970,7066,1946,6042,3994,8090,90,4186,2138,6234,1114,5210,3162,7258,602,4698,2650,6746,1626,5722,3674,7770,346,4442,2394,6490,1370,5466,3418,7514,858,4954,2906,7002,1882,5978,3930,8026,218,4314,2266,6362,1242,5338,3290,7386,730,4826,2778,6874,1754,5850,3802,7898,474,4570,2522,6618,1498,5594,3546,7642,986,5082,3034,7130,2010,6106,4058,8154,58,4154,2106,62 02,1082,5178,3130,7226,570,4666,2618,6714,1594,5690,3642,7738,314,4410,2362,6458,1338,5434,3386,7482,826,4922,2874,6970,1850,5946,3898,7994,186,4282,2234,6330,1210,5306,3258,7354,698,4794,2746,6842,1722,5818,3770,7866,442,4538,2490,6586,1466,5562,3514,7610,954,5050,3002,7098,1978,6074,4026,8122,122,4218,2170,6266,1146,5242,3194,7290,634,4730,2682,6778,1658,5754,3706,7802,378,4474,2426,6522,1402,5498,3450,7546,890,4986,2938,7034,1914,6010,3962,8058,250,4346,2298,6394,1274,5370,3322,7418,762,4858,2810,6906,1786,5882,3834,7930,506,4602,2554,6650,1530,5626,3578,7674,1018,5114,3066,7162,2042,6138,4090,8186,6,4102,2054,6150,1030,5126,3078,7174,518,4614,2566,6662,1542,5638,3590,7686,262,4358,2310,6406,1286,5382,3334,7430,774,4870,2822,6918,1798,5894,3846,7942,134,4230,2182,6278,1158,5254,3206,7302,646,4742,2694,6790,1670,5766,3718,7814,390,4486,2438,6534,1414,5510,3462,7558,902,4998,2950,7046,1926,6022,3974,8070,70,4166,2118,6214,1094,5190,3142,7238,582,4678,2630,6726,1606,5702,3654,7750,326,4422,2374,6470,1350,5446,3398,7494,838,4934,2886,6982,1862,5958,3910,8006,198,4294,2246,6342,1222,5318,3270,7366,710,4806,2758,6854,1734,5830,3782,7878,454,4550,2502,6598,1478,5574,3526,7622,966,5062,3014,7110,1990,6086,4038,8134,38,4134,2086,6182,1062,5158,3110,7206,550,4646,2598,6694,1574,5670,3622,7718,294,4390,2342,6438,1318,5414,3366,7462,806,4902,2854,6950,1830,5926,3878,7974,166,4262,2214,6310,1190,5286,3238,7334,678,4774,2726,6822,1702,5798,3750,7846,422,4518,2470,6566,1446,5542,3494,7590,934,5030,2982,7078,1958,6054,4006,8102,102,4198,2150,6246,1126,5222,3174,7270,614,4710,2662,6758,1638,5734,3686,7782,358,4454,2406,6502,1382,5478,3430,7526,870,4966,2918,7014,1894,5990,3942,8038,230,4326,2278,6374,1254,5350,3302,7398,742,4838,2790,6886,1766,5862,3814,7910,486,4582,2534,6630,1510,5606,3558,7654,998,5094,3046,7142,2022,6118,4070,8166,22,4118,2070,6166,1046,5142,3094,7190,534,4630,2582,6678,1558,5654,3606,7702,278,4374,2326,6422,1302,5398,3350,7446,790,4886,2838,6934,1814,5910,3862,7958,150,4246,2198,6294,1174,5270,3222,7318,662,4758,2710,6806,1686,5782,3734,7830,406,4502,2454,6550,1430,5526,3478,7574,918,5014,2966,7062,1942,6038,3990,8086,86,4182,2134,6230,1110,5206,3158,7254,598,4694,2646,6742,1622,5718,3670,7766,342,4438,2390,6486,1366,5462,3414,7510,854,4950,2902,6998,1878,5974,3926,8022,214,4310,2262,6358,1238,5334,3286,7382,726,4822,2774,6870,1750,5846,3798,7894,470,4566,2518,6614,1494,5590,3542,7638,982,5078,3030,7126,2006,6102,4054,8150,54,4150,2102,6198,1078,5174,3126,7222,566,4662,2614,6710,1590,5686,3638,7734,310,4406,2358,6454,1334,5430,3382,7478,822,4918,2870,6966,1846,5942,3894,7990,182,4278,2230,6326,1206,5302,3254,7350,694,4790,2742,6838,1718,5814,3766,7862,438,4534,2486,6582,1462,5558,3510,7606,950,5046,2998,7094,1974,6070,4022,8118,118,4214,2166,6262,1142,5238,3190,7286,630,4726,2678,6774,1654,5750,3702,7798,374,4470,2422,6518,1398,5494,3446,7542,886,4982,2934,7030,1910,6006,3958,8054,246,4342,2294,6390,1270,5366,3318,7414,758,4854,2806,6902,1782,5878,3830,7926,502,4598,2550,6646,1526,5622,3574,7670,1014,5110,3062,7158,2038,6134,4086,8182,14,4110,2062,6158,1038,5134,3086,7182,526,4622,2574,6670,1550,5646,3598,7694,270,4366,2318,6414,1294,5390,3342,7438,782,4878,2830,6926,1806,5902,3854,7950,142,4238,2190,6286,1166,5262,3214,7310,654,4750,2702,6798,1678,5774,3726,7822,398,4494,2446,6542,1422,5518,3470,7566,910,5006,2958,7054,1934,6030,3982,8078,78,4174,2126,6222,1102,5198,3150,7246,590,4686,2638,6734,1614,5710,3662,7758,334,4430,2382,6478,1358,5454,3406,7502,846,4942,2894,6990,1870,5966,3918,8014,206,4302,2254,6350,1230,5326,3278,7374,718,4814,2766,6862,1742,5838,3790,7886,462,4558,2510,6606,1486,5582,3534,7630,974,5070,3022,7118,1998,6094,4046,8142,46,4142,20 94,6190,1070,5166,3118,7214,558,4654,2606,6702,1582,5678,3630,7726,302,4398,2350,6446,1326,5422,3374,7470,814,4910,2862,6958,1838,5934,3886,7982,174,4270,2222,6318,1198,5294,3246,7342,686,4782,2734,6830,1710,5806,3758,7854,430,4526,2478,6574,1454,5550,3502,7598,942,5038,2990,7086,1966,6062,4014,8110,110,4206,2158,6254,1134,5230,3182,7278,622,4718,2670,6766,1646,5742,3694,7790,366,4462,2414,6510,1390,5486,3438,7534,878,4974,2926,7022,1902,5998,3950,8046,238,4334,2286,6382,1262,5358,3310,7406,750,4846,2798,6894,1774,5870,3822,7918,494,4590,2542,6638,1518,5614,3566,7662,1006,5102,3054,7150,2030,6126,4078,8174,30,4126,2078,6174,1054,5150,3102,7198,542,4638,2590,6686,1566,5662,3614,7710,286,4382,2334,6430,1310,5406,3358,7454,798,4894,2846,6942,1822,5918,3870,7966,158,4254,2206,6302,1182,5278,3230,7326,670,4766,2718,6814,1694,5790,3742,7838,414,4510,2462,6558,1438,5534,3486,7582,926,5022,2974,7070,1950,6046,3998,8094,94,4190,2142,6238,1118,5214,3166,7262,606,4702,2654,6750,1630,5726,3678,7774,350,4446,2398,6494,1374,5470,3422,7518,862,4958,2910,7006,1886,5982,3934,8030,222,4318,2270,6366,1246,5342,3294,7390,734,4830,2782,6878,1758,5854,3806,7902,478,4574,2526,6622,1502,5598,3550,7646,990,5086,3038,7134,2014,6110,4062,8158,62,4158,2110,6206,1086,5182,3134,7230,574,4670,2622,6718,1598,5694,3646,7742,318,4414,2366,6462,1342,5438,3390,7486,830,4926,2878,6974,1854,5950,3902,7998,190,4286,2238,6334,1214,5310,3262,7358,702,4798,2750,6846,1726,5822,3774,7870,446,4542,2494,6590,1470,5566,3518,7614,958,5054,3006,7102,1982,6078,4030,8126,126,4222,2174,6270,1150,5246,3198,7294,638,4734,2686,6782,1662,5758,3710,7806,382,4478,2430,6526,1406,5502,3454,7550,894,4990,2942,7038,1918,6014,3966,8062,254,4350,2302,6398,1278,5374,3326,7422,766,4862,2814,6910,1790,5886,3838,7934,510,4606,2558,6654,1534,5630,3582,7678,1022,5118,3070,7166,2046,6142,4094,8190]。
当N＝2048，
序列1c为：
Shorten_Q＝[0,1024,512,1536,256,1280,768,1792,128,1152,640,1664,384,1408,896,1920,64,1088,576,1600,320,1344,832,1856,192,1216,704,1728,448,1472,960,1984,32,1056,544,1568,288,1312,800,1824,160,1184,672,1696,416,1440,928,1952,96,1120,608,1632,352,1376,864,1888,224,1248,736,1760,480,1504,992,2016,16,1040,528,1552,272,1296,784,1808,144,1168,656,1680,400,1424,912,1936,80,1104,592,1616,336,1360,848,1872,208,1232,720,1744,464,1488,976,2000,48,1072,560,1584,304,1328,816,1840,176,1200,688,1712,432,1456,944,1968,112,1136,624,1648,368,1392,880,1904,240,1264,752,1776,496,1520,1008,2032,8,1032,520,1544,264,1288,776,1800,136,1160,648,1672,392,1416,904,1928,72,1096,584,1608,328,1352,840,1864,200,1224,712,1736,456,1480,968,1992,40,1064,552,1576,296,1320,808,1832,168,1192,680,1704,424,1448,936,1960,104,1128,616,1640,360,1384,872,1896,232,1256,744,1768,488,1512,1000,2024,24,1048,536,1560,280,1304,792,1816,152,1176,664,1688,408,1432,920,1944,88,1112,600,1624,344,1368,856,1880,216,1240,728,1752,472,1496,984,2008,56,1080,568,1592,312,1336,824,1848,184,1208,696,1720,440,1464,952,1976,120,1144,632,1656,376,1400,888,1912,248,1272,760,1784,504,1528,1016,2040,4,1028,516,1540,260,1284,772,1796,132,1156,644,1668,388,1412,900,1924,68,1092,580,1604,324,1348,836,1860,196,1220,708,1732,452,1476,964,1988,36,1060,548,1572,292,1316,804,1828,164,1188,676,1700,420,1444,932,1956,100,1124,612,1636,356,1380,868,1892,228,1252,740,1764,484,1508,996,2020,20,1044,532,1556,276,1300,788,1812,148,1172,660,1684,404,1428,916,1940,84,1108,596,1620,340,1364,852,1876,212,1236,724,1748,468,1492,980,2004,52,1076,564,1588,308,1332,820,1844,180,1204,692,1716,436,1460, 948,1972,116,1140,628,1652,372,1396,884,1908,244,1268,756,1780,500,1524,1012,2036,12,1036,524,1548,268,1292,780,1804,140,1164,652,1676,396,1420,908,1932,76,1100,588,1612,332,1356,844,1868,204,1228,716,1740,460,1484,972,1996,44,1068,556,1580,300,1324,812,1836,172,1196,684,1708,428,1452,940,1964,108,1132,620,1644,364,1388,876,1900,236,1260,748,1772,492,1516,1004,2028,28,1052,540,1564,284,1308,796,1820,156,1180,668,1692,412,1436,924,1948,92,1116,604,1628,348,1372,860,1884,220,1244,732,1756,476,1500,988,2012,60,1084,572,1596,316,1340,828,1852,188,1212,700,1724,444,1468,956,1980,124,1148,636,1660,380,1404,892,1916,252,1276,764,1788,508,1532,1020,2044,2,1026,514,1538,258,1282,770,1794,130,1154,642,1666,386,1410,898,1922,66,1090,578,1602,322,1346,834,1858,194,1218,706,1730,450,1474,962,1986,34,1058,546,1570,290,1314,802,1826,162,1186,674,1698,418,1442,930,1954,98,1122,610,1634,354,1378,866,1890,226,1250,738,1762,482,1506,994,2018,18,1042,530,1554,274,1298,786,1810,146,1170,658,1682,402,1426,914,1938,82,1106,594,1618,338,1362,850,1874,210,1234,722,1746,466,1490,978,2002,50,1074,562,1586,306,1330,818,1842,178,1202,690,1714,434,1458,946,1970,114,1138,626,1650,370,1394,882,1906,242,1266,754,1778,498,1522,1010,2034,10,1034,522,1546,266,1290,778,1802,138,1162,650,1674,394,1418,906,1930,74,1098,586,1610,330,1354,842,1866,202,1226,714,1738,458,1482,970,1994,42,1066,554,1578,298,1322,810,1834,170,1194,682,1706,426,1450,938,1962,106,1130,618,1642,362,1386,874,1898,234,1258,746,1770,490,1514,1002,2026,26,1050,538,1562,282,1306,794,1818,154,1178,666,1690,410,1434,922,1946,90,1114,602,1626,346,1370,858,1882,218,1242,730,1754,474,1498,986,2010,58,1082,570,1594,314,1338,826,1850,186,1210,698,1722,442,1466,954,1978,122,1146,634,1658,378,1402,890,1914,250,1274,762,1786,506,1530,1018,2042,6,1030,518,1542,262,1286,774,1798,134,1158,646,1670,390,1414,902,1926,70,1094,582,1606,326,1350,838,1862,198,1222,710,1734,454,1478,966,1990,38,1062,550,1574,294,1318,806,1830,166,1190,678,1702,422,1446,934,1958,102,1126,614,1638,358,1382,870,1894,230,1254,742,1766,486,1510,998,2022,22,1046,534,1558,278,1302,790,1814,150,1174,662,1686,406,1430,918,1942,86,1110,598,1622,342,1366,854,1878,214,1238,726,1750,470,1494,982,2006,54,1078,566,1590,310,1334,822,1846,182,1206,694,1718,438,1462,950,1974,118,1142,630,1654,374,1398,886,1910,246,1270,758,1782,502,1526,1014,2038,14,1038,526,1550,270,1294,782,1806,142,1166,654,1678,398,1422,910,1934,78,1102,590,1614,334,1358,846,1870,206,1230,718,1742,462,1486,974,1998,46,1070,558,1582,302,1326,814,1838,174,1198,686,1710,430,1454,942,1966,110,1134,622,1646,366,1390,878,1902,238,1262,750,1774,494,1518,1006,2030,30,1054,542,1566,286,1310,798,1822,158,1182,670,1694,414,1438,926,1950,94,1118,606,1630,350,1374,862,1886,222,1246,734,1758,478,1502,990,2014,62,1086,574,1598,318,1342,830,1854,190,1214,702,1726,446,1470,958,1982,126,1150,638,1662,382,1406,894,1918,254,1278,766,1790,510,1534,1022,2046,1,1025,513,1537,257,1281,769,1793,129,1153,641,1665,385,1409,897,1921,65,1089,577,1601,321,1345,833,1857,193,1217,705,1729,449,1473,961,1985,33,1057,545,1569,289,1313,801,1825,161,1185,673,1697,417,1441,929,1953,97,1121,609,1633,353,1377,865,1889,225,1249,737,1761,481,1505,993,2017,17,1041,529,1553,273,1297,785,1809,145,1169,657,1681,401,1425,913,1937,81,1105,593,1617,337,1361,849,1873,209,1233,721,1745,465,1489,977,2001,49,1073,561,1585,305,1329,817,1841,177,1201,689,1713,433,1457,945,1969,113,1137,625,1649,369,1393,881,1905,241,1265,753,1777,497,1521,1009,2033,9,1033,521,1545,265,1289,777,1801,137,1161,649,1673,393,1417,905,1929,73,1097,585,1609,329,1353,841,1865,201,1225,713,1737,457,1481,969,1993,41,1065,553,1577,297,1321,809,1833,169,1193,681,1705,425,1449,937,1961,105,1129,617,1641,361,1385,873,1897,233,1257,745,176 9,489,1513,1001,2025,25,1049,537,1561,281,1305,793,1817,153,1177,665,1689,409,1433,921,1945,89,1113,601,1625,345,1369,857,1881,217,1241,729,1753,473,1497,985,2009,57,1081,569,1593,313,1337,825,1849,185,1209,697,1721,441,1465,953,1977,121,1145,633,1657,377,1401,889,1913,249,1273,761,1785,505,1529,1017,2041,5,1029,517,1541,261,1285,773,1797,133,1157,645,1669,389,1413,901,1925,69,1093,581,1605,325,1349,837,1861,197,1221,709,1733,453,1477,965,1989,37,1061,549,1573,293,1317,805,1829,165,1189,677,1701,421,1445,933,1957,101,1125,613,1637,357,1381,869,1893,229,1253,741,1765,485,1509,997,2021,21,1045,533,1557,277,1301,789,1813,149,1173,661,1685,405,1429,917,1941,85,1109,597,1621,341,1365,853,1877,213,1237,725,1749,469,1493,981,2005,53,1077,565,1589,309,1333,821,1845,181,1205,693,1717,437,1461,949,1973,117,1141,629,1653,373,1397,885,1909,245,1269,757,1781,501,1525,1013,2037,13,1037,525,1549,269,1293,781,1805,141,1165,653,1677,397,1421,909,1933,77,1101,589,1613,333,1357,845,1869,205,1229,717,1741,461,1485,973,1997,45,1069,557,1581,301,1325,813,1837,173,1197,685,1709,429,1453,941,1965,109,1133,621,1645,365,1389,877,1901,237,1261,749,1773,493,1517,1005,2029,29,1053,541,1565,285,1309,797,1821,157,1181,669,1693,413,1437,925,1949,93,1117,605,1629,349,1373,861,1885,221,1245,733,1757,477,1501,989,2013,61,1085,573,1597,317,1341,829,1853,189,1213,701,1725,445,1469,957,1981,125,1149,637,1661,381,1405,893,1917,253,1277,765,1789,509,1533,1021,2045,3,1027,515,1539,259,1283,771,1795,131,1155,643,1667,387,1411,899,1923,67,1091,579,1603,323,1347,835,1859,195,1219,707,1731,451,1475,963,1987,35,1059,547,1571,291,1315,803,1827,163,1187,675,1699,419,1443,931,1955,99,1123,611,1635,355,1379,867,1891,227,1251,739,1763,483,1507,995,2019,19,1043,531,1555,275,1299,787,1811,147,1171,659,1683,403,1427,915,1939,83,1107,595,1619,339,1363,851,1875,211,1235,723,1747,467,1491,979,2003,51,1075,563,1587,307,1331,819,1843,179,1203,691,1715,435,1459,947,1971,115,1139,627,1651,371,1395,883,1907,243,1267,755,1779,499,1523,1011,2035,11,1035,523,1547,267,1291,779,1803,139,1163,651,1675,395,1419,907,1931,75,1099,587,1611,331,1355,843,1867,203,1227,715,1739,459,1483,971,1995,43,1067,555,1579,299,1323,811,1835,171,1195,683,1707,427,1451,939,1963,107,1131,619,1643,363,1387,875,1899,235,1259,747,1771,491,1515,1003,2027,27,1051,539,1563,283,1307,795,1819,155,1179,667,1691,411,1435,923,1947,91,1115,603,1627,347,1371,859,1883,219,1243,731,1755,475,1499,987,2011,59,1083,571,1595,315,1339,827,1851,187,1211,699,1723,443,1467,955,1979,123,1147,635,1659,379,1403,891,1915,251,1275,763,1787,507,1531,1019,2043,7,1031,519,1543,263,1287,775,1799,135,1159,647,1671,391,1415,903,1927,71,1095,583,1607,327,1351,839,1863,199,1223,711,1735,455,1479,967,1991,39,1063,551,1575,295,1319,807,1831,167,1191,679,1703,423,1447,935,1959,103,1127,615,1639,359,1383,871,1895,231,1255,743,1767,487,1511,999,2023,23,1047,535,1559,279,1303,791,1815,151,1175,663,1687,407,1431,919,1943,87,1111,599,1623,343,1367,855,1879,215,1239,727,1751,471,1495,983,2007,55,1079,567,1591,311,1335,823,1847,183,1207,695,1719,439,1463,951,1975,119,1143,631,1655,375,1399,887,1911,247,1271,759,1783,503,1527,1015,2039,15,1039,527,1551,271,1295,783,1807,143,1167,655,1679,399,1423,911,1935,79,1103,591,1615,335,1359,847,1871,207,1231,719,1743,463,1487,975,1999,47,1071,559,1583,303,1327,815,1839,175,1199,687,1711,431,1455,943,1967,111,1135,623,1647,367,1391,879,1903,239,1263,751,1775,495,1519,1007,2031,31,1055,543,1567,287,1311,799,1823,159,1183,671,1695,415,1439,927,1951,95,1119,607,1631,351,1375,863,1887,223,1247,735,1759,479,1503,991,2015,63,1087,575,1599,319,1343,831,1855,191,1215,703,1727,447,1471,959,1983,127,1151,639,1663,383,1407,895,1919,255,1279,767,1791,511,1535,1023,2047]。
序列2c为：
Shorten_Q_half＝[1,1025,513,1537,257,1281,769,1793,129,1153,641,1665,385,1409,897,1921,65,1089,577,1601,321,1345,833,1857,193,1217,705,1729,449,1473,961,1985,33,1057,545,1569,289,1313,801,1825,161,1185,673,1697,417,1441,929,1953,97,1121,609,1633,353,1377,865,1889,225,1249,737,1761,481,1505,993,2017,17,1041,529,1553,273,1297,785,1809,145,1169,657,1681,401,1425,913,1937,81,1105,593,1617,337,1361,849,1873,209,1233,721,1745,465,1489,977,2001,49,1073,561,1585,305,1329,817,1841,177,1201,689,1713,433,1457,945,1969,113,1137,625,1649,369,1393,881,1905,241,1265,753,1777,497,1521,1009,2033,9,1033,521,1545,265,1289,777,1801,137,1161,649,1673,393,1417,905,1929,73,1097,585,1609,329,1353,841,1865,201,1225,713,1737,457,1481,969,1993,41,1065,553,1577,297,1321,809,1833,169,1193,681,1705,425,1449,937,1961,105,1129,617,1641,361,1385,873,1897,233,1257,745,1769,489,1513,1001,2025,25,1049,537,1561,281,1305,793,1817,153,1177,665,1689,409,1433,921,1945,89,1113,601,1625,345,1369,857,1881,217,1241,729,1753,473,1497,985,2009,57,1081,569,1593,313,1337,825,1849,185,1209,697,1721,441,1465,953,1977,121,1145,633,1657,377,1401,889,1913,249,1273,761,1785,505,1529,1017,2041,5,1029,517,1541,261,1285,773,1797,133,1157,645,1669,389,1413,901,1925,69,1093,581,1605,325,1349,837,1861,197,1221,709,1733,453,1477,965,1989,37,1061,549,1573,293,1317,805,1829,165,1189,677,1701,421,1445,933,1957,101,1125,613,1637,357,1381,869,1893,229,1253,741,1765,485,1509,997,2021,21,1045,533,1557,277,1301,789,1813,149,1173,661,1685,405,1429,917,1941,85,1109,597,1621,341,1365,853,1877,213,1237,725,1749,469,1493,981,2005,53,1077,565,1589,309,1333,821,1845,181,1205,693,1717,437,1461,949,1973,117,1141,629,1653,373,1397,885,1909,245,1269,757,1781,501,1525,1013,2037,13,1037,525,1549,269,1293,781,1805,141,1165,653,1677,397,1421,909,1933,77,1101,589,1613,333,1357,845,1869,205,1229,717,1741,461,1485,973,1997,45,1069,557,1581,301,1325,813,1837,173,1197,685,1709,429,1453,941,1965,109,1133,621,1645,365,1389,877,1901,237,1261,749,1773,493,1517,1005,2029,29,1053,541,1565,285,1309,797,1821,157,1181,669,1693,413,1437,925,1949,93,1117,605,1629,349,1373,861,1885,221,1245,733,1757,477,1501,989,2013,61,1085,573,1597,317,1341,829,1853,189,1213,701,1725,445,1469,957,1981,125,1149,637,1661,381,1405,893,1917,253,1277,765,1789,509,1533,1021,2045,3,1027,515,1539,259,1283,771,1795,131,1155,643,1667,387,1411,899,1923,67,1091,579,1603,323,1347,835,1859,195,1219,707,1731,451,1475,963,1987,35,1059,547,1571,291,1315,803,1827,163,1187,675,1699,419,1443,931,1955,99,1123,611,1635,355,1379,867,1891,227,1251,739,1763,483,1507,995,2019,19,1043,531,1555,275,1299,787,1811,147,1171,659,1683,403,1427,915,1939,83,1107,595,1619,339,1363,851,1875,211,1235,723,1747,467,1491,979,2003,51,1075,563,1587,307,1331,819,1843,179,1203,691,1715,435,1459,947,1971,115,1139,627,1651,371,1395,883,1907,243,1267,755,1779,499,1523,1011,2035,11,1035,523,1547,267,1291,779,1803,139,1163,651,1675,395,1419,907,1931,75,1099,587,1611,331,1355,843,1867,203,1227,715,1739,459,1483,971,1995,43,1067,555,1579,299,1323,811,1835,171,1195,683,1707,427,1451,939,1963,107,1131,619,1643,363,1387,875,1899,235,1259,747,1771,491,1515,1003,2027,27,1051,539,1563,283,1307,795,1819,155,1179,667,1691,411,1435,923,1947,91,1115,603,1627,347,1371,859,1883,219,1243,731,1755,475,1499,987,2011,59,1083,571,1595,315,1339,827,1851,187,1211,699,1723,443,1467,955,1979,123,1147,635,1659,379,1403,891,1915,251,1275,763,1787,507,1531,1019,2043,7,1031,519,1543,263,1287,775,1799,135,1159,647,1671,391,1415,903,1927,71,1095,583,1607,327,1351,839,1863,199,1223,711,1735,455,1479,967,1991,39,1063,551,1575,295,1319,807,1831,167,1191,679,1703,423,1447,935,1959,103,1127,615,1639,3 59,1383,871,1895,231,1255,743,1767,487,1511,999,2023,23,1047,535,1559,279,1303,791,1815,151,1175,663,1687,407,1431,919,1943,87,1111,599,1623,343,1367,855,1879,215,1239,727,1751,471,1495,983,2007,55,1079,567,1591,311,1335,823,1847,183,1207,695,1719,439,1463,951,1975,119,1143,631,1655,375,1399,887,1911,247,1271,759,1783,503,1527,1015,2039,15,1039,527,1551,271,1295,783,1807,143,1167,655,1679,399,1423,911,1935,79,1103,591,1615,335,1359,847,1871,207,1231,719,1743,463,1487,975,1999,47,1071,559,1583,303,1327,815,1839,175,1199,687,1711,431,1455,943,1967,111,1135,623,1647,367,1391,879,1903,239,1263,751,1775,495,1519,1007,2031,31,1055,543,1567,287,1311,799,1823,159,1183,671,1695,415,1439,927,1951,95,1119,607,1631,351,1375,863,1887,223,1247,735,1759,479,1503,991,2015,63,1087,575,1599,319,1343,831,1855,191,1215,703,1727,447,1471,959,1983,127,1151,639,1663,383,1407,895,1919,255,1279,767,1791,511,1535,1023,2047]。
序列3c为；
Shorten_Z＝[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,4832,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176,8,4104,2056,6152,1032,5128,3080,7176,520,4616,2568,6664,1544,5640,3592,7688,264,4360,2312,6408,1288,5384,3336,7432,776,4872,2824,6920,1800,5896,3848,7944,136,4232,2184,6280,1 160,5256,3208,7304,648,4744,2696,6792,1672,5768,3720,7816,392,4488,2440,6536,1416,5512,3464,7560,904,5000,2952,7048,1928,6024,3976,8072,72,4168,2120,6216,1096,5192,3144,7240,584,4680,2632,6728,1608,5704,3656,7752,328,4424,2376,6472,1352,5448,3400,7496,840,4936,2888,6984,1864,5960,3912,8008,200,4296,2248,6344,1224,5320,3272,7368,712,4808,2760,6856,1736,5832,3784,7880,456,4552,2504,6600,1480,5576,3528,7624,968,5064,3016,7112,1992,6088,4040,8136,40,4136,2088,6184,1064,5160,3112,7208,552,4648,2600,6696,1576,5672,3624,7720,296,4392,2344,6440,1320,5416,3368,7464,808,4904,2856,6952,1832,5928,3880,7976,168,4264,2216,6312,1192,5288,3240,7336,680,4776,2728,6824,1704,5800,3752,7848,424,4520,2472,6568,1448,5544,3496,7592,936,5032,2984,7080,1960,6056,4008,8104,104,4200,2152,6248,1128,5224,3176,7272,616,4712,2664,6760,1640,5736,3688,7784,360,4456,2408,6504,1384,5480,3432,7528,872,4968,2920,7016,1896,5992,3944,8040,232,4328,2280,6376,1256,5352,3304,7400,744,4840,2792,6888,1768,5864,3816,7912,488,4584,2536,6632,1512,5608,3560,7656,1000,5096,3048,7144,2024,6120,4072,8168,24,4120,2072,6168,1048,5144,3096,7192,536,4632,2584,6680,1560,5656,3608,7704,280,4376,2328,6424,1304,5400,3352,7448,792,4888,2840,6936,1816,5912,3864,7960,152,4248,2200,6296,1176,5272,3224,7320,664,4760,2712,6808,1688,5784,3736,7832,408,4504,2456,6552,1432,5528,3480,7576,920,5016,2968,7064,1944,6040,3992,8088,88,4184,2136,6232,1112,5208,3160,7256,600,4696,2648,6744,1624,5720,3672,7768,344,4440,2392,6488,1368,5464,3416,7512,856,4952,2904,7000,1880,5976,3928,8024,216,4312,2264,6360,1240,5336,3288,7384,728,4824,2776,6872,1752,5848,3800,7896,472,4568,2520,6616,1496,5592,3544,7640,984,5080,3032,7128,2008,6104,4056,8152,56,4152,2104,6200,1080,5176,3128,7224,568,4664,2616,6712,1592,5688,3640,7736,312,4408,2360,6456,1336,5432,3384,7480,824,4920,2872,6968,1848,5944,3896,7992,184,4280,2232,6328,1208,5304,3256,7352,696,4792,2744,6840,1720,5816,3768,7864,440,4536,2488,6584,1464,5560,3512,7608,952,5048,3000,7096,1976,6072,4024,8120,120,4216,2168,6264,1144,5240,3192,7288,632,4728,2680,6776,1656,5752,3704,7800,376,4472,2424,6520,1400,5496,3448,7544,888,4984,2936,7032,1912,6008,3960,8056,248,4344,2296,6392,1272,5368,3320,7416,760,4856,2808,6904,1784,5880,3832,7928,504,4600,2552,6648,1528,5624,3576,7672,1016,5112,3064,7160,2040,6136,4088,8184,4,4100,2052,6148,1028,5124,3076,7172,516,4612,2564,6660,1540,5636,3588,7684,260,4356,2308,6404,1284,5380,3332,7428,772,4868,2820,6916,1796,5892,3844,7940,132,4228,2180,6276,1156,5252,3204,7300,644,4740,2692,6788,1668,5764,3716,7812,388,4484,2436,6532,1412,5508,3460,7556,900,4996,2948,7044,1924,6020,3972,8068,68,4164,2116,6212,1092,5188,3140,7236,580,4676,2628,6724,1604,5700,3652,7748,324,4420,2372,6468,1348,5444,3396,7492,836,4932,2884,6980,1860,5956,3908,8004,196,4292,2244,6340,1220,5316,3268,7364,708,4804,2756,6852,1732,5828,3780,7876,452,4548,2500,6596,1476,5572,3524,7620,964,5060,3012,7108,1988,6084,4036,8132,36,4132,2084,6180,1060,5156,3108,7204,548,4644,2596,6692,1572,5668,3620,7716,292,4388,2340,6436,1316,5412,3364,7460,804,4900,2852,6948,1828,5924,3876,7972,164,4260,2212,6308,1188,5284,3236,7332,676,4772,2724,6820,1700,5796,3748,7844,420,4516,2468,6564,1444,5540,3492,7588,932,5028,2980,7076,1956,6052,4004,8100,100,4196,2148,6244,1124,5220,3172,7268,612,4708,2660,6756,1636,5732,3684,7780,356,4452,2404,6500,1380,5476,3428,7524,868,4964,2916,7012,1892,5988,3940,8036,228,4324,2276,6372,1252,5348,3300,7396,740,4836,2788,6884,1764,5860,3812,7908,484,4580,2532,6628,1508,5604,3556,7652,996,5092,3044,7140,2020,6116,4068,8164,20,4116,2068,6164,1044,5140,3092,7188,532,4628,2580,6676,1556,5652,3604,7700,276,4372,2324,6420,1300,5396,3348,7444,788,4884,2836,6932,1812,5908,3860,7956,148,4244,2196,6 292,1172,5268,3220,7316,660,4756,2708,6804,1684,5780,3732,7828,404,4500,2452,6548,1428,5524,3476,7572,916,5012,2964,7060,1940,6036,3988,8084,84,4180,2132,6228,1108,5204,3156,7252,596,4692,2644,6740,1620,5716,3668,7764,340,4436,2388,6484,1364,5460,3412,7508,852,4948,2900,6996,1876,5972,3924,8020,212,4308,2260,6356,1236,5332,3284,7380,724,4820,2772,6868,1748,5844,3796,7892,468,4564,2516,6612,1492,5588,3540,7636,980,5076,3028,7124,2004,6100,4052,8148,52,4148,2100,6196,1076,5172,3124,7220,564,4660,2612,6708,1588,5684,3636,7732,308,4404,2356,6452,1332,5428,3380,7476,820,4916,2868,6964,1844,5940,3892,7988,180,4276,2228,6324,1204,5300,3252,7348,692,4788,2740,6836,1716,5812,3764,7860,436,4532,2484,6580,1460,5556,3508,7604,948,5044,2996,7092,1972,6068,4020,8116,116,4212,2164,6260,1140,5236,3188,7284,628,4724,2676,6772,1652,5748,3700,7796,372,4468,2420,6516,1396,5492,3444,7540,884,4980,2932,7028,1908,6004,3956,8052,244,4340,2292,6388,1268,5364,3316,7412,756,4852,2804,6900,1780,5876,3828,7924,500,4596,2548,6644,1524,5620,3572,7668,1012,5108,3060,7156,2036,6132,4084,8180,12,4108,2060,6156,1036,5132,3084,7180,524,4620,2572,6668,1548,5644,3596,7692,268,4364,2316,6412,1292,5388,3340,7436,780,4876,2828,6924,1804,5900,3852,7948,140,4236,2188,6284,1164,5260,3212,7308,652,4748,2700,6796,1676,5772,3724,7820,396,4492,2444,6540,1420,5516,3468,7564,908,5004,2956,7052,1932,6028,3980,8076,76,4172,2124,6220,1100,5196,3148,7244,588,4684,2636,6732,1612,5708,3660,7756,332,4428,2380,6476,1356,5452,3404,7500,844,4940,2892,6988,1868,5964,3916,8012,204,4300,2252,6348,1228,5324,3276,7372,716,4812,2764,6860,1740,5836,3788,7884,460,4556,2508,6604,1484,5580,3532,7628,972,5068,3020,7116,1996,6092,4044,8140,44,4140,2092,6188,1068,5164,3116,7212,556,4652,2604,6700,1580,5676,3628,7724,300,4396,2348,6444,1324,5420,3372,7468,812,4908,2860,6956,1836,5932,3884,7980,172,4268,2220,6316,1196,5292,3244,7340,684,4780,2732,6828,1708,5804,3756,7852,428,4524,2476,6572,1452,5548,3500,7596,940,5036,2988,7084,1964,6060,4012,8108,108,4204,2156,6252,1132,5228,3180,7276,620,4716,2668,6764,1644,5740,3692,7788,364,4460,2412,6508,1388,5484,3436,7532,876,4972,2924,7020,1900,5996,3948,8044,236,4332,2284,6380,1260,5356,3308,7404,748,4844,2796,6892,1772,5868,3820,7916,492,4588,2540,6636,1516,5612,3564,7660,1004,5100,3052,7148,2028,6124,4076,8172,28,4124,2076,6172,1052,5148,3100,7196,540,4636,2588,6684,1564,5660,3612,7708,284,4380,2332,6428,1308,5404,3356,7452,796,4892,2844,6940,1820,5916,3868,7964,156,4252,2204,6300,1180,5276,3228,7324,668,4764,2716,6812,1692,5788,3740,7836,412,4508,2460,6556,1436,5532,3484,7580,924,5020,2972,7068,1948,6044,3996,8092,92,4188,2140,6236,1116,5212,3164,7260,604,4700,2652,6748,1628,5724,3676,7772,348,4444,2396,6492,1372,5468,3420,7516,860,4956,2908,7004,1884,5980,3932,8028,220,4316,2268,6364,1244,5340,3292,7388,732,4828,2780,6876,1756,5852,3804,7900,476,4572,2524,6620,1500,5596,3548,7644,988,5084,3036,7132,2012,6108,4060,8156,60,4156,2108,6204,1084,5180,3132,7228,572,4668,2620,6716,1596,5692,3644,7740,316,4412,2364,6460,1340,5436,3388,7484,828,4924,2876,6972,1852,5948,3900,7996,188,4284,2236,6332,1212,5308,3260,7356,700,4796,2748,6844,1724,5820,3772,7868,444,4540,2492,6588,1468,5564,3516,7612,956,5052,3004,7100,1980,6076,4028,8124,124,4220,2172,6268,1148,5244,3196,7292,636,4732,2684,6780,1660,5756,3708,7804,380,4476,2428,6524,1404,5500,3452,7548,892,4988,2940,7036,1916,6012,3964,8060,252,4348,2300,6396,1276,5372,3324,7420,764,4860,2812,6908,1788,5884,3836,7932,508,4604,2556,6652,1532,5628,3580,7676,1020,5116,3068,7164,2044,6140,4092,8188]。
当N＝1024，
序列1d为：
Shorten_Q＝[0,512,256,768,128,640,384,896,64,576,320,832,192,704,448,960,32,544,288,800,160,672,416,928,96,608,352,864,224,736,480,992,16,528,272,784,144,656,400,912,80,592,336,848,208,720,464,976,48,560,304,816,176,688,432,944,112,624,368,880,240,752,496,1008,8,520,264,776,136,648,392,904,72,584,328,840,200,712,456,968,40,552,296,808,168,680,424,936,104,616,360,872,232,744,488,1000,24,536,280,792,152,664,408,920,88,600,344,856,216,728,472,984,56,568,312,824,184,696,440,952,120,632,376,888,248,760,504,1016,4,516,260,772,132,644,388,900,68,580,324,836,196,708,452,964,36,548,292,804,164,676,420,932,100,612,356,868,228,740,484,996,20,532,276,788,148,660,404,916,84,596,340,852,212,724,468,980,52,564,308,820,180,692,436,948,116,628,372,884,244,756,500,1012,12,524,268,780,140,652,396,908,76,588,332,844,204,716,460,972,44,556,300,812,172,684,428,940,108,620,364,876,236,748,492,1004,28,540,284,796,156,668,412,924,92,604,348,860,220,732,476,988,60,572,316,828,188,700,444,956,124,636,380,892,252,764,508,1020,2,514,258,770,130,642,386,898,66,578,322,834,194,706,450,962,34,546,290,802,162,674,418,930,98,610,354,866,226,738,482,994,18,530,274,786,146,658,402,914,82,594,338,850,210,722,466,978,50,562,306,818,178,690,434,946,114,626,370,882,242,754,498,1010,10,522,266,778,138,650,394,906,74,586,330,842,202,714,458,970,42,554,298,810,170,682,426,938,106,618,362,874,234,746,490,1002,26,538,282,794,154,666,410,922,90,602,346,858,218,730,474,986,58,570,314,826,186,698,442,954,122,634,378,890,250,762,506,1018,6,518,262,774,134,646,390,902,70,582,326,838,198,710,454,966,38,550,294,806,166,678,422,934,102,614,358,870,230,742,486,998,22,534,278,790,150,662,406,918,86,598,342,854,214,726,470,982,54,566,310,822,182,694,438,950,118,630,374,886,246,758,502,1014,14,526,270,782,142,654,398,910,78,590,334,846,206,718,462,974,46,558,302,814,174,686,430,942,110,622,366,878,238,750,494,1006,30,542,286,798,158,670,414,926,94,606,350,862,222,734,478,990,62,574,318,830,190,702,446,958,126,638,382,894,254,766,510,1022,1,513,257,769,129,641,385,897,65,577,321,833,193,705,449,961,33,545,289,801,161,673,417,929,97,609,353,865,225,737,481,993,17,529,273,785,145,657,401,913,81,593,337,849,209,721,465,977,49,561,305,817,177,689,433,945,113,625,369,881,241,753,497,1009,9,521,265,777,137,649,393,905,73,585,329,841,201,713,457,969,41,553,297,809,169,681,425,937,105,617,361,873,233,745,489,1001,25,537,281,793,153,665,409,921,89,601,345,857,217,729,473,985,57,569,313,825,185,697,441,953,121,633,377,889,249,761,505,1017,5,517,261,773,133,645,389,901,69,581,325,837,197,709,453,965,37,549,293,805,165,677,421,933,101,613,357,869,229,741,485,997,21,533,277,789,149,661,405,917,85,597,341,853,213,725,469,981,53,565,309,821,181,693,437,949,117,629,373,885,245,757,501,1013,13,525,269,781,141,653,397,909,77,589,333,845,205,717,461,973,45,557,301,813,173,685,429,941,109,621,365,877,237,749,493,1005,29,541,285,797,157,669,413,925,93,605,349,861,221,733,477,989,61,573,317,829,189,701,445,957,125,637,381,893,253,765,509,1021,3,515,259,771,131,643,387,899,67,579,323,835,195,707,451,963,35,547,291,803,163,675,419,931,99,611,355,867,227,739,483,995,19,531,275,787,147,659,403,915,83,595,339,851,211,723,467,979,51,563,307,819,179,691,435,947,115,627,371,883,243,755,499,1011,11,523,267,779,139,651,395,907,75,587,331,843,203,715,459,971,43,555,299,811,171,683,427,939,107,619,363,875,235,747,491,1003,27,539,283,795,155,667,411,923,91,603,347,859,219,731,475,987,59,571,315,827,187,699,443,955,123,635,379,891,251,763,507,1019,7,519,263,775,135,647,391,903,71,583,327,839,199,711,455,967,39,551,295,807,167,679,423,935,103,615,359,871,231,743,487,999,23,535,279,791,151,663,407,919,87,599,343,855,215,727,471,983,55,567,311,823,18 3,695,439,951,119,631,375,887,247,759,503,1015,15,527,271,783,143,655,399,911,79,591,335,847,207,719,463,975,47,559,303,815,175,687,431,943,111,623,367,879,239,751,495,1007,31,543,287,799,159,671,415,927,95,607,351,863,223,735,479,991,63,575,319,831,191,703,447,959,127,639,383,895,255,767,511,1023]。
序列2d为：
Shorten_Q_half＝[1,513,257,769,129,641,385,897,65,577,321,833,193,705,449,961,33,545,289,801,161,673,417,929,97,609,353,865,225,737,481,993,17,529,273,785,145,657,401,913,81,593,337,849,209,721,465,977,49,561,305,817,177,689,433,945,113,625,369,881,241,753,497,1009,9,521,265,777,137,649,393,905,73,585,329,841,201,713,457,969,41,553,297,809,169,681,425,937,105,617,361,873,233,745,489,1001,25,537,281,793,153,665,409,921,89,601,345,857,217,729,473,985,57,569,313,825,185,697,441,953,121,633,377,889,249,761,505,1017,5,517,261,773,133,645,389,901,69,581,325,837,197,709,453,965,37,549,293,805,165,677,421,933,101,613,357,869,229,741,485,997,21,533,277,789,149,661,405,917,85,597,341,853,213,725,469,981,53,565,309,821,181,693,437,949,117,629,373,885,245,757,501,1013,13,525,269,781,141,653,397,909,77,589,333,845,205,717,461,973,45,557,301,813,173,685,429,941,109,621,365,877,237,749,493,1005,29,541,285,797,157,669,413,925,93,605,349,861,221,733,477,989,61,573,317,829,189,701,445,957,125,637,381,893,253,765,509,1021,3,515,259,771,131,643,387,899,67,579,323,835,195,707,451,963,35,547,291,803,163,675,419,931,99,611,355,867,227,739,483,995,19,531,275,787,147,659,403,915,83,595,339,851,211,723,467,979,51,563,307,819,179,691,435,947,115,627,371,883,243,755,499,1011,11,523,267,779,139,651,395,907,75,587,331,843,203,715,459,971,43,555,299,811,171,683,427,939,107,619,363,875,235,747,491,1003,27,539,283,795,155,667,411,923,91,603,347,859,219,731,475,987,59,571,315,827,187,699,443,955,123,635,379,891,251,763,507,1019,7,519,263,775,135,647,391,903,71,583,327,839,199,711,455,967,39,551,295,807,167,679,423,935,103,615,359,871,231,743,487,999,23,535,279,791,151,663,407,919,87,599,343,855,215,727,471,983,55,567,311,823,183,695,439,951,119,631,375,887,247,759,503,1015,15,527,271,783,143,655,399,911,79,591,335,847,207,719,463,975,47,559,303,815,175,687,431,943,111,623,367,879,239,751,495,1007,31,543,287,799,159,671,415,927,95,607,351,863,223,735,479,991,63,575,319,831,191,703,447,959,127,639,383,895,255,767,511,1023]。
序列3d为；
Shorten_Z＝[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,48 32,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176,8,4104,2056,6152,1032,5128,3080,7176,520,4616,2568,6664,1544,5640,3592,7688,264,4360,2312,6408,1288,5384,3336,7432,776,4872,2824,6920,1800,5896,3848,7944,136,4232,2184,6280,1160,5256,3208,7304,648,4744,2696,6792,1672,5768,3720,7816,392,4488,2440,6536,1416,5512,3464,7560,904,5000,2952,7048,1928,6024,3976,8072,72,4168,2120,6216,1096,5192,3144,7240,584,4680,2632,6728,1608,5704,3656,7752,328,4424,2376,6472,1352,5448,3400,7496,840,4936,2888,6984,1864,5960,3912,8008,200,4296,2248,6344,1224,5320,3272,7368,712,4808,2760,6856,1736,5832,3784,7880,456,4552,2504,6600,1480,5576,3528,7624,968,5064,3016,7112,1992,6088,4040,8136,40,4136,2088,6184,1064,5160,3112,7208,552,4648,2600,6696,1576,5672,3624,7720,296,4392,2344,6440,1320,5416,3368,7464,808,4904,2856,6952,1832,5928,3880,7976,168,4264,2216,6312,1192,5288,3240,7336,680,4776,2728,6824,1704,5800,3752,7848,424,4520,2472,6568,1448,5544,3496,7592,936,5032,2984,7080,1960,6056,4008,8104,104,4200,2152,6248,1128,5224,3176,7272,616,4712,2664,6760,1640,5736,3688,7784,360,4456,2408,6504,1384,5480,3432,7528,872,4968,2920,7016,1896,5992,3944,8040,232,4328,2280,6376,1256,5352,3304,7400,744,4840,2792,6888,1768,5864,3816,7912,488,4584,2536,6632,1512,5608,3560,7656,1000,5096,3048,7144,2024,6120,4072,8168,24,4120,2072,6168,1048,5144,3096,7192,536,4632,2584,6680,1560,5656,3608,7704,280,4376,2328,6424,1304,5400,3352,7448,792,4888,2840,6936,1816,5912,3864,7960,152,4248,2200,6296,1176,5272,3224,7320,664,4760,2712,6808,1688,5784,3736,7832,408,4504,2456,6552,1432,5528,3480,7576,920,5016,2968,7064,1944,6040,3992,8088,88,4184,2136,6232,1112,5208,3160,7256,600,4696,2648,6744,1624,5720,3672,7768,344,4440,2392,6488,1368,5464,3416,7512,856,4952,2904,7000,1880,5976,3928,8024,216,4312,2264,6360,1240,5336,3288,7384,728,4824,2776,6872,1752,5848,3800,7896,472,4568,2520,6616,1496,5592,3544,7640,984,5080,3032,7128,2008,6104,4056,8152,56,4152,2104,6200,1080,5176,3128,7224,568,4664,2616,6712,1592,5688,3640,7736,312,4408,2360,6456,1336,5432,3384,7480,824,4920,2872,6968,1848,5944,3896,7992,184,4280,2232,6328,1208,5304,3256,7352,696,4792,2744,6840,1720,5816,3768,7864,440,4536,2488,6584,1464,5560,3512,7608,952,5048,3000,7096,1976,6072,4024,8120,120,4216,2168,6264,1144,5240,3192,7288,632,4728,2680,6776,1656,5752,3704,7800,376,4472,2424,6520,1400,5496,3448,7544,888,4984,2936,7032,1912,6008,3960,8056,248,4344,2296,6392,1272,5368,3320,7416,7 60,4856,2808,6904,1784,5880,3832,7928,504,4600,2552,6648,1528,5624,3576,7672,1016,5112,3064,7160,2040,6136,4088,8184]。
当N＝512，
序列1e为：
Shorten_Q＝[0,256,128,384,64,320,192,448,32,288,160,416,96,352,224,480,16,272,144,400,80,336,208,464,48,304,176,432,112,368,240,496,8,264,136,392,72,328,200,456,40,296,168,424,104,360,232,488,24,280,152,408,88,344,216,472,56,312,184,440,120,376,248,504,4,260,132,388,68,324,196,452,36,292,164,420,100,356,228,484,20,276,148,404,84,340,212,468,52,308,180,436,116,372,244,500,12,268,140,396,76,332,204,460,44,300,172,428,108,364,236,492,28,284,156,412,92,348,220,476,60,316,188,444,124,380,252,508,2,258,130,386,66,322,194,450,34,290,162,418,98,354,226,482,18,274,146,402,82,338,210,466,50,306,178,434,114,370,242,498,10,266,138,394,74,330,202,458,42,298,170,426,106,362,234,490,26,282,154,410,90,346,218,474,58,314,186,442,122,378,250,506,6,262,134,390,70,326,198,454,38,294,166,422,102,358,230,486,22,278,150,406,86,342,214,470,54,310,182,438,118,374,246,502,14,270,142,398,78,334,206,462,46,302,174,430,110,366,238,494,30,286,158,414,94,350,222,478,62,318,190,446,126,382,254,510,1,257,129,385,65,321,193,449,33,289,161,417,97,353,225,481,17,273,145,401,81,337,209,465,49,305,177,433,113,369,241,497,9,265,137,393,73,329,201,457,41,297,169,425,105,361,233,489,25,281,153,409,89,345,217,473,57,313,185,441,121,377,249,505,5,261,133,389,69,325,197,453,37,293,165,421,101,357,229,485,21,277,149,405,85,341,213,469,53,309,181,437,117,373,245,501,13,269,141,397,77,333,205,461,45,301,173,429,109,365,237,493,29,285,157,413,93,349,221,477,61,317,189,445,125,381,253,509,3,259,131,387,67,323,195,451,35,291,163,419,99,355,227,483,19,275,147,403,83,339,211,467,51,307,179,435,115,371,243,499,11,267,139,395,75,331,203,459,43,299,171,427,107,363,235,491,27,283,155,411,91,347,219,475,59,315,187,443,123,379,251,507,7,263,135,391,71,327,199,455,39,295,167,423,103,359,231,487,23,279,151,407,87,343,215,471,55,311,183,439,119,375,247,503,15,271,143,399,79,335,207,463,47,303,175,431,111,367,239,495,31,287,159,415,95,351,223,479,63,319,191,447,127,383,255,511]。
序列2e为：
Shorten_Q_half＝[1,257,129,385,65,321,193,449,33,289,161,417,97,353,225,481,17,273,145,401,81,337,209,465,49,305,177,433,113,369,241,497,9,265,137,393,73,329,201,457,41,297,169,425,105,361,233,489,25,281,153,409,89,345,217,473,57,313,185,441,121,377,249,505,5,261,133,389,69,325,197,453,37,293,165,421,101,357,229,485,21,277,149,405,85,341,213,469,53,309,181,437,117,373,245,501,13,269,141,397,77,333,205,461,45,301,173,429,109,365,237,493,29,285,157,413,93,349,221,477,61,317,189,445,125,381,253,509,3,259,131,387,67,323,195,451,35,291,163,419,99,355,227,483,19,275,147,403,83,339,211,467,51,307,179,435,115,371,243,499,11,267,139,395,75,331,203,459,43,299,171,427,107,363,235,491,27,283,155,411,91,347,219,475,59,315,187,443,123,379,251,507,7,263,135,391,71,327,199,455,39,295,167,423,103,359,231,487,23,279,151,407,87,343,215,471,55,311,183,439,119,375,247,503,15,271,143,399,79,335,207,463,47,303,175,431,111,367,239,495,31,287,159,415,95,351,223,479,63,319,191,447,127,383,255,511]。
序列3e为：
Shorten_Z＝[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552, 896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,4832,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176]。

Claims (26)

  1. 一种极化码编码的速率匹配方法，其特征在于，包括：
    获取K个待编码比特、目标码长M、母码长度N，N＝2 n，M、K、n和N为正整数；
    根据所述母码长度N对所述K个待编码比特进行极化编码，得到编码后的比特序列；
    根据速率匹配序列对所述编码后的比特序列进行速率匹配，获取速率匹配后的序列；
    输出所述速率匹配后的序列。
  2. 如权利要求1所述的方法，其特征在于，所述根据所述母码长度N对所述K个待编码比特进行极化编码为：所述K个待编码比特映射到M-P个非打孔的极化子信道中可靠度较高的K个极化子信道，使得K个待编码比特所在的极化子信道的可靠度高于其余M-P-K个极化子信道的可靠度，P为预冻结的极化子信道的数目，P大于或等于0。
  3. 如权利要求1或2所述的方法，其特征在于，所述根据速率匹配序列对所述编码后的比特序列进行速率匹配获取速率匹配后的序列之前，对所述编码后的比特序列进行交织。
  4. 一种极化码解码的解速率匹配方法，其特征在于，包括：
    获取包括M个比特的待解码信息序列，其中包括K个待解码比特；
    确定母码长度N，母码长度N，N＝2 n，M、K、n和N为正整数；
    根据所述母码长度N和速率匹配序列对所述待解码信息序列进行解速率匹配和解码，获取解码结果；
    输出所述解码结果。
  5. 如权利要求1～4中任意一项所述的方法，其特征在于，
    所述速率匹配序列为第一序列的子序列，所述第一序列为附录一中的序列1或序列2或序列3，其中所述序列1包括N 2个极化子信道号，所述序列2包括N 2/2个极化子信道号，所述序列3中包括N 2个大小不同的值，所述极化子信道号由0开始，N 2大于或等于N。
  6. 如权利要求1～5中任意一项所述的方法，其特征在于，
    当N＝8192，所述速率匹配序列为说明书中的序列1a或序列2a或序列3a；或者
    当N＝4096，所述速率匹配序列为说明书中的序列1b或2b或3b；或者
    当N＝2048，所述速率匹配序列为说明书中的序列1c或2c或3c；或者
    当N＝1024，所述速率匹配序列为说明书中的序列1d或2d或3d；或者
    当N＝512，所述速率匹配序列为说明书中的序列1e或2e或3e；
    其中所述序列1a、所述序列1b、所述序列1c、所述序列1d、所述序列1e为所述序列1的子序列；
    所述序列2a、所述序列2b、所述序列2c、所述序列2d、所述序列2e为所述序列2的子序列；
    所述序列3a、所述序列3b、所述序列3c、所述序列3d、所述序列3e为所述序列3的子序列。
  7. 如权利要求5或6所述的方法，其特征在于，所述速率匹配的方式为打孔操作，
    若所述第一序列为所述序列1，所述打孔操作为：根据所述目标码长M取所述速率匹配序列中后M个极化子信道号对应子信道承载的比特或根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
    若所述第一序列为所述序列2，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
    若所述第一序列为所述序列3，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中最小的N-M个值所在的极化子信道承载的比特或根据所述目标码长M取所述速率匹配序列中最大的M个值所在的极化子信道承载的比特。
  8. 如权利要求1～7中任意一项所述的方法，其特征在于，所述K个比特中，包括循环冗余校验比特。
  9. 如权利要求1～8中任意一项所述的方法，其特征在于，所述K个比特中，包括奇偶校验比特。
  10. 一种极化Polar码速率匹配装置，其特征在于，包括：
    获取单元，K个待编码比特、目标码长M，所述K为正整数；
    编码单元，用于确定母码长度N，根据所述母码长度N对所述K个待编码比特进行极化编码，得到编码后的比特序列，N＝2 n，n为正整数，K≤N；以及
    速率匹配单元，用于根据速率匹配序列对所述编码后的比特序列进行速率匹配，获取长度为M的速率匹配后的比特序列。
  11. 如权利要求10所述的装置，其特征在于，所述根据所述母码长度N对所述K个待编码比特进行极化编码为：所述K个待编码比特映射到M-P个非打孔的极化子信道中可靠度较高的K个极化子信道，使得K个待编码比特所在的极化子信道的可靠度高于其余M-P-K个极化子信道的可靠度，P为预冻结的极化子信道的数目，P大于或等于0。
  12. 如权利要求10或11所述的装置，其特征在于，所述根据速率匹配序列对所述编码后的比特序列进行速率匹配获取长度为M的速率匹配后的比特序列之前，对所述编码后的比特序列进行交织。
  13. 一种Polar码解速率匹配装置，其特征在于，包括：
    获取单元1001，用于获取待解码信息序列；
    处理单元1002，用于确定母码长度N，根据母码长度N和速率匹配序列对所述待解码信息序列序列进行解速率匹配的操作和解码解调操作，从而得到并输出解码结果。
  14. 如权利要求10～13中任意一项所述的装置，其特征在于，所述速率匹配序列为第一序列的子序列，所述第一序列为附录一中的序列1或序列2或序列3，其中所述序列1包括N 2个极化子信道号，所述序列2包括N 2/2个极化子信道号，所述序列3中包括N 2个大小不同的值，所述极化子信道号由0开始，N 2大于或等于N。
  15. 如权利要求10～14中任意一项所述的装置，其特征在于，
    当N＝8192，所述速率匹配序列为说明书中的序列1a或序列2a或序列3a；或者
    当N＝4096，所述速率匹配序列为说明书中的序列1b或2b或3b；或者
    当N＝2048，所述速率匹配序列为说明书中的序列1c或2c或3c；或者
    当N＝1024，所述速率匹配序列为说明书中的序列1d或2d或3d；或者
    当N＝512，所述速率匹配序列为说明书中的序列1e或2e或3e；
    其中所述序列1a、所述序列1b、所述序列1c、所述序列1d、所述序列1e为所述序列1的子序列；
    所述序列2a、所述序列2b、所述序列2c、所述序列2d、所述序列2e为所述序列2的子序列；
    所述序列3a、所述序列3b、所述序列3c、所述序列3d、所述序列3e为所述序列3的子序列。
  16. 如权利要求14或15所述的装置，其特征在于，所述速率匹配的方式为打孔操作，
    若所述第一序列为所述序列1，所述打孔操作为：根据所述目标码长M取所述速率匹配序列中后M个极化子信道号对应子信道承载的比特或根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
    若所述第一序列为所述序列2，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中前N-M个极化子信道号对应的子信道承载的比特；或者
    若所述第一序列为所述序列3，所述打孔操作为：根据所述目标码长M删除所述速率匹配序列中取值最小的N-M个值所在的极化子信道承载的比特或根据所述目标码长M取所述速率匹配序列中取值最大的M个值所在的极化子信道承载的比特。
  17. 如权利要求10～16中任意一项所述的装置，其特征在于，所述K个比特中包括循环冗余校验比特。
  18. 如权利要求10～17中任意一项所述的装置，其特征在于，所述K个比特中包括奇偶校验比特。
  19. 一种极化Polar码速率匹配装置，其特征在于，包括：处理器，用于执行如权利要求1～3和5～9中任一项所述的方法。
  20. 如权利要求19所述的装置，其特征在于，还包括：存储器，用于存储由所述处理器执行如权利要求1～3和5～9中任一项所述的方法时所用的程序指令。
  21. 一种Polar码速率匹配装置，其特征在于，包括：
    输入接口电路，用于获取K个待编码比特；
    逻辑电路，用于基于所述K个待编码比特按照如权利要求1～3和5～9中任意一项所述的方法得到速率匹配后的序列；
    输出接口电路，用于输出所述速率匹配后的序列。
  22. 一种极化Polar码解速率匹配装置，其特征在于，包括：处理器，用于执行如权利要求4～9任一项所述的方法。
  23. 如权利要求19所述的装置，其特征在于，还包括：存储器，用于存储由所述处 理器执行如权利要求4～9中任一项所述的方法时所用的程序指令。
  24. 一种Polar码解速率匹配装置，其特征在于，包括：
    输入接口电路，用于获取待解码信息序列；
    逻辑电路，用于按照如权利要求4～9中任意一项所述的方法对所述待解码信息序列序列进行解速率匹配的操作和解码解调操作，从而得到解码结果；
    输出接口电路，用于输出所述解码结果。
  25. 一种计算机可读介质，其特征在于，存储有计算机程序指令，当所述计算机程序指令由计算机执行时，使得如权利要求1-9中任意一项所述的方法被执行。
  26. 一种程序产品，其特征在于，所述程序产品包括计算机程序指令，当所述计算机程序指令由计算机执行时，使得如权利要求1-9中任意一项所述的方法被执行。
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