JP2000516072A - Method and apparatus for encoding and decoding data - Google Patents

Abstract

SUMMARY The present invention provides a method and apparatus for performing data encoding and decoding in a coded orthogonal frequency division multiplexing (or COFDM) system. In the present invention, an n-bit data word is encoded as a code word of 2 ^m symbols (2, 4, 8, etc.). The code word is selected from the coset set of linear subcodes of the code with the specified generator matrix, with respect to the desired low peak-to-average envelope power ratio (i.e., PMEPR) characteristics of the transmission on the COFDM path. The code word thus selected limits PMEPR to 3 or less, for values of m greater than 3, has specified error control characteristics, and uses analytical circuit techniques (such as combinatorial logic). Can be included in a number of different codewords to allow implementation in a form that uses sigma and data transmission at useful rates. Also, to achieve a higher code rate, other code words can be selected to achieve higher maximum PMEPR or lower error detection capability.

Description

DETAILED DESCRIPTION OF THE INVENTION                 Method and apparatus for encoding and decoding data Technical field   The present invention relates to a method and apparatus for encoding and decoding data, and to such a method. A communication system incorporating the method and apparatus, and in particular, Coded orthogonal frequency division multiplexing (ie, coded orthogonal frequency divis) ion multiplexing (hereinafter sometimes abbreviated as COFDM). Method, apparatus and system for transmitting data using transmission frequency division multiplexing It is about the system.Background art   With the increasing widespread use of powerful computer-based equipment and equipment, Demand for high-speed, high-capacity communications based on both wired and wireless technologies is growing rapidly Has increased. One way to address this growing demand is to use the highest possible To design a communication system to operate at the symbol rate. However, The increase in symbol rate is typically such that the spacing between symbols is correspondingly reduced, Therefore, the inter-symbol interference (ie, inter-symbo l Interference (hereinafter abbreviated as ISI) and the resulting error generate. In the case of a wired connection path, ISI, for example, What happens from signal reflections when the interfaces between them do not exactly match There is. In the case of a wireless connection path, the majority of ISI sources are empty Multipath propagation with signal reflection from an object (such as a building) between Dynamic wireless communication extends to a mobile device by moving the mobile device using the communication path. The continuous and complex variation of the multiple propagation paths Susceptible to SI.   Actively researched to increase communication system capacity and take measures against ISI The technology being developed is multi-carrier (or multi-tone) like COFDM operation This is a frequency division multiplex operation. In a COFDM system, multiple symbols are Simultaneously transmitted, each symbol consists of several carrier signals with closely spaced frequencies. One corresponding modulation (eg, phase shift keying) is controlled. symbol The transmission wave frequency so that has an appropriate relationship to the rate transmitted on each carrier. By selecting the intercept, each individual transmitted wave receives interference from adjacent transmitted waves. (In such a case, the transmitted waves are orthogonal to each other. ). Between one symbol period (ie, one code word) The set of symbols transmitted simultaneously for each transmission wave is the data symbol ( Selected to encode one group of data words). Set contains redundancy (i.e., code words are more than data words) With a symbol). This redundancy introduces an error in the received symbol upon reception. Used for error detection and correction of those errors if necessary.   The COFDM system is a sequential processing system by transmitting a plurality of symbols in parallel. Is used to transmit individual symbols continuously, Since it is possible to achieve the desired data rate with a low symbol rate, the ISI Partially address the problem. A relatively low symbol rate is A relatively long duration, i.e., a relatively long period between symbol transitions, This reduces the effects of ISI. Error detection and And correction capability depends on the received symbol due to signal strength fluctuations and other causes. By allowing some errors to occur to be detected and corrected, Enhance resistance to data corruption.   However, current COFDM systems have a typical Peak-to-mean envelope power ratio (i.e., peak-to-mean envelope power  ratio, and may be abbreviated as PMEPR below). You encounter difficulties in maximizing the potential benefits of power. this The signals actually have their phases changing at each symbol transition and are dense. Low, but still variable, because it is the sum of several signals at frequencies Transient apparent peaks spaced at substantially longer intervals of moving amplitude Tend to show. Transmitter peaks without clipping or other distortion Must be adjusted to play. This allows the equipment to be Signal level well below its maximum capability during most of the time interval. Can work with As a result, the geographic range of the device is The rate is significantly lower than expected or covers a desired range. Although very powerful equipment is prepared, its operation rate is low.   Principle by careful selection of code words that encode possible data words It is generally known that this problem can be reduced. This technology See, for example, Electronics Letters, 8th December 1994, vol. 30, no. 209 "Block codi" by A.E.Jones, T.A.Wilkinson and S.K.Barton3, 8-2099 ng scheme for reduction of peak to mean envelope power ratio of multicar rier transmission schemes (peak-average envelope power ratio for multicarrier systems) Block coding method for reducing the number of pixels). However, PME Good code error correction when selecting code words to reduce PR It is also important to make sure you have the property. In addition, high throughput The implementation of the communication system of the present invention requires long code words and a fast encoding / decoding process. It would be desirable to include the use of lossiers. That is, especially during decoding (about 16 The use of long code words (typically words or more) is typically If the number of valid code words to be examined is large, a known simple lookup Use of a table uses compact (and low-power, inexpensive) circuitry It does not help to perform quick actions. Thus, for example, a data word or code .For example, performing combinational logic to perform interconversion on word symbols Or at least limit the size of the lookup table used Encoding and decoding operations are definable in terms of such analytical procedures It is desirable.   In addition, make selections to define the code used in a typical system. Another problem arises from the small amount of code words that are targeted. Co Possible even with binary symbols as the number of deword symbols increases The number of unique code words increases very rapidly. Three (ternary) or four ( 4 For symbols with possible values, the number of available codes is very short. The word is also very large, so simply to minimize PMEPR But simply searching for all possible code words is not practical in real time. It is possible. Attempts to meet error correction and implementation requirements are also particularly desirable. , But at the same time, PMEPR, error detection / correction and encoding / A code word set that meets favorable criteria for ease of decryption is actually It is unknown whether it exists or not, so we simply add the difficulty of such a search. I can't do it. Many different data words (i.e., data containing a relatively large number of symbols) Data words) to achieve an acceptable high data rate. A large number of code words for use in the selected code This difficulty is exacerbated by the desire to be available.   That certain types of code can have some desirable properties Is known. IEEE Transactions on Communications, vol.39, no.7, July 1991, pp.1031-1033, B.M.Popovic's paper "Synthesis of Power Effic Ient Multitone Signals with Flat Amplitude Spectrum Is less than 6dBV (equivalent to PMEPR) What Bainarima for building multi-tone multi-frequency signal having a) crest factor Or teach that polyphase complementary sequences can also be used. example For example, IRE Transactions on Information Theory, vol. IT-7, Apr. 1961, pp.82-8 Binary complement of 'Complementary series' by M.J.E.Golay in 7 places As seen in the case of sequence pairs, some of these sequences Examples are disclosed. However, Provic's paper above mentioned that multipath signal fluctuations And it is aimed at situations like multi-frequency jams. Required in these situations Can continuously use a single codeword to generate a signal Wear. Thus, Provic's paper describes PMEPR in the actual code that carries the information. To specify multiple sequences to achieve the 6dBV limit for Do not provide guidance on what No other requirements are mentioned. The Propovic dissertation states, Construction of multifrequency signals with peak amplitudes is still old without an analytical solution This is a problem. "   Golay complementary sequence pairs and the codes defined using them Note that this is different from Golay code, and should be confused. There is no. Golay complementary sequence pairs and Golay codes are each identical It was originally defined by a researcher and is therefore named after that researcher. They are just called.   Very large despite the potentially huge search space for possible code words The PEMPRs are related in such a way that different code words can actually be defined. Provide predictable low limits, specify error detection and correction characteristics, and It is an object of the present invention to provide an encoding and decoding method and apparatus which can be implemented in a computer. This is the purpose of Ming.Disclosure of the invention   In accordance with one aspect of the invention, a coded orthogonal frequency division multiplexing scheme ( That is, transmission using a multi-carrier frequency division multiplex communication system such as COFDM). For 2^mTo encode a data word as a single symbol code word Is provided. The method determines the values of the data words and the desired Regarding the peak average envelope power ratio characteristic of One or more of the linear subcodes of the code having a generator matrix From the coset set, determine the desired peak average envelope power ratio for the above transmission. The data words according to the code words selected in the form of Encode. However, in this case, z is from 0 to 2^mIs an integer value up to -1, and y is m X is an integer value from 0 to 0, and x is a factor represented as a binary number of (m + 1) bits. Indicates a bit unit multiplier, and the result of the division is expressed as a one-digit value.   In accordance with another aspect of the invention, a coded orthogonal frequency division multiplex communication method is provided. Transmission using a multi-carrier frequency division multiplexing scheme such as the equation (i.e., COFDM). 2 for sending^mHow to encode a data word as a code word of one symbol A law is provided. The method comprises determining the values of the data words and the location of the transmission. Regarding the desired peak average envelope power ratio characteristics, One or more of the linear subcodes of the code having a generator matrix From the coset set, determine the desired peak average envelope power ratio for the above transmission. The data words according to the code words selected in the form of Encoding and decoding each codeword using an analytic encoder Including the step of: However, in this case, z is from 0 to 2^mAn integer value up to -1 , Y is an integer value from m to 0, and x is represented as a binary number of (m + 1) bits. Indicates the bit unit multiplier of the factor, and the result of the division is expressed as a single digit value.   In accordance with yet another aspect of the present invention, the coded orthogonal frequency division multiplexing Use a multi-carrier frequency division multiplexing scheme such as 2 where m is at least 4^mAs a symbolic code word A method is provided for encoding a data word. The method is based on Value and the desired peak average envelope power ratio characteristic of the above transmission. regarding,One or more of the linear subcodes of the code having a generator matrix From the coset set, determine the desired peak average envelope power ratio for the above transmission. The data words according to the code words selected in the form of Encode. However, in this case, z is from 0 to 2^mIs an integer value up to -1, and y is m X is an integer value from 0 to 0, and x is a factor represented as a binary number of (m + 1) bits. Indicates a bit unit multiplier, and the result of the division is expressed as a one-digit value.   In accordance with yet another aspect of the invention, a coded orthogonal frequency division multiplexor is provided. Use multi-carrier frequency division multiplexing such as duplex communication (i.e., COFDM). Where m is at least 4 and j is at least 3,^j 2 with different possible values of^mData words as code words for A method for encoding a code is provided. The method includes determining the values of the data words, And for the desired peak average envelope power ratio characteristic of the transmission, One or more of the linear subcodes of the code having a generator matrix From the coset set, determine the desired peak average envelope power ratio for the above transmission. The data words according to the code words selected in the form of Encode. However, in this case, z is from 0 to 2^mIs an integer value up to -1, and y is m X is an integer value from 0 to 0, and x is a factor represented as a binary number of (m + 1) bits. Indicates a bit unit multiplier, and the result of the division is expressed as a one-digit value.   According to another aspect of the present invention, for transmission over a communication path, j is a positive integer. Each variable is 2^j2 with different possible values of^mAs the code word of the symbol A method for encoding a data word is provided. The method comprises the steps of: Value and z is from 0 to 2^mHas an integer value of -1 and y is an integer from m to 0 Has a value and x is a bitwise multiplication of a factor represented as a (m + 1) -bit binary number. And the result of the division is represented as a single digit value, One or more of the linear subcodes of the code having a generator matrix Data according to the code with the code word of value j contained in the set of cosets -Encode the word.   According to yet another aspect of the invention, z is between 0 and 2^mHas an integer value up to -1, y Has an integer value from m to 0, and x is a factor expressed as (m + 1) -bit binary number. Indicates the bit-wise multiplication of the child, the result of the division is represented as a single digit value, 2 where the code symbol is j> 2^jWith different possible values of Within a set of codewords having a generator matrix of^mIndividual A method is provided for identifying a code word of a symbol. In the method, The matrix for the trix row has multiple fast Hadamard transforms or equivalent Expression 2^mApply to the symbol input word, combine the result of the conversion and By identifying the words, the coefficients for the rows of the matrix are derived. You.   In accordance with yet another aspect of the present invention, an orthogonal frequency division multiplexing system (ie, COFDM) for transmission using a multi-carrier frequency division multiplex communication system.^m An encoder is provided that encodes the data word as the code word of the symbol. It is. The encoder receives a data word, and the value of the data word, And z is from 0 to 2^mY has an integer value from m to 0. Where x is a bit-wise multiplication of a factor expressed as a (m + 1) -bit binary number, Assuming that the result of the division is represented as a single digit value, One or more of the linear subcodes of the code having a generator matrix From the coset set, determine the desired peak average envelope power ratio for the above transmission. The code received according to the code word selected in the form of a complete coset Means for encoding a word is provided.   The circuit embodying the invention is relatively simple and compact. Thus, for example, The identification of code words used for encryption is only for relatively long code words. No need to waste time searching through millions of code words. same As such, decoding also includes all valid code words that may be present (long Requires a huge lookup table (not practical for code words) do not do. Time wastage that was previously believed to be necessary or at the limit of potential Practical embodiments of the invention as opposed to costly searches and large look-up tables Simplicity and compactness illustrate the significant advances provided by the present invention .BRIEF DESCRIPTION OF THE FIGURES   Data is encoded using coded orthogonal frequency division multiplexing (COFDM). Method of encoding and decoding data in a transmitting communication system according to the invention The method and apparatus will be described with reference to the accompanying drawings as follows.   FIG. 1 is a block diagram illustrating multipath propagation between a base station and a mobile communication device.   FIG. 2 is a block diagram illustrating the effect of multipath propagation on the propagation of a single symbol. You.   FIG. 3 is a block diagram illustrating intersymbol interference derived from multipath propagation.   4a to 4h show symbols according to the use of the multi-carrier frequency division multiplexing scheme. It is a block diagram which shows the reduction of inter-cell interference.   FIG. 5 shows an encoder and a modulator used in a COFDM system. It is a block diagram.   FIG. 6 is a block diagram of a COFDM system incorporating the encoder and modulator of FIG. FIG.   FIG. 7 shows the general form of a COFDM signal waveform without any PMEPR restrictions. FIG.   FIG. 8 encodes a 5-bit data word as an 8-bit code word 1 is a block diagram illustrating an encoder and a decoder incorporating the present invention.   FIG. 9 encodes an 8-bit data word as a 16-bit code word. FIG. 2 is a block diagram showing an encoder to be converted.   FIG. 10 is for decoding the code words generated by the encoder of FIG. FIG. 3 is a block diagram illustrating a decoder incorporating the present invention.   FIG. 11 is a block diagram for decoding a code word generated by the encoder of FIG. FIG. 4 is a block diagram illustrating an alternative embodiment of a decoder that can be used.   FIG. 12 shows a 13 bit data word as a 16 symbol quaternary code word. FIG. 1 is a block diagram illustrating an encoder incorporating the present invention for encoding a code.   FIG. 13 decodes a quaternary code word generated by the encoder of FIG. FIG. 2 is a block diagram illustrating a decoder incorporating the present invention.   FIG. 14 shows an 18 bit data word as a 16 symbol octal code word. FIG. 1 is a block diagram illustrating an encoder incorporating the present invention for encoding a code.   FIG. 15 decodes the octal code word generated by the encoder of FIG. FIG. 2 is a block diagram illustrating a decoder incorporating the present invention.Embodiment of the Invention   FIGS. 1 to 3 show multi-path transport under conditions of mobile communication in an urban environment. It is a figure for showing a related problem. The base station 10 is connected via radio, i.e. It communicates with a mobile communication device such as a mobile telephone 12 at the wave wavelength. Base station 10 and telephone 12 There is a direct signal path 14 between them, but in the form of buildings and other structures Due to the presence of multiple reflectors (for microwave signals) multiple reflected signal paths 16 Also exists. Typically, the lengths of these paths are different and, therefore, There are various propagation delays to arrive. For example, the general The signal is the sum of the various signals arriving at the phone through different paths.   The transmitted signal modulates the carrier (e.g., by phase shift keying) It can be considered to include consecutive symbols. As shown in FIG. Signal received over one symbol as a result of signals over multiple paths with transit delay The resulting hybrid signal 18 is an exact replica of the transmitted signal 20 for that symbol. Absent. In particular, the received hybrid signal typically has a longer duration than the transmitted signal. Is long and has a large amplitude aftermath after the exact duplication of the transmitted signal is completed Have a minute. As shown in FIG. 3, this aftermath portion nominally ends at time T. If the pulse extends for a significant part of the symbol period, the aftermath It overlaps the following symbol and interferes with its reception. As a result, subsequent symbols are May be received differently. That is, the symbol detected by the receiving device Values are correct as sent, even in the absence of other sources of noise. May not be properly expressed. This type of signal destruction is referred to as intersymbol interference (ISI). Known.   The effect of ISI is that the received symbol is multipath propagated, as shown in FIG. The duration of the transmitted symbol so that it extends beyond the delay-induced aftermath. It can be reduced by increasing the time. The ensuing symbol rate As a countermeasure against the decrease, a plurality of carriers each modulating one of the plurality of carriers. It has been proposed to transmit symbols simultaneously. Therefore, FIGS. As shown, if eight carriers are used, the symbol period on each carrier is , Assuming that the same overall symbol rate is maintained (as shown in FIG. 3). 8) eight times that required for a single carrier. Frequency between multiple carriers If the interval is an integer multiple of the reciprocal of the symbol period, the It is possible to recover each symbol stream without interference from other carriers It is. This technique is known as orthogonal frequency division multiplexing (or OFDM). Have been.   Error detection and correction by using block coding in addition to using multiple carriers It has been proposed to grant positive abilities. In this case, as shown in FIG. A continuous stream of symbols (in this example, binary symbols) consists of five symbol Is conceptually divided into contiguous groups or words that contain files. 5 symbols Is a group of 5 bits that output the corresponding 8-bit code word. 8 bit (5B / 8B) encoder 30. This word word Generate eight carriers at eight consecutive frequencies at intervals of interval Δf Eight phase shift controllers for controlling one bank of oscillators 34 Applied to one bank of the controller 32. The individual bits of the code word In each case, the corresponding phase shift controller is either positive or negative. Given the phase shift ΔΦ, and depending on its current value, Therefore, the generated carrier signal is modulated.   The modulated output of oscillator 34 is combined in summer 36 and the transmitted signal Is generated. In practice, the functions of the oscillator 34 and the adder 36 As indicated by the dotted line 38, the inverse fast Fourier transform (ie, IFF T) is conveniently combined in a digital signal processor implementing T). Thus, Referring to FIG. 6, data to be transmitted is received at block 40. , Is converted into a parallel format by the serial / parallel converter 42. If the parallel data is The data encoded by the encoder 30 and encoded by the modulator / IFFT 32/3 8 to control the generation of the multi-carrier signal. This multi-carrier signal is a D / A converter The data is converted to analog form at the data 44, amplified and sent to the transmitter 46. It is.   Reception is essentially the reverse of the above process as follows. The signal is transmitted by the receiver 48 , Amplified and converted to digital form by the A / D converter 50. You. The digital signal is a digital signal processor that implements a fast Fourier transform (FFT). The demodulator 52 is provided with a demodulator 52 provided with a Is adjusted. Demodulated data reverses the encoding applied by encoder 30 And more typically passed to a decoder 54 that performs error detection and correction functions. You. Thereafter, the parallel-sequential converter 56 converts the data into a sequential format and outputs the data to an output unit 58. Send to   The selection of the code performed by the encoder 30 determines the efficient operation of the transmitter 46. Has important implications for Multi-carrier signals have equal amplitude and equal spacing. Contains the sum of several sinusoids with digit frequencies. As a result, the maximum The logarithmic value is greatly affected by the relative phases of the constituent sine waves. These relative files Phase should be transmitted during the next symbol period at the end of each symbol period. It is changed by the phase shift controller 32 according to the value of the code word. Will be updated. A specific set of relative phases, in other words the current code word According to the value, the amplitude of the transmitted signal fluctuates but does not change as shown in FIG. There will always be a distinct peak that occasionally appears with small value intervals. This The peak average envelope power ratio (PMEPR) of the transmitted signal, such as Is relatively large.   In order for such a signal to be transmitted without distortion, the amplitude of the transmitted signal must be When most of the dynamic range is reached, Transmitter 46 uses a linear amplifier, which is relatively efficient. It is necessary to operate at a low rate.   The set of code words actually used to encode the data for transmission It has been suggested that appropriate choices reduce this problem. Extreme of PMEPR The use of block coding to avoid the use of code words to create T. EEE 45th Vehicular Technology Conference, July 1995, pp. 825-829. A. Wilkinson and A.E. "Minimisation of the peak to mean env" by Jones elope power ratio of multicarrier transmission schemes of block coding ( The maximum peak-to-average power envelope ratio of the multi-carrier transmission system with block coding This proposal suggests that PMEPR can be reduced. But at the same time enable efficient implementation of encoding and decoding and demodulation That provide the required capabilities to detect and correct errors in the He points out the difficulty of choosing a set of word words. Address this issue Possible ways of doing this are discussed, but no actual solution is given. this A modification to the approach was made at the 1996 IEEE 46th Vehicular Technology Conference April 1996, pp. "Combined coding for error" by the same author at 904-908 control and increased robustness to system nonlinearities in OFDM (OFD Combinatorial Coding for Error Control in M and System Nonlinearity In this modification, the desired error correction The linear block code is selected to have positive and detection characteristics, then the PM Code to maintain equivalent error detection / correction characteristics while reducing EPR To exploit code linearity and redundancy to convert code values to new values Have been. In the above paper, 4B / 7B, 4B / 8B and 11B / 15B Code is given as an example, in which case it is modulated according to each code word All possible code words in PMEPR order for the multicarrier signal obtained Procedures that involve listing values identify the required transformations. ing. Such an exhaustive search approach (for example, an order of 15 bits) (E.g.) can be executed on relatively short code words, The number of possible code words to be increased rapidly increases with the length of the code word. You. Potentially over 4,000,000,000, even for 32-bit code words Code words must be listed, which can be time consuming if successful And, in practice, probably should be avoided. Code word Ensures that a proper subset of the code is actually present in the set being inspected The inability to do so can further amplify the above problem.   Popovic's paper cited above describes a single binary or polyphase complementary sequence. Suggests the use of However, the 1996 paper cited above does not Kens is not easy to correct in relation to their error correction / detection capabilities Suggest that.   Thus, error correction and detection, modulated according to the code word concerned The maximum PMEPR of the multi-carrier signal, and (eg, including the use of combinatorial logic) Meet desired criteria for simple and efficient implementation of practical encoders and decoders The problem of identifying sets containing many different code words still exists You. The magnitude of this problem is an 8-bit code word as shown in FIG. By examining an example of encoding a 5-bit data word using It can be recognized. In the general case, there are 256 possible code words, of which Of 32 must be selected to encode 32 possible data words Absent. What specific code word combinations may affect compliance with the above criteria If there is no knowledge of the effect, all 32 possible 256 A subset of the code words must be considered. That is, those possible sub The set covers the number of orders 256! / (224! * 32!) = 1041. If an 8-bit Assuming that the data word must be encoded using 16 bits, A possible subset to consider is 65536! / (65280 * 256!) = 10726 Number of orders.   FIG. 8 shows a 5B / 8B encoder 60 and a decoder 80 having the following characteristics. Is shown. That is, Error detection: minimum Hamming distance (i.e., a valid codeword The value cannot be changed, for example by interference, to convert it to a valid code word. The minimum number of symbols that must be obtained is 2 (to generalize this, 2^mPieces The distance for a binary codeword containing the symbol^m-2), Maximum PMEPR of a multi-carrier signal modulated according to an 8-bit code word Is, for example, 3 dB for a 6 dBV power level; The block code for the data to be encoded must reach a valid high data rate Include enough different code words to allow for generation ,and, Encoder 60 and especially decoder 80 are easily implemented using combinatorial logic What you can do, It is.   Referring to FIG. 8, encoder 60 has five data bits at its input. Receive. These lower 4 bits are respectively assigned to the corresponding binary multipliers 62-6. 8 and the fifth most significant bit is connected to the selector 70. Each multiplier 62-6 8 is a multiplier 62: 0101 0101, a multiplier 64: 0011 0011, and a multiplier 66: 0000 111 With each fixed 8-bit value of 1 and multiplier 68: 1111 1111, It is configured to multiply the received input bits. In practice, each multiplier is 8 One logical AND gate is used to divide the input bits into each of the fixed 8-bit values. Implemented to combine with bits separately. The outputs of these four multipliers are It is connected to an exclusive OR circuit (XOR) 72 in units of Bitwise binary addition of the default values (ie, the addition modulo 2 of the corresponding significant bits) Be executed.   Depending on whether the most significant value of the input bit is 0 or 1, the selector 70 Select one of two possible 8-bit values: 0001 0010 or 0000 0110 You. The selected value is sent to the fifth input of the XOR circuit 72 and the 8-bit code ・ Word is output. The encoder 60 of FIG. 8 has 32 possible 5-bit data bits. The data words are encoded into 8-bit code words as shown in Table 1 below.  Decoder 80 receives an 8-bit code word at its input, and Is applied to an 8-bit secondary Reed-Muller (RM) decoder 82. This recovery For example, F.J. is described in North-Holland, Amsterdam, 1986, pp. 385-388. MacWi lliams, N.J.A. Sloane's The Theory of Error-Correcting Codes -Theory of Correction Codes) '. -Use a decoding algorithm. This decoder combines the decoding with at least the code word. Errors affecting the number of bits in the input code word up to the limit on code length. Perform error detection / correction simultaneously. The RM decoder 82 constructs the output value of the RM decoder. And generating two sets of output signals. A "secondary" set of signals 84 (3 in this case) Signals) identify the “family” or cosets of the output values and the “primary” Bit 86 (in this case, four signals) identifies a particular output value within that coset. You. The first four signals 86 are the data output by the entire decoder 80. Create directly the decoded value for at least the lower 4 bits of the word. Secondary Are connected to a selector 88, where they are output by an overall decoder 80. The value of the fifth most significant bit of the input data word is generated. In particular, Selector 88 compares the three secondary signals with the two stored 3-bit values. . If the secondary signal 84 has the value 101, the selector determines the value of the most significant output bit as binary. Giving the number 0 and giving the value 110 gives a binary number 1. If signal 84 is not 101 or 110 If so, the RM decoder converts the received 8-bit code word into Table 1 above. Recognizes that the code is not listed in Is indicated by a signal at the error output 90.   Alternative decoding, described below with reference to FIG. 11, in terms of a 16-bit codeword The technique uses a valid code word (a code word that can be output by an encoder). ) Only.   The error detection and correction capability of the secondary RM decoder 82 itself is different from that of the RM decoder 82. It is inherited by the entire decoder 80 by being incorporated. Thus, as described above, The minimum Hamming distance of the node is 2 (2 when m = 3).^m-2= 2). The decoder 80 Up to 2 affecting the bits forming the code word^m-2-1 error detected Can be Therefore, when m = 3, the detectable error is 1. Descriptive Assuming, for convenience, that the code word is 8 bits, the decoder 80 can be used in the general case. The number of correctable errors is 2^m-3Since it is -1 and equal to zero if m = 3, Error correction cannot be guaranteed in practice. Single in received codeword For bit errors, if one of two valid 8-bit codes is In some cases, changing the single bit position creates the same receiving code word. Will be affected. In practice, the above decoding techniques may be effective Arbitrarily choose one of the possible codes, so correcting the error May be (although not guaranteed). For longer code words (see below) Error correction is possible in the case.   PMEP of a multi-carrier signal modulated according to the 8-bit code word of Table 1. R is 3 dB or less. Decoder 80 uses an analysis technique (ie, code (Using systematic patterns within a set of words) It can be implemented at least partially using combinatorial logic. Selector 88 is Backup table, which contains the entire set of codewords. Moreover, the embodiment of the look-up table is not essential.   For the codes in Table 1, the acceptable data rates (ie, code words There are 32 code words that provide 5 data bits per second). Decoder 80 Although the code itself in the table includes a known secondary RM decoder 82, the code of Table 1 Note that the code is not included. The secondary Reed-Muller code is , A non-linear subset of the second order Reed-Muller code of length 8, as described above. It is a novel feature and has a combination with many advantages. Such codes have been recognized to date Never. Similarly, the decoder 80 provides a 7B / 8B secondary Reed-Muller code Prior art 8-bit secondary RM decoder for decoding data encoded according to Do not use   The desired combination of features implemented by encoder 60 and decoder 80 of FIG. The 5B / 8B code having the characteristic is derived as follows and has a relatively long code word. Extended to generate code in code. The starting point for the 5B / 8B code is " An 8 bit value generator matrix defining the "base" code as follows: (0000 1111) (X₁) (0011 0011) (X_Two) (0101 0101) (X_Three) (1111 1111) (X ') Each row of this matrix is represented by X for convenience in the following description.₁, X_Two, X_ThreeAnd the mark X ' No. is attached.   The basis code is a combination of all 16 linear combinations of the rows of this generator matrix. , But the format of the combination is as follows. a₁X₁(+) A_TwoX_Two(+) A_ThreeX_Three(+) A'X ' However, the symbol (+) indicates bit-wise addition modulo 2, a₁, A_Two, A_ThreeAnd a 'is 0 or Is a coefficient that takes a value of 1. (Note: the sign (+) indicates a factor in this specification including the accompanying drawings. In some cases, it may be represented as a cross in a circle.) This base code The code is linear, that is, any two of the code words The result of the bitwise XOR operation on the word is one other code in the code. Is a word. The subcode of this code is the code word of the base code. Any subset (and in particular may be whole code) Is a subcode whose code is linear in itself. In practice, the subcode is Enough of the codewords in the base code to make up the system Must be included. The actual ratio is determined by various parameters, The length of the word and the number of possible values for each symbol in the codeword ( In the case of a binary number, it depends on two values. Present binary code word of length 8 , The line containing eight of the sixteen code words in the base code The shape subcode may be sufficient. Longer code words and more For possible symbol values, one-half (1/4, 1/8, 1/16, etc.) Smaller percentages may be sufficient.   Row X₁, X_TwoAnd X_ThreeBitwise multiplication of all possible pairs of₁* X_Two, X₁* X_Three And X_Two* Adds three additional rows derived by X3 By combining with the matrix, an 8-bit generator array is created. (0000 0011) (X₁* X_Two) (0000 0101) (X₁* X_Three) (0001 0001) (X_Two* X_Three) The bitwise multiplication, denoted by the symbol *, is the corresponding position of the two values to be multiplied. Multiplies the bits in the position, or performs bitwise logic on the two values Perform an AND operation.   These additional rows are bitwise exclusive OR (XOR) operations (ie, bitwise Combined by the addition modulo 2), (for a code word of length 8) A next value or a second coset representative value is generated. The first coset representative value is the row (X₁* X_Two)When( X_Two* X_Three) Is generated as follows. (0001 0010) (X₁* X_Two) (+) (X₁* X_Three) The remaining coset representative value is the row identifier X₁* X_TwoAnd X_ThreeAll of the subscripts of By identifying the possible permutations, it is generated as follows. (0001 0100) (X₁* X_Three) (+) (X_Two* X_Three) (0000 0110) (X₁* X_Two) (+) (X₁* X_Three) (X₁* X_Two) Is (X_Two* X₁) And the same value as (X_Two* X_Three) Is (X_Three* X_Two), So (X₁* X_Two) (+) (X_Two* X_Three) And (X_Three* X_Two) (+) (X_Two* X₁) Have the same value and no difference. As well To the permutation (X₁* X_Three) (+) (X_Three* X_Two) Indicates which additional rows are included in each coset representative value The equivalent expression (X₁* X_Three) (+) (X_Two* X_ThreeReplaced by) I have.   Finally, each of these coset representative values is calculated by a bitwise exclusive OR operation. Combined with all code words in the base code, the base as shown in Table 2 below A total of 48 code words that make up a set of three cosets of codes Generated.   All of these 48 code words have a combination of the aforementioned characteristics, Which of these 48 codewords to choose in the general case Can also be. In practice, use many code words that are integer powers of two Preferably, the incoming data stream is blocked for encoding. Is simplified, and the code words are represented by the same coset representative. It is easy to select and implement code words in the 16 groups. Obedience Therefore, the number of selected coset representative values is larger than the number of available coset representative values. Not the largest power of two. In the current case, three are available , The corresponding maximum power of 2 is 2¹That is, 2. Select two coset representative values Selecting provides 32 code words. Therefore, all data The corresponding maximum binary data word length that can encode a word is shown in Table 1. 5 bits.   In the case of the encoder 60 and the decoder 80 shown in FIG. As shown, the coset representative value (X₁* X_Two) (+) (X_Two* X_Three) (I.e., 0001 0010) And (X₁* X_Two) (+) (X₁* X_Two) (I.e. 0000 0110) is arbitrary Has been selected. It is also selected according to the value of the most significant bit of the input data word. The values themselves selected by the factor 70 are these two coset representative values.   A particular input of a particular coset representative value, in fact a particular input bit position (most significant bit) The assignment of the selector 70 and the decoder 80 is optional. The selectors 88 must be consistent. Provided by RM decoder 82 The three secondary signals 84 provided are added to the additional row (X₁ * X_Two), (X₁* X_Three) And (X_Two* X_Three). 3 bits stored in selector 88 Each individual bit of the value corresponds to each of these rows. The value of each 3-bit is Associated with a specific value for the upper data bit, one of three additional rows Is the remainder assigned to that data bit value (at encoder 60). Indicates whether it is currently present in the class representative value. Therefore, a data bit having the value 0 Is the coset representative value (X₁* X_Two) (+) (X_Two* X_Three) Is assigned, and the corresponding line ( X₁* X_Two) (+) (X_Two* X_Three) In the coset representative value is related to its data bit value. The position of one bit of the 3-bit value 101 stored in the selector 88 is used as a target. Is shown.   Also, as shown in FIG. 8, in the encoder 60, the four least significant data Row X of generator matrix to bits_Three, X_Two, X₁And X 'are multiplied. RM decoder 8 2. The primary signal 86 provided by 2 is the coefficient a for each of these rows._Three, A_Two, A₁, corresponds to a '. The assignment of a particular input data bit to each of these rows is Assuming that the same assignment is used in encoder 60 and decoder 80, Optional. This means that a particular 8-bit code word The possibility of assignment to a 5-bit data word depends on the individual to each of these rows. Are governed by the input data bit allocation of Which data bits control the selection of the coset representative in selector 70 Is assigned by the selection and the selection of the coset that makes that selection. That's what it means.   As mentioned above, the code word in Table 2 is row X₁, X_Two, X_ThreeAnd X ' Sets of cosets of linear subcodes (in this case, all codes) with a detector matrix I do. The use of this type of base code gives the desired error detection and correction characteristics. You. The selection of the linear subcode is performed, for example, by using combinational logic, Encoding and decoding can be performed in a compact manner. Surplus The corollary is generally to convert the base codes to different parts of the total space of possible code words. It is considered to be systematically shifting or translating. Remainder of linear subcode Certain undesirable (such as all zeros or all ones) by extra use It is possible to avoid code words, so that in particular the value of PMEPR is Although higher, the error control characteristics of the base code are maintained to some extent. like that Selection of code words from the coset set allows for better control of PMEPR You. This is because, as described above, each coset of the base code has a widely similar related P This is because they tend to include code words with MEPR. Coset representative value is higher In certain cases, such as those having an identified format as described above, PMEPR is 3 dB Is observed.   The encoding and decoding capabilities of the present invention are suitable for relatively long code words as follows: Can be used. To retain the advantages of error control and ease of implementation , The code word length is preferably an integer power of two. Thus, the length of the next code word greater than 8 is 16. 16 bit The generator matrix for the code word is as follows. (0000 0000 1111 1111) (X₁) (0000 1111 0000 1111) (X_Two) (0011 0011 0011 0011) (X_Three) (0101 0101 0101 0101) (X_Four) (1111 1111 1111 1111) (X ') The 16-bit base code consists of all 32 lines in this generator matrix row. Combinations of forms, including combinations of forms, include: a₁X_Two(+) A_TwoX_Two(+) A_ThreeX_Three(+) A_FourX_Four(+) A'X ' Where a₁, A_Two, A_Three, A_FourAnd a ′ take a value of 0 or 1.   Row X₁, X_Two, X_ThreeAnd X_FourDerived by bitwise multiplication of all possible pairs of The following three additional rows are combined with the above 16-bit generator matrix. Produces the following corresponding generator array: (0000 0000 0000 1111) (X₁* X_Two) (0000 0000 0011 0011) (X₁* X_Three) (0000 0000 0101 0101) (X₁* X_Four) (0000 0011 0000 0011) (X_Two* X_Three) (0000 0101 0000 0101) (X_Two* X_Four) (0001 0001 0001 0001) (X_Three* X_Four)   These additional rows are combined by a bit-wise exclusive-OR (XOR) operation, Generate two quadratic coset representative values. The first coset representative value is the row (X₁* X_Two), (X_Two* X_Three ) And (X_Three* X_Four) Is generated as follows. (0001 0010 0001 1101) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X_Three* X_Four) The remaining coset representative value is the row identifier X₁, X_Two, X_ThreeAnd X_FourAll of the subscripts By identifying the possible permutations, it is generated as follows. (0001 0100 0001 1011) (X₁* X_Two) (+) (X_Two* X_Four) (+) (X_Three* X_Four) (0000 0110 0011 0101) (X₁* X_Three) (+) (X_Two* X_Three) (+) (X_Two* X_Four) (0001 0100 0010 0111) (X₁* X_Three) (+) (X_Three* X_Four) (+) (X_Two* X_Four) (0000 0110 0101 0011) (X₁* X_Four) (+) (X_Two* X_Four) (+) (X_Two* X_Three) (0001 0010 0100 0111) (X₁* X_Four) (+) (X_Three* X_Four) (+) (X_Two* X_Three) (0001 0001 0010 1101) (X₁* X_Two) (+) (X₁* X_Three) (+) (X_Three* X_Four) (0001 0001 0100 1011) (X₁* X_Two) (+) (X₁* X_Four) (+) (X_Three* X_Four) (0000 0011 0110 0101) (X_Two* X_Three) (+) (X₁* X_Three) (+) (X₁* X_Four) (0000 0101 0110 0011) (X_Two* X_Four) (+) (X₁* X_Four) (+) (X₁* X_Three) (0000 0101 0011 1001) (X₁* X_Three) (+) (X₁* X_Two) (+) (X_Two* X_Four) (0000 0011 0101 1001) (X_Two* X_Three) (+) (X₁* X_Two) (+) (X₁* X_Four) Specify which additional rows are included in each coset representative value as in the case described above. For purposes, some permutations have been replaced by equivalent expressions.   Finally, each of these coset representative values is calculated by a bitwise exclusive OR operation. Combined with all code words in the 16-bit base code, 384 codes that make up a set consisting of 12 cosets of such base codes A word is generated. All of these code words have the following properties corresponding to the properties described above. Error detection: minimum Hamming distance 4 (2 when m = 4)^m-2= 4). Error correction The encoder has up to three (2) that affect the bits that make up one code word.^{m- Two} -1 = 3) error is detected and one such error (2^m-3Correct -1 = 1) Can be The PME of the multi-carrier signal modulated according to the 16-bit code word of Table 3 PR does not exceed 3 dB. A sufficient number of commands to be able to achieve an acceptable data rate Words can be used.   (Use multiple code words that are integer powers of two and have the same coset representative. Select the code words in one of the 32 groups) , The maximum power of 2 not exceeding 12 (which is the number of coset representative values) is 2^ThreeThat is, 8 It is. Thus, in the preferred case, the remainder of each of the 32 code words Eight of the classes are selected, 256 corresponding to an 8-bit long data word. Code words are provided.   FIG. 9 illustrates the use of the codewords of Table 3 to generate an 8-bit data bit according to the present invention. 8B / 16B encoder 1 for encoding words as 16-bit code words 00 is shown. The encoder provides 16 bits for the 5 least significant bits of the input data word. Set generator matrix row X_Four, X_Three, X_Two, X₁And X ' Includes binary multipliers 102-110. The three most significant bits of the input data word The selector 112 supplies the combination of those three bits. One of the eight possible coset representative values is selected according to the offset value. Shown in FIG. In one embodiment, of the 12 possible coset representatives listed in Table 3, The first eight have been selected for use in selector 112. 5 binary powers The outputs of the arithmetic units 102-1105 and the selector 112 are bit-wise exclusive OR ( XOR) circuit 114 to generate a 16-bit code word. You.   FIG. 10 illustrates decoding of a 16-bit code word back into a corresponding data word. 1 shows a first embodiment of a decoder 120 that performs the following. Code words have large values of m Is a 16-bit secondary RM decoder 12 corresponding to the 8-bit secondary RM decoder 82 of FIG. 2 is supplied. The five primary signals 124 generated by the RM decoder 122 are , The five least significant bits of the data word output by the global decoder 120 Provide the decoded value for These signals 124 are 16-bit generator matrices. Rix Row X_Four, X_Three, X_Two, X₁Coefficient a for X and X '_Four, A_Three, A_Two, A₁And a ' Respond. Six additional rows (X₁* X_Two), (X₁* X_Three), (X₁* X_Four), (X_Two* X_Three), (X_Two * X_Four) And (X_Three* X_Four) Correspond to the six secondary signals 126 from the RM decoder. Selector 12 to generate the values of the three most significant bits of the dynamic output data word. 8 Specifically, the selector 128 determines the values of the six secondary signals 126. With the eight 6-bit values shown in FIG. These 6 bits are needed Subject to appropriate procedures implemented in the form of software or firmware accordingly Thus, it can be stored or derived. Each 6-bit value has three most significant data Data associated with a particular combination of values for the One combination into three data bit values of that combination (encoder 10 Indicates (at 0) whether it is in the assigned coset representative. Second letter If signal 126 has one of the eight stored values, selector 128 The corresponding three data bit values are provided as the upper output bits. The signal is Otherwise, the received 16-bit code word is the first eight bits of Table 3 It is regarded by the RM decoder 122 as not being included in the cosets. An event is indicated by a signal at the data error output section 130 of the selector 128 Is done.   Has a function to process error correction code, d = 2^m-2Is the decoder 120 Any decoder having a minimum Hamming distance d such that the transmitted code word Correctly decodes a received code word whose distance from is d / 2. But However, decoder 120 is not a "minimum distance" decoder. The minimum distance decoder is Even if the distance is more than d / 2, the smallest distance from the received code word It has the additional property of decoding code words with separation. In addition, data Generation of a signal in the error output unit 130 is performed by a secondary read by the RM decoder 122. -The output of the Maller code word is an indeterminate result and the encoder 100 Means that it cannot be what is actually output.   FIG. 11 shows the valid code word (ie, the output by encoder 100). 7 shows an alternative 16-bit decoder 140 that outputs only the resulting code word). This Is the minimum distance decoder for this code word set.   The method used in decoder 140 is based on IEEE Trans. Inform. Theory, 1986 , Vol. IT-32, no.1, pp. J.H. Conway and N.J.A. Sloane "Soft decoding techniques for codes and lattices, including the G olay code and the Leech lattice Software decoding technology). Code "is an example of a decoding method. For each coset representative value, in order, The code word is XORed with its coset representative and the result is the length 16 Decoded as element RM (1,4) of the primary Reed-Muller code and at the same time Is assigned a “score”. The determination of which code word to decode is The coset with the highest representative score (if any, any coset of A) is done by choosing   Decoding of the elements of RM (1,4) is described in F.J.Ma, North-Holland, Amsterdam, 1986. cWilliams and N.J.A. 'The Theory of Error-Correcting Co' by Sloane Fast Hadamard Transform Method Described in 'des (Error Correcting Code Theory)' Use Length 2^mThe fast Hadamard transform for a binary sequence of Is length 2^mAssociated with the element pair RM (l, m) of the primary Reed-Muller code of^m Generate transformation components. The largest of all components Are the “scores” for decoding, and the decoding algorithm Component pair to determine which of the pair Select the RM (l, m) element corresponding to the largest component (with sign).   Referring to FIG. 11, decoder 140 exclusions each 16-bit code word. Received by a logical or (XOR) gate 142 where the coset storage from the code word Perform a bitwise subtraction modulo 2 of the value in mechanism 144. (Modulo 2 performances In the case of arithmetic, bitwise subtraction and addition are equal, and the bitwise exclusive OR gate is It can be used for any operation. The function of gate 142 is aligned with the description below. It is defined as subtraction for compatibility. ) The coset storage unit 144 includes the encoder 1 The same eight coset representative values as those selected by the selector 112 of 00 Including. The memory 144 also stores all eight possible 3-bit values in input 1 3. Includes a 3-bit counter 146 that continuously cycles in response to 48 increment signals. . Which of the eight remainder class representative values is exclusive of the 3-bit current value of this counter It is supplied to the OR gate 142, that is, the same residue class as in the case of the encoder. Controls the correspondence between table values and 3-bit values. Thus, the counter 146 is Circulates through the eight possible values of x, each of the eight coset representative values becomes To the user 142 in order.   16-digit binary number output by gate 142 for the current coset representative value The word is sent to a replacement circuit 150 where the fast Hadamard method is applied. To match element values, the word values +1 and -1 are each 2 Are replaced by the base values 0 and 1.   Fast Hadamard Transform (see MacWilliams and Sloane cited above) or Modified version (by Designs, Codes and Cryptography, vol.7, 1996, pp.187-214 A.E. Ashikhnrin and S.N. 'Fast decoding algorithms for Litsyn' first order Reed-Muller and related codes 16) output from the circuit 150 to perform the The symbol word is sent to a so-called “Green machine” 152. Green machine The output of 152 is E₀To E₁₅From the conversion component with the sign Naru (length 2^mSubscript 15 is 2 for the general form of^m-1 Will be replaced). Component E₀Is the row of the code generator matrix Component E corresponding to the leftmost symbol₁₅Corresponds to the rightmost symbol You. These components are sent to a comparator 154. The comparator 154 converts Compares the absolute value of the component and calculates the maximum absolute value | E_z| And its corresponding (0 or (Up to 15). If there is more than one component with the largest absolute value, One of them is arbitrarily selected. The comparator has the value E_zAnd z to latch 156 Output, and the latch 156, along with those values, 6. Receive the corresponding 3-bit index value given by 6.   For the first coset representative, the latch 156 stores the received Ez, z and coset Simply store the index. (For the same 16-bit input code word) With the increment of the counter 146, each successive coset representative value is added to the XOR gate 142. Supplied to_zAnd z are passed to latch 156. like that Transformation component E_zIs greater than the component currently latched , Then the latch returns the newly received value with its associated z and coset Select with the index.   When the counter 146 has cycled through all eight possible values, the contents of the latch 156 Are collected in the output data decoded in block 158 of FIG. z The value is a binary number a where z is 4 bits.₁a_Twoa_Threea_Four4 decoded linesets when represented by a ' Number a_Four, A_Three, A_TwoAnd a₁Is determined directly. The fifth coefficient a 'is the last latched Value E_zIs zero if positive, 1 if negative, and the coset index is Configure the upper output bit.   Especially if the decoding is performed in hardware An embodiment of the fast Hadamard transform has been described above as being performed continuously By giving each a separate Green machine, The required quick Hadamard transform can also be calculated in parallel.   The advantage of the decoder 120 described above is that it operates faster than the decoder 140. Have a point in general. Therefore, the decoder 120 must be main restored to gain speed benefits. It is desirable to use it as a signal. However, the signal 126 from the RM decoder 122 Is encoded by the encoder 100 as indicated by the data error output unit 130. If it is found that it corresponds to a code word that cannot be output The decoded code word can be sent to the decoder 140. Of the decoders 120 and 140 Such a choice between the frequency of occurrence of the signal at the data error output unit 130 Can be adapted based on the current noise level as represented by You. It is a sign that such signals are relatively frequent and the current communication path is noisy. If indicated, select decoder 140 for immediate use for each codeword On the other hand, the decoder 120 can be continuously used in parallel to monitor the communication quality. it can. If the signal at the data error output 130 decreases, the decoder 120 Can again be selected as the primary decoder.   Shown in Table 3 of the 65,536 possible 16-bit code words A particular 384 16-bit code word has all of the different characteristics described above ( I.e. a large minimum Hamming distance, a low value of the PMEPR of the actual implementation, a useful number Possible code words) at the same time, and thus have a significant and very advantageous utility. give. In particular, combining error control characteristics that do not exceed the desired level with low PMEPR Finding a set of code words for the combined relatively long words is a trick. It was an irreparable problem, and there was no certainty that a solution existed. The present invention , Not only such combinations, but also the convenience of implementation and the satisfactory data rate Provide an appropriately sized code word population that can be achieved.   Code sets with corresponding properties for relatively long code word lengths are: Easily identified by extending the above procedure to create Table 3 Can be. Thus, a generator matrix for a 32-bit code word Is as follows. (0000 0000 0000 0000 1111 1111 1111 1111) (X₁) (0000 0000 1111 1111 0000 0000 1111 1111) (X_Two) (0000 1111 0000 1111 0000 1111 0000 1111) (X_Three) (0011 0011 0011 0011 0011 0011 0011 0011) (X_Four) (0101 0101 0101 0101 0101 0101 0101 0101) (X_Five) (1111 1111 1111 1111 1111 1111 1111 1111) (X ') The 32-bit base code is used to calculate all 64 linears in the rows of this generator matrix. Including combinations. That is, a combination of the following formats is used. a₁X₁(+) A_TwoX_Two(+) A_ThreeX_Three(+) A_FourX_Four(+) A_FiveX_Five(+) A'X ' Where a₁, A_Two, A_Three, A_Four, A_FiveAnd a ′ take a value of 0 or 1, respectively.   Put the 32-bit generator matrix in row X₁, X_Two, X_Three, X_FourAnd X_FiveAll possible Additional rows and combinations derived by bit-wise multiplication of various combinations This creates a corresponding generator array. The additional lines are as follows It is. (0000 0000 0000 0000 0000 0000 1111 1111) (X₁* X_Two) (0000 0000 0000 0000 0000 1111 0000 1111) (X₁* X_Three) (0000 0000 0000 0000 0011 0011 0011 0011 0011) (X₁* X_Four) (0000 0000 0000 0000 0101 0101 0101 0101) (X₁* X_Five) (0000 0000 0000 1111 0000 0000 0000 1111) (X_Two* X_Three) (0000 0000 0011 0011 0000 0000 0011 0011) (X_Two* X_Four) (0000 0000 0101 0101 0000 0000 0101 0101) (X_Two* X_Five) (0000 0011 0000 0011 0000 0011 0000 0011) (X_Three* X_Four) (0000 0101 0000 0101 0000 0101 0000 0101) (X_Three* X_Five) (0001 0001 0001 0001 0001 0001 0001 0001 0001) (X_Four* X_Five) These additional rows are combined by a bitwise exclusive OR operation, and a (length 32 60 word quadratic coset representatives are generated. Like this , The first residue class representative value is the row (X₁* X_Two), (X_Two* X_Three), (X_Three* X_Four), (X_Four* X_FiveCombine) The remaining coset representative value is generated by the row identifier X₁, X_Two, X_Three, X_Fourand X_FiveAll possible permutations of the subscript of the expression (X₁* X_Two) (+) (X_Two * X_Three) (+) (X_Three* X_Four) (+) (X_Four* X_Five).   These coset representative values are converted to a 32-bit base by a bit-wise exclusive OR operation. Each of the code words is combined with all the code words and the remainder of the 60 base codes A total of 3840 possible code words that make up the extra set are generated. These particular code words provide the following combination of properties: ・ The minimum Hamming distance of the code is (2 when m = 5)^m-2= 8) 8. -Up to seven error correction decoders affect the bits that make up the code word (2^m-2-1 = 7) errors and up to three (2^m-3Correct the error of -1 = 3) Can be A multi-carrier signal modulated according to these particular 32-bit code words , It has no more than 3 dB of PMEPR. Decoders for use with this code shall be operated using analysis techniques And can be implemented, for example, using combinational logic, at least in part. Can be. Codes available to enable an acceptable data rate to be achieved A word exists. Selected from over 4,000,000,000 possible code words, 32 bits long These particular 3840 32-bit code words are described in the extensive variety of above. At the same time, so that there are practical benefits associated with them .   (Use multiple code words that are integer powers of two, each representing the same coset representative Select a group of 64 code words with values) , The largest power of 2 not exceeding 60 (which is the number of coset representative values) is 25 32 . Thus, in the preferred case, 32 code words each of 32 code words 2048 cosets selected, corresponding to 11-bit binary data word length Are created. In this case, the encoder has six inputs. Force data bit a₁, A_Two, A_Three, A_Four, A_FiveAnd a 'is a 32-bit generator matrix Su row X₁, X_Two, X_Three, X_Four, X_FiveX 'and the input data The remaining 5 bits of the data word are 32 bits according to the combined value of those 5 bits. The selected coset representative value is supplied to a selector for selection. For decoding, a 32-bit secondary RM decoder is used to determine (the data word to be decoded). ), And generates six primary signals and ten secondary signals. This 10 secondary signals store 32 possible 10-bit values with the 10 signals Select to generate the remaining 5 bits of the 11-bit data word to be decoded Sent to Decoding of FIG. 11 using the fast Hadamard transform method as an alternative It is also possible to use a decoder similar to the unit 140.   Length 2^mGeneralized generator matrix for a particular codeword in It will be as follows.Where z is from 0 to 2^mY has an integer value from m to 0. Where x is both represented as (m + 1) -bit binary numbers ((2.z + 1) and 2^yAs A) indicates a bitwise multiplication of the two coefficients (ie, a bitwise logical AND), and The result of the operation is represented by one digit. The first row of this matrix where (y = m) is row X₁Give The second line of (y = m-1) is the line X_TwoAnd the penultimate line (y = 1) is the line X_m And the last row that is (y = 0) gives row X '.   Thus, for example, for a 64-bit long codeword (where m = 6), (y = m− 1 = 5), the first symbol (z = 0) on the second row X2 is ((2.0 + 1) x2^m-1) / 2^m-1 I.e. (1x32) / 32 It is. Expressing the factors involved in the bitwise multiplication as (m + 1) -bit binary numbers, the result is , (0000001x0100000) / 32, This is expressed as (0000000) / 32 or one digit zero. The second symbol (z = 1) on the second row is (2.1 + 1) x2^m-1) / 2^m-1, I.e. (3x32) / 32, which is likewise Generates (0000011x0100000) / 32 or 1 digit 0. The 16th symbol (z = 15) in this row is (2.15 + 1) x2^m-1) / 2^m-1, That is (31x32) / 32, which is likewise (00111111x0100000) / 32 or 1-digit zero is generated. The 17th symbol (z = 16) in this line is (2.16 + 1) x2^m-1) / 2^m-1, That is, (33x32) / 32, which is (0100001x0100000) / 32 or 0100000) / 32 or one digit 1 is generated. The same result is obtained for the 18th through 32nd symbols.   For the 33rd symbol (z = 32), the matrix is ((2.32 + 1) x2^m-1) / 2^m-1Give For example, this yields (65x32) / 32 or (1000001x0100000) / 32 or a single digit zero. Similarly, a result of 0 is obtained for the subsequent 15 symbols.   For the 49th symbol (z = 48), the matrix is ((2.48 + 1) x2^m-1) / 2^m-1Give For example, this yields (97x32) / 32 or (11000001x0100000) / 32 or a single digit 1 . Similar results are obtained for the last 15 symbols below. in this way And the second line as a whole (0000 0000 0000 0000 1111 1111 1111 1111 0000 0000 0000 0000 1111 111111 11 1111).   Third row X_ThreeFor (y = m−2 = 4), the first symbol is ((2.0 + 1) × 2^m-2) / 2^m-2sand That is, (1x16) / 16, and (0000001x0010000) / 16 or 0 is obtained. The following 7 A similar result is obtained for the symbol. The ninth symbol is ((2.8 + 1) x2^m-2) / 2^m-2That is, (17x16) / 16, and (0010001x0010000) / 16 or 1 is obtained. Similar results are obtained for the tenth through sixteenth symbols. That Later, eight zeros and eight one blocks alternate, and the row as a whole is (0000 0000 1111 1111 0000 0000 1111 1111 0000 0000 1111 1111 0000 0000 1 111 1111). The other rows of the generator matrix for the 64-bit code word are similar Derived.   2^mThe bit basis code is a generator matrix that is a combination of the following forms: Include all linear combinations of the rows of. a₁X₁ (+) a_TwoX_Two(+) A_ThreeX_Three(+) .... (+) a_mX_m(+) A'X ' Where a₁, A_Two, A_Three, ...., a_mAnd a ′ take a value of 0 or 1.   The corresponding generator array is 2^mRow X with bit generator matrix₁, X_Two, X_Three, ... .X_mAdditional rows and pairs derived by bitwise multiplication of all possible pairs of Created by matching. These additional rows are used for bit-wise exclusive OR operation. Therefore, they are combined to generate m! / 2 secondary coset representative values of the base code. You That is, the first coset representative value is the row (X₁* X_Two), (X_Two* X_Three), ... and (X_m-1* X_m) The remaining coset representatives, which are generated by combining Expression (X₁* X_Two) (+) (X_Two* X_Three) (+) .... (+) (X_m-1* X_m) In row identifier X₁, X_Two, X_Three , .... X_mGenerated by identifying all possible permutations of the subscript It is.   Each of these coset representative values is calculated by a bit-wise exclusive OR operation.^mBit Combined with all code words in the base code of the M! / 2 total m! (2^{m + 1}) / 2 possible code words Generated. These code words have the following property combinations: ・ The minimum Hamming distance of the cord is 2^m-2It is. The error correction decoder uses code word Up to 2 affecting the bits that make up the code^m-2-1 error detected, up to 2^{m -3} -1 error can be corrected. A multi-carrier signal modulated according to these code words is a PME of 3 dB or more Has no PR. The decoder used for this code can be operated using analytical techniques Preferably, it can be implemented at least partially using combinatorial logic. Codes available to enable an acceptable data rate to be achieved A word exists.   Total number of binary code words available for each of the code word length ranges And other parameters are as shown below. Other parameters are The number of cosets, two of which can be encoded using the available number of code words Maximum number of data words in integer power, data word length in that case, code ・ The number of primary and secondary signals provided by the RM decoder in terms of word length, and And the selector values stored in the encoder and the decoder as shown in FIGS. Is a number.   For purposes of simplicity and clarity, the above description should have only two symbol values: It is intended for encoding using binary codes. However, the present invention also provides To the encoding using a larger number of symbol values of the power of j. For example, It is also possible to use symbol values of the fourth order of J = 2 and the eighth order of j = 3. Like that In that case, the same generator matrix as described above is used. For example, (m = 3 and 2^m For a codeword containing 8 symbols (= 8), the generator matrix is become. (0000 1111) (X₁) (0011 0011) (X_Two) (0101 0101) (X_Three) (1111 1111) (X ')   An 8-symbol quaternary base code (four possible symbol values) has the form: Contains all 256 linear combinations of the rows of the generator matrix, which are the best combinations. a₁X₁(+) A_TwoX_Two(+) A_ThreeX_Three(+) A'X ' However, (+) is symbol unit addition modulo 2.^j(Ie modulo 4 for quaternary numbers) And a₁, A_Two, A_ThreeAnd a 'are coefficients taking all integer values from 0 to 3. You. This base code is linear, that is, any of the code words The result of the symbol-wise addition modulo 2 for the two is another code word in the code. Mode.   The corresponding generator array consists of combining the generator matrix with additional rows. Is created. These additional lines are line X₁, X_TwoAnd X_ThreeOf every possible pair of Symbolwise multiplication or X₁* X_Two, X₁* X_ThreeAnd X_Two* X_Three, And quaternary encoding It is derived by multiplying by 4 in the case of 2,8 octal encoding. Therefore, 8 symbol The additional lines in the case of quaternary encoding are as follows. (0000 0022) 2 (X₁* X_Two) (0000 0202) 2 (X₁* X_Three) (0002 0002) 2 (X_Two* X_Three) The additional line is a symbol unit addition modulo 2_jCombined by m! / 2 secondary A coset representative value is generated. For quadruple code words of length 8, the first coset Typical values are shown in row 2 (X₁* X_Two) And 2 (X_Two* X_Three) Produced by combining It is. (0002 0020) 2 (X₁* X_Two) (+) 2 (X_Two* X_Three) The remaining coset representative values are given by Equation 2 (X₁* X_Two) (+) 2 (X_Two* X_ThreeRow identifier X in)₁, X_TwoAnd X_Three By identifying all possible permutations of the subscript of Is done. (0002 0200) 2 (X₁* X_Three) (+) 2 (X_Two* X_Three) (0000 0220) 2 (X₁* X_Two) (+) 2 (X₁* X_Three)   Ultimately, each of these coset representative values is a symbol-wise addition modulo 2^jBy Is combined with all code words in the base code, and m! / 2 M! (2^{j (m + 1)}) / 2 code words are generated . In the 8-symbol quaternary case, the 768 code values and And coefficient a₁, A_Two, A_ThreeTable 4 below shows examples of the corresponding values of and a ′. Table 4 And CR₁= 2 (X₁* X_Two) (+) 2 (X_Two* X_Three), CR_Two= 2 (X₁* X_Three) (+) 2 (X_Two* X_Three) And CR_Three = 2 (X₁* X_Two) (+) 2 (X₁* X_Three).  2^mAn individual symbol quaternary base code is a generator matrix that is a combination of the following forms: Includes all linear combinations in the rows of the box. a₁X₁(+) A_TwoX_Two(+) A_ThreeX_Three(+) .... (+) a_mX_m(+) A'X ' Where a₁, A_Two, A_Three, ...., a_mAnd a ′ take a value of 0, 1, 2 or 3, respectively. versus The corresponding generator array is row X₁, X_Two, X_Three, ..., X_mAll possible pairs of symbol Combining additional rows derived by unit-wise multiplication with a generator matrix Created by These additional lines are connected by symbol unit addition modulo 4. And m! / 2 quadratic coset representative values are generated. Thus, the first coset representative value is , Row 2 (X₁* X_Two), 2 (X_Two* X_Three), ..., and 2 (X_m-1* X_mGenerated by combining The remaining coset representative value is given by Equation 2 (X₁* X_Two) (+) 2 (X_Two* X_Three) (+) .... (+) 2 (X_m-1* X_m Row identification X in)₁, X_Two, X_Three, ..., X_mAll possible permutations of the subscript Generated by identifying.   Each of these coset representative values is 2^mBase M! / 2 cosets of the base code, combined with all code words in the code M! (4^{(m + 1)}/ 2 code words are generated.   Use other length codewords or different numbers of symbols (for example, octal) A similar code for a given encoding can be defined in a similar manner. general And the number of symbol values is 2^j, The base code is a combination of the following forms: Contains all linear combinations of rows of the generator matrix. a₁X₁(+) A_TwoX_Two(+) A_ThreeX_Three(+) .... (+) a_mX_m(+) A'X ' Where a₁, A_Two, A_Three, ...., a_mAnd a ′ are 0, 1, 2,.^jTake the value of -1 , (+) Is symbol unit addition modulo 2^jRepresents The corresponding generator array is X₁, X_Two, X_Three, ..., X_mSymbolwise multiplication of all possible pairs of^j-1 Is added by 2^mCombined with generator matrix Created by These additional lines are symbol unit addition modulo 2^jBy And m! / 2 quadratic coset representative values are generated. Thus, the first coset Table value is row 2^j-1(X₁* X_Two), 2^j-1(X_Two* X_Three), ..., and 2^j-1(X_m-1* X_m) And the remaining coset representative is given by Equation 2.^j-1(X₁* X_Two) (+) 2^j-1(X_Two* X_Three) (+ ) .... (+) 2^j-1(X_m-1* X_mRow identification X in)₁, X_Two, X_Three, ..., X_mSubscription Generated by identifying all possible permutations of the   Each of these coset representative values is a symbol unit addition modulo 2^jBy 2^mBase M! / 2 cosets of the base code, combined with all code words in the code M! (2^{j (m + 1)}) / 2 code words are generated.   Discusses error detection and error correction characteristics for 4, 8 and higher order codes The minimum Hamming distance may affect the code word Does not fully describe the scope of the error. This is an error in a single symbol Is up to 2_j-1May move to the adjacent symbol value of Value 2^j-1And 0 are considered adjacent). The actual effect is that each At least two errors in different symbols including only one move to the default value It is at least as serious or more serious than the case. For example, from value 6 to value 1 A single quaternary symbol change (movement through three adjacent values), one from value 6 to 7 Another octal symbol change from 7 to 0 (both move to adjacent value) More serious. Therefore, a useful additional criterion is to identify any valid code words. Minimum symbol value shift required to convert to another valid codeword Is the minimum Lee distance defined as a value. (In the above binary case, the Hamming distance The minimum distance was not discussed since the distance was the same. )   Defined above for quaternary, octal and higher order cases The resulting code word has the following advantageous property combinations: ・ The minimum Hamming distance of the cord is 2^m-2It is. An error correction decoder based on the Hamming distance Up to 2 that affect Bol^m-2-1 can be detected arbitrarily, and , 2_m-3Such errors up to -1 can be corrected. ・ The minimum distance between cords is 2^m-1It is. Error correction decoder based on Lie distance Is up to 2^m-1With a symbol value shift of -1 (same number of codeword symbols Error), and 2^m-2Such up to -1 Movement can be corrected. In fact, the minimum distance decoder based on Lie distance is Certain additional patterns of errors affecting the symbols that make up the code word Arbitrarily (i.e., some of the characteristics of the decoder based on the Hamming distance Prepare). A multi-carrier signal modulated according to these code words has a P No MEPR. ・ Based on either the Hamming distance or the Lee distance used for this code Decoders can be operated using analytical techniques, for example, at least partially It can be implemented using combinatorial logic. Codes available to enable an acceptable data rate to be achieved A word exists.   In practice, these codes represent the information represented by the data bits. Is transmitted over a path including an encoder and a decoder. Binary code The assignment of the data bits to the row coefficients of the generator matrix is Optional, but in all other cases (quaternary, octal, etc.) It is desirable to make careful assignments of data bits.   For a quaternary code, each of the following (selected) data bit pairs is Row coefficient a as shown in Table 5₁, A_Two, A_Three, ..., a_mAnd a '. This association is an example of a Gray code, which includes changes between two adjacent coefficient values. Error (3 and 0 are considered adjacent) causes a single data bit error. Has the advantageous property of Thus, the relatively Small errors can generate large data word errors at the end of transmission. Little ability.   As in the case of the binary case, a number of codewords that are integer powers of two. It is preferable to use an incoming data stream for encoding. Simplifies the division of a stream into blocks and allows codewords to have the same coset representative It is easy to classify the data into two groups. Thus, the remainder to be selected The number of class representative values is the maximum power of two (2) that does not exceed the number of available remainder class representative values.^r ). Next, one coset representative value is selected using the r data bits. In accordance with the Gray code in Table 5 above, the remaining pairs of data bits Number symbol a₁, A_Two, A_Three, ..., a_mAnd converted to a '.   FIG. 12 is similar to encoder 100 of FIG. 9, but in accordance with the present invention, Using a code word derived from the generator array as described above, 13 bits The data word of the octet as a code word of 16 quaternary symbols. 13B / 16Q encoder 160 shown in FIG. (0000 0000 1111 1111) (X₁) (0000 1111 0000 1111) (X_Two) (0011 0011 0011 0011) (X_Three) (0101 0101 0101 0101) (X_Four) (1111 1111 1111 1111) (X ') (0000 0000 0000 2222) 2 (X₁* X_Two) (0000 0000 0022 0022) 2 (X₁* X_Two) (0000 0000 0202 0202) 2 (X₁* X_Four) (0000 0022 0000 0022) 2 (X_Two* X_Three) (0000 0202 0000 0202) 2 (X_Two* X_Four) (0002 0002 0002 0002) 2 (X_Three* X_Four) In particular, this encoder uses eight cosets with coset representatives as follows: (0002 0020 0002 2202) 2 (X₁* X_Two) (+) 2 (X_Two* X_Three) (+) 2 (X_Three* X_Four) (0002 0200 0002 2022) 2 (X₁* X_Two) (+) 2 (X_Two* X_Four) (+) 2 (X_Three* X_Four) (0000 0220 0022 0202) 2 (X₁* X_Three) (+) 2 (X_Two* X_Three) (+) 2 (X_Two* X_Four) (0002 0200 0020 0222) 2 (X₁* X_Three) (+) 2 (X_Three* X_Four) (+) 2 (X_Two* X_Four) (0000 0220 0202 0022) 2 (X₁* X_Four) (+) 2 (X_Two* X_Four) (+) 2 (X_Two* X_Three) (0002 0020 0200 0222) 2 (X₁* X_Four) (+) 2 (X_Three* X_Four) (+) 2 (X_Two* X_Three) (0002 0002 0020 2202) 2 (X₁* X_Two) (+) 2 (X₁* X_Three) (+) 2 (X_Three* X_Four) (0002 0002 0200 2022) 2 (X₁* X_Two) (+) 2 (X₁* X_Four) (+) 2 (X_Three* X_Four) Thus, 2^rHas a value of 8, so r is 3.   Referring to FIG. 12, the 13B / 16Q encoder 160 converts the input data word The least significant 10 consecutive pairs of bits in the corresponding quaternary number according to Table 5 above Includes five Gray encoders 162-170 that encode into symbols. Each encoder 1 62-170 outputs the output quaternary symbol to that of the five multipliers 172-180. Feed one to each of the 16 symbol generator matrix rows X_Four, X_Three, X_Two, X₁and Each corresponding row of X 'is multiplied. Most significant 3 bits of input data word Is supplied to the selector 182, and according to the combination value of those three bits, One of the eight possible coset representatives is selected. Five multipliers 172-1 80 and the output of the selector 182 are output by a symbol unit modulo 4 adder 184. Of the 16 quaternary symbols output by the encoder 160 A code word is generated.   FIG. 13 shows a code of 16 quaternary symbols generated by the encoder 100. 16Q / 1 based on Lie distance to decode words into 13-bit data words 3A shows a 3B minimum distance decoder 200. The method used in decoder 200 is illustrated in FIG. Based on the "super code" cited in connection with the eleven decoders 140 Major fixes to work with quaternary and larger codes Is added. For each coset representative value, in order, for example, a symbol unit The remainder class representative value is subtracted from the received code word using modulo 4 , The result of which is decoded as an element of a length 16 quaternary base code, As described below, a “score” is assigned to the coset representative value. Which co Determining whether or not to decode to a word depends on whether the representative value has the highest score Select any one of them), and within the range of the coset, This is done by selecting the element of the identified base code.   The fast Hadamard transform is used for decoding the elements of the base code. However However, in contrast to decoder 140, decoder 200 is as if receiving base codes. Collectively decode the result of the above subtraction as if it were a stripped codeword. The symbol so that it has a value of +1 or -1 (as in the case of a decoder) Applies a fast Hadamard transform to multiple input vectors that are not constrained. Each input vector For a torque, the largest conversion factor is selected. According to the sign of the conversion coefficient, 1 One or more of the associated pairs of the next Reed-Muller code RM (1, m) are output. this The method is similar to the procedure for decoding the received RM (1, m) codeword. However, the output for each input vector determines the decoding of that vector by RM (1, m). Not expressed as a element.   Referring to FIG. 13, a decoder 200 outputs a quaternary signal in a modulo-4 subtractor 202. Receives a code word of the number 16 symbols and modulo the remainder from the code word The value stored in the memory 204 is subtracted. This coset storage mechanism is an encoder Same eight cosets from which selection is made by 160 selectors 182 Includes table values. The storage unit 204 also stores each of the eight remainder class representative values in order in the subtractor. In response to the increment signal at input 208, It has a 3-bit counter 206 that continuously cycles through the eight possible 3-bit values.   16-th quaternary number output by the subtractor 202 with respect to the current time coset representative value The bol word S is passed to the vector generator 210. The vector generator 210 For the word S, 32 16-symbol input vector sequences D Sent to a Green machine 212 that performs a fast Hadamard transform on each of the vectors You. In this case, the generator 210 has five coefficients b₁, B_Two, B_Three, B_FourAnd 0 for each of b ' Or 1 so that their coefficients as a whole are 32 possible combinations. It will be everything. For each such combination, follow the relation The value B is derived. B = b₁X₁(+) B_TwoX_Two(+) B_ThreeX_Three(+) B_FourX_Four(+) B'X ' Next, D is derived by the following equation. D = | 2X '-((S-B) mod 4) | -X' In this equation, subtractions with values such as X ', S and B are denoted by | ... | Is performed in symbol units as the modulo operation being performed.   The output of the Green machine 212 is, for each vector D, E₀To E₁₅Sign up to There are 16 conversion components attached. These components are the comparator 214, the absolute values of the transform components are compared and the maximum absolute value | Ez | And the corresponding value z (from 0 to 15) is identified. Components with the same absolute value If there are a plurality, one of them is arbitrarily selected. Next, the comparator 214 calculates the value z and E_zIs output to the latch 216, and the latch 216 additionally outputs the signal from the generator 210. Coefficient b₁, B_Two, B_Three, B_FourAnd the value of b ′ and the counter 206 of the storage 204 Receive the corresponding 3-bit index value provided.   1st coset representative value and coefficient b₁, B_Two, B_Three, B_FourAnd the combination of the values of b ' The latch 216 receives the received E_z, Z, the five coefficients and the coset index Simply remember. 31 additional vectors D are generated by the generator for the first coset representative value As derived by 210, the counter 206 is also incremented and each successive remainder is incremented. As more than 32 vectors D for coset representative values are derived, E_zAnd z Each value, along with the coefficient and coset index values, are passed to latch 216. You. Such a transformation component E_zIs the component currently latched If the absolute value is greater, latch 216 stores the newly received value in relation to it. Connected z, coset index and coefficient b₁, B_Two, B_Three, B_FourAnd the value of b ', select.   When counter 206 has cycled through all eight possible values, the contents of latch 216 Are collected in the output data decoded in block 218 of FIG. z Value, E_zSign and coefficient b₁, B_Two, B_Three, B_FourAnd b 'together The value of the least significant 10 bits is determined directly as described below, and the coset index is Construct the three most significant output bits.   To determine the value of the least significant 10 output bits, the coefficient c 'is E_z1 if positive, positive Is set to 0 and the value of z is c₁c_Twoc_Threec_FourExpressed as a 4-bit binary number Is done. a_Four, A_Three, A_Two, A₁And the five decoded row coefficients a ′ are given by It is determined. a₁= b₁+ 2c₁ a_Two= b_Two+ 2c_Two a_Three= b_Three+ 2c_Three a_Four= b_Four+ 2c_Four a '= b' + 2c ' The decoded data bits are based on these row coefficients by reference to the Gray code in Table 5. Can be derived in pairs from   For an octal code, three data bits are line-aligned, as shown in Table 6 below. Number a₁, A_Two, A_Three, ..., a_mAnd a '.   As in the case described above, the number of selected coset representative values depends on the available coset representatives. The largest power of two that does not exceed the number of values (2^r). Next, r data bits are One coset representative value is selected using the octal Glay code in Table 6 above. A continuous pair of remaining data bits is an octal symbol a₁, A_Two, A_Three, ..., a_mYou And a '.   FIG. 14 is similar to encoder 100 of FIG. 9 and encoder 160 of FIG. However, according to the present invention, the code derived as above from the following generator array: Words are used to convert an 18-bit data word into an octal 16 symbol code. 18 shows an 18B / 160 encoder 160 that encodes as dewords. (0000 0000 1111 1111) (X₁) (0000 1111 0000 1111) (X_Two) (0011 0011 0011 0011) (X_Three) (0101 0101 0101 0101) (X_Four) (1111 1111 1111 1111) (X ') (0000 0000 0000 4444) 4 (X₁* X_Two) (0000 0000 0044 0044) 4 (X₁* X_Three) (0000 0000 0404 0404) 4 (X₁* X_Four) (0000 0044 0000 0044) 4 (X_Two* X_Three) (0000 0404 0000 0404) 4 (X_Two* X_Four) (0004 0004 0004 0004) 4 (X_Three* X_Four) In particular, the encoder uses eight cosets with the following coset representatives: (0004 0040 0004 4404) 4 (X₁* X_Two) (+) 4 (X_Two* X_Three) (+) 4 (X_Three* X_Four) (0004 0400 0004 4044) 4 (X₁* X_Two) (+) 4 (X_Two* X_Four) (+) 4 (X_Three* X_Four) (0000 0440 0044 0404) 4 (X₁* X_Three) (+) 4 (X_Two* X_Three) (+) 4 (X_Two* X_Four) (0004 0400 0040 0444) 4 (X₁* X_Three) (+) 4 (X_Three* X_Four) (+) 4 (X_Two* X_Four) (0000 0440 0404 0044) 4 (X₁* X_Four) (+) 4 (X_Two* X_Four) (+) 4 (X_Two* X_Three) (0004 0040 0400 0444) 4 (X₁* X_Four) (+) 4 (X_Three* X_Four) (+) 4 (X_Two* X_Three) (0004 0004 0040 4404) 4 (X₁* X_Two) (+) 4 (X₁* X_Three) (+) 4 (X_Three* X_Four) (0004 0004 0400 4044) 4 (X₁* X_Two) (+) 4 (X₁* X_Four) (+) 4 (X_Three* X_Four) Thus, 2^rHas a value of 8, so r is 3.   Referring to FIG. 14, the 18B / 160 encoder 230 converts the input data word Is a octal number corresponding to a set of three consecutive bits of the least significant 15 bits according to Table 6 above. Includes five Gray encoders 232-240 for encoding into symbols. Each encoder 2 32-240 outputs its octal symbol to that of the five multipliers 242-250. Feed one to each of the 16 symbol generator matrix rows X_Four, X_Three, X_Two, X₁and Each corresponding row of X 'is multiplied. Most significant 3 bits of input data word Is supplied to the selector 252, and according to the combination value of those three bits, One of the eight possible coset representatives is selected. Five multipliers 242-2 50 and the output of the selector 252 are output by the symbol unit modulo 8 adder 254. 16-symbol octal code output by the encoder 230 -A word is generated.   FIG. 15 corresponds to the decoding shown in FIG. Convert the generated 16-symbol octal code word to an 18-bit data word 1 shows a 160 / 18B minimum Lie distance decoder 260.   Referring to FIG. 15, a decoder 260 performs octal octal conversion in a modulo-8 subtractor 262. Receives a code word of the number 16 symbols and modulo the remainder from the code word The value stored in the memory 264 is subtracted. This coset storage mechanism is an encoder Same eight cosets from which the selection is made by the selector 252 of 230 Includes table values. The coset storage 264 also subtracts each of the eight coset representative values. In response to the increment signal at input 268, the A 3-bit counter 266 that continuously cycles through all eight possible 3-bit values Have.   Octal 16-thin output by subtractor 262 with respect to the current time coset representative value The bol word S is passed to the vector generator 270. The vector generator 270 For the word S, 1024 16-symbol input vector sequences D Green machine 272 performing a fast Hadamard transform on each of these vectors Send to In this case, the generator 270 has five coefficients b₁, B_Two, B_Three, B_FourAnd each of b ' Are assigned 0, 1, 2 or 3 so that their coefficients collectively are 1024 All possible combinations. For each such combination, A value B according to the following relational expression is derived. B = b₁X₁(+) B_TwoX_Two(+) B_ThreeX_Three(+) B_FourX_Four(+) B'X ' Next, a vector D is derived from the following equation. D = | 4x '-((S-B) mod 8) | -2X' As before, in this equation subtraction with values such as X ′, S and B Performed symbol-wise as a modulo operation, indicated by |.   The output of the Green machine 272 is, for each vector D, E₀To E₁₅Sign up to There are 16 conversion components attached. These components are the comparator 274, the absolute values of the transform components are compared, and the maximum absolute value | E_z| And And the corresponding value z (from 0 to 15) is identified. Components with the same absolute value If there are a plurality, one of them is arbitrarily selected. Next, the comparator 274 calculates the value z and E_zTo latch 216, which in addition generates 270 From the coefficient b₁, B_Two, B_Three, B_FourAnd the value of b 'and the counter 266 in the storage 264 Accordingly, the corresponding 3-bit index value supplied is received.   1st coset representative value and coefficient b₁, B_Two, B_Three, B_FourAnd the combination of the values of b ' The latch 276 receives the received E_z, Z, 5 coefficients and coset indexes Simply remember. 1024 additional vectors D are generated for the first coset representative value. As derived by generator 270, counter 266 is also incremented and each successive As more than 1024 vectors D relating to the coset representative value are derived, E_zYou And z, together with the coefficient and coset index values, Passed to 6. Such a transformation component E_zIs currently latched If it has an absolute value greater than the , Its associated z, coset index and coefficient b₁, B_Two, B_Three, B_FourAnd b ' Select with value.   When the counter 276 cycles through all eight possible values, the contents of the latch 276 Are collected in the output data decoded in block 278 of FIG. z Value, E_zSign and coefficient b₁, B_Two, B_Three, B_FourAnd b 'together The value of the least significant 15 bits is determined directly as described below, and the coset index is Construct the three most significant output bits.   To determine the value of the 15 least significant output bits, the coefficient c 'is E_z1 if positive, positive Is set to 0 and the value of z is c₁c_Twoc_Threec_FourExpressed as a 4-bit binary number Is done. a_Four, A_Three, A_Two, A₁And the five decoded row coefficients a ′ are given by It is determined. a₁= b₁+ 4c₁ a_Two= b_Two+ 4c_Two a_Three= b_Three+ 4c_Three a_Four= b_Four+ 4c_Four a '= b' + 4c ' Next, the decoded data bits are derived from the above coefficients by referring to the Gray code in Table 6. Is done.   An encoder corresponding to that shown in FIGS. 12 and 14 is replaced with a code of another length (m ≠ 4). Can be similarly constructed for codewords or higher order (j> 3) code. Noh. As described above, the coset representative value is selected using the r data bits. , A contiguous group of remaining data bits of size j according to the Gray code table Each symbol a₁, A_Two, A_Three, ..., a_mAnd converted to a '. Symbol a₁, A_Two , A_Three... a_mAnd each row X of the generator matrix of the base code in a '₁, X_Two, X_Three, ..., X_mAnd X 'are multiplied, and the result is symbolized to the selected coset representative. Unit modulo 2^jIt is added by calculation.   A decoder corresponding to that shown in FIGS. A similar construction can be made for higher order codes. Symbol length is 2^mso Each symbol is 2^jIn the general case of a code word with possible values of , The subtractor modulo 2 from the received code word^jRemainder class by operation Decrease table value. Coefficient b₁, B_Two, ..., b_mAnd each of b 'is from 0 to 2^j-1Integer up to -1 Have a number, so their coefficients are 2^{(j-1) (m + 1)}Possible combinations of It will cover everything. For each such combination, Values B and D are derived. B = b₁X₁(+) B_TwoX_Two(+) ... (+) b_mX_m(+) B'X ' D = | 2^j-1X '-((S-B) mod2^j) | -2^j-2X ' However, the symbol (+) is symbol unit addition modulo 2.^jRepresents Each lower output video When determining the value of the bit, z is the binary number c₁c_Two... if expressed as cm Row coefficient a₁, A_Two, ..., a_mAnd a ′ are determined by the following equation: a₁= b₁+2^j-1c₁ a_Two= b_Two+2^j-1c_Two ... a_m= b_m+2^j-1c_m a '= b' + 2^j-1c '   The decoder is, as described above, a fixed decision decoder, i.e. 0 And 2^jTake an integer value between -1. Use a similar decoder as a flexible decision decoder You can also. In this case, received according to the size of the analog signal received Codeword symbols are 0 and 2^jIs represented as being a real number between. this The value S is the modulo 2 of the coset representative value (of integer values)^jObtained by subtraction. Up Following the decoding procedure, a real-valued transformation component is generated, The one with the largest absolute value is selected. Subsequent decryption procedures are described above. It is similar to that of   The decoders 200 and 260 of FIGS. 13 and 15 Provide data directly. The codewords that receive the coset representatives , And decodes the result as an element of the associated base code. Do not explicitly identify code words in the code as intermediate steps. all For the value B and all stored coset representative values, the value Z and the maximum absolute value are Transformation component E_zTo get the value of this base codeword, One of the element pairs of the primary Reed-Muller code RM (1, m) with the corresponding value B It is possible to select Degree 2^mSylvester-Hadamard line The column z of column H (see below) is replaced to generate the vector c (most of the matrix The left column is column 0). E_zIs positive, the + symbol in c becomes 0, and the-symbol Is converted to 1 and if negative, the + symbol in c becomes 1 and the-symbol in c becomes 0. Is converted. c now contains the required RM (1, m) element. Selected by the decoder Can be derived by the following equation: Two^j-1C (+) B Row coefficient a to be decoded₁, A_Two, ..., a_mAnd a 'apply Gaussian elimination in a known manner By doing this, the code word and row X of the generator matrix₁, X_Two, ... X_mAnd from X '. Decode code corresponding to the output from the encoder The code is symbol unit modulo 2^jThe base code word is To the coset representative value identified by the marked coset index Is obtained by Sylvester-Hadamard matrix H of degree 2_TwoIsIt is. That is, all elements are the same except for the lower right negative. Next highest Matrix H of powers of 2_FourTo derive_TwoH to the right and down of_TwoAnd its negative (-H_Two ) Is placed at the lower right. Thus, H_FourIs It is. This process subsequently proceeds with the higher power of two Sylvester-Hadamard matrix. It is repeated to derive. For example, H₈To derive_FourH to the right and down of_FourDuplicate And its negative (-H_Four) Is placed at the lower right.   For example, a 13-bit data word 1 1011 0110 0001 is 13B / 16Q encoded Let us consider the case applied to the input of unit 160. Corresponding row factor a_Four, A_Three, A_Two, a₁And a 'have the values 2, 1, 3, 0 and 1, respectively, and are the codes in the base code Give the word 1320 0213 1320 0213 The selected coset representative value is 0002 0002  0020 2202. Therefore, the output quaternary code word is 1322 0211 13 00 is 2011. During communication from the encoder 160 to the decoder 200, the code word Assume that the code is broken and has a value of 1322 0212 1300 2031. In this case receive Transmitted code word has a Lee distance of 3 from transmitted code word .   The subtractor 202 of the decoder converts each coset representative value into the received codeword or Decrease. In the case of a 110-residue coset, the result of this subtraction is 1320 0210 1320 0233 Which is passed to the vector generator 210. The generators are each Green Generate 32 vectors D that are combined in machine 212 to produce a row vector E 16 conversion components E₀To E₁₅Generate Each such vector E In the corresponding row vector D, the 16th-order Sylvester-Adamer Matrix H₁₆Is the result obtained by multiplyingThese vectors and the corresponding transform components E are given by You. (B1, b2, b3, b4, b ') = (0,0,0,0,0):   B = (0000 0000 0000 0000), D = (0,0, -1,1,1, -1,0,1,0,0, -1,1,1, -1.0,0),   E = (1, -1, -1,9, -1, -7,1, -1,1, -1,1,1, -1,1,1, -1). (B1, b2, b3, b4, b ') = (0,0,0,0,1):   B = (1111 1111 1111 1111), D = (1, -1,0,0,0,0,1,0,1, -1,0,0,0,0, -1, -1),   E = (-1,5,1,3,1,3, -1,5,3,1, -3, -1, -3, -1,3,1). (B1, b2, b3, b4, b ') = (0,0,0,1,0):   B = (0101 0101 0101 0101), D = (0, -1, -1,0,1,0,0,0,0, -1, -1,0,1,0,0, -1),   E = (-3,3,3,5, -5, -3, -3,3,1,1,1,1,1,1,1,1) ..... (B1, b2, b3, b4, b ') = (0,1,1,0,0):   B = (0011 1122 0011 1122), D = (0,0,0,0,0,0,0, -1,0,0,0,0,0,0,0,0),   E = (-1,1,1, -1,1, -1, -1, -1, -1,1,1, -1, -1, -1, -1, -1,1). (B1, b2, b3, b4, b ') = (0,1,1,0,1):   B = (1122 2233 1122 2233), D = (1, -1,1, -1, -1,1, -1,0,1, -1,1, -1, -1,1,1,1 ),   E = (1,3, -1, -3, -1,13,1,3, -3, -1,3,1,3,1, -3, -1). (B1, b2, b3, b4, b ') = (0,1,1,1,0):   B = (0112 1223 0112 1223), D = (0, -1,0, -1,0,1,0,0,0, -1,0, -1,0,1,0,1),   E = (-1,1,1, -1, -7,7, -1,1, -1,1,1, -1,1, -1, -1,1). ... (B1, b2, b3, b4, b ') = (1,1,1,1,0)   B = (0112 1223 1223 2330), D = (0, -1,0, -1,0,1,0,0,1,0,1,0, -1,0,1,0),   E = (1,3, -1, -3, -1,5,1,3, -3, -1,3,1, -5,1, -3, -1). (B1, b2, b3, b4, b ') = (1,1,1,1,1):   B = (1223 2330 23303001), D = (1,0,1,0, -1,0, -1,1,0,1,0,1,0, -1,0, -1),   E = (1, -1, -1,1,7,1,1, -1,1, -1, -1,1, -1,9,1, -1).   The conversion coefficient of the maximum absolute value for this coset is B = (1122 22331122 2233) E generated for z = 5_Five= 13. This coefficient is also stored in the storage unit 20 of the decoder. From the transform coefficients generated for any of the other seven coset representative values in Large in absolute value. H₁₆Column 5 is (+-+-+-++-+-+-+). E_FiveIs positive Therefore, the + symbol is replaced by 0 and the-symbol is replaced by 1, so that C = (0101 1010  0101 1010). Therefore, the base codeword selected by the decoder is 2 C (+) B = (1320 0213 1320 0213), and this is the code output from the encoder 160. Was deemed a word, and the communication error was corrected.   The encoder 200 and the decoder 260 shown in FIGS. Based on the Lie distance for quaternary, octal and higher order codes The minimum distance decoder may be implemented in the form of dedicated hardware. in this case , As described above in connection with the decoder 140 of FIG. Some or all, for example by preparing a Green machine for each one Can be calculated in columns. Alternatively, use a general purpose processor and Appropriate software procedures incorporating functionality expressed in C language programs ー It is also possible to execute ja. This program is -Each symbol has a length determined by the parameter M (equal to m above) A code word with a number of values determined by parameter J (equal to j above) ,and, -List of coset representative values And input -The position in the above list of coset representative values for which the maximum absolute value of the conversion coefficient is to be detected, A base codeword partially but identified from the position z of the transform coefficient; and, -Corresponding row factor a₁, A_Two,…, A_ThreeAnd the value of a ', Is output. Only intended for use with codes of quaternary or higher order However, this program is also useful for binary codes.   The following is an example of the source code of such a program.   The secondary binary RM decoders 82 and 122 shown in FIGS. It can be implemented in hardware, or a C language program such as Use a general-purpose processor that executes appropriate software procedures, such as It can also be implemented by doing.   Length 2^mLength 2 for symbol quaternary code^m-1Binary decoding for code Decoding can also be accomplished by using a detector. This form is (quaternary signal Ease of implementation and operating speed (by avoiding the need for a circuit to act on the signal) It is desirable from the viewpoint of. Such decryption is performed on rows with the values 2,1,3,0 and 1. Coefficient a_Four, A_Three, A_Two, A₁Base code words 1320 0213 1320 0213 And coset representative values 0002 0002 0020 2202 (used already in the previous example By reference to the 16-symbol quaternary code word 1322 0211 1300 2011. Is illustrated. This code word representing the binary data word 11011 0110 0001 Code is received as a code with the error 1322 0212 1300 2031 Assume that Each symbol of the received codeword is first represented by the Gray in Table 5. According to the code, they are converted into binary symbol pairs, and the binary symbols are Assigned to each half of a 32-bit word. In this way, first For the quaternary symbol (1), the corresponding binary symbol pair is 01, The first symbol (0) of the pair of bits constitutes the first symbol of a 32-bit word. , The second symbol of this pair (1) constitutes the seventeenth symbol of the word I do. The second quaternary number is 3, the corresponding binary symbol pair is 10, The first symbol (1) of this pair comprises the second symbol of a 32-bit word. The second symbol (0) of this pair replaces the 18th symbol of the word Constitute. Thus, processing continues for the remaining quaternary code words, 0111 0101 0100 1010 1011 0111 1000 1001 Is generated.   This binary word is supplied to a 32-bit secondary binary RM decoder. this Is similar to the case of the decoder 122 of FIG. Has been extended. This decoder produces two sets of binary output signals. That One is the coefficient u₁, U_Two, U_Three, U_Four, U_FiveAnd six primary signals representing u ′ and the second Is a set of coefficients u₁₂, U₁₃, U₁₄, U₁₅, U_{twenty three}, U_{twenty four}, U_{twenty five}, U₃₄, U₃₅And u₄₅To These are the ten secondary signals represented. For a 32-bit word, the output signal of the RM decoder The signals have the following values: u₁= 1 u₁₂= 0 u_{twenty three}= 1 u_Two= 0 u₁₃= 1 u_{twenty four}= 1 u_Three= 0 u₁₄= 1 u_{twenty five}= 0 u_Four= 1 u₁₅= 0 u₃₄= 1 u_Five= 1 u₃₅= 0 u '= 0 u₄₅= 1   These primary and secondary signals are sent to a binary to quaternary converter, where Thus the row coefficient a corresponding to the input quaternary data word₁, A_Two, A_Three, A_FourAnd a 'made Is done. a₁= (2u_Two+ u₁₂(1 + 2u₁)) Mod4 [= 0] a_Two= (2u_Three+ u₁₃(1 + 2u₁)) Mod4 [= 3] a_Three= (2u_Four+ u₁₄(1 + 2u₁)) Mod4 [= 1] a_Four= (2u_Five+ u₁₅(1 + 2u₁)) Mod4 [= 2] a '= 2u' + u₁                 [= 1] The values in [] are related to the specific values of the primary and secondary signals given above. The results of the engagement are shown. 5 decoded row coefficients a_Four, A_Three, A_Two, A₁And it for a ' It is shown that quaternions 2, 1, 3, 0 and 1 are generated respectively.   The coefficient converter also calculates the coefficient a according to the following relation:₁₂, A₁₃, A₁₄, A_{twenty three}, A_{twenty four}You And a₃₄Are generated. a₁₂= (U_{twenty three}+ u₁₂u₁₃) Mod2 [= 1] a₁₃= (U_{twenty four}+ u₁₂u₁₄) Mod2 [= 1] a₁₄= (U_{twenty five}+ u₁₂u₁₅) Mod2 [= 0] a_{twenty three}= (U₃₄+ u₁₃u₁₄) Mod2 [= 0] a_{twenty four}= (U₃₅+ u₁₃u₁₅) Mod2 [= 0] a₃₄= (U₄₅+ u₁₄u₁₅) Mod2 [= 1] These additional signals are based on which a 16 symbol quaternary code word is generated. Six additional rows 2 (X₁* X_Two), 2 (X₁* X_Three), 2 (X₁* X_Four), 2 (X_Two * X_Three), 2 (X_Two* X_Four) And 2 (X_Three* X_Four) Indicates which of the combinations was combined. For data word 11011 0110 0001, the coset representative value is 2 (X₁* X_Two) (+) 2 (X₁* X_Three ) (+) 2 (X_Three* X_Four) And the associated coefficient a₁₂, A₁₃And a₃₄Is 1 to indicate this Value.   Six coefficient signals a₁₂, A₁₃, A₁₄, A_{twenty three}, A_{twenty four}And a₃₄Using the combination of FIG. By the same selector means as the selector 128 in the decoder 120 of FIG. The three most significant bits of the data word can be derived. The lower 10 bits are Row coefficient a decoded with reference to Gray code 5₁, A_Two, A_Three, A_FourAnd a 'from a' Derived.   It should be noted that the length 2^{m + 1}Bit of code 2 using a binary decoder^mThe decoding of a symbolic quaternary code word is as follows: It requires reference to the Gray code table for two purposes. One purpose is as shown in FIG. Error detection as described above in connection with encoders 160 and 230 of FIG. To assist in the transmission of data, including output and correction functions, over a complete communication path. You. The second purpose allows the use of a binary decoder circuit, whereby the quaternary Saves money and effort in implementing circuits that process signals directly to increase operating speed That is.   Binary decoder for the general case of a quaternary code word containing 2m symbols To apply, as described above, the code word uses the Gray code in Table 5. Corresponding 2^{m + 1}Symbolically converted to a bit binary code word. This , Each symbol in each pair of binary symbols of the Gray code is Assigned to each half of the binary code word. This binary code ・ Word is coefficient u₁, U_Two, ..., u_{m + 1}, U ', u₁₂, U₁₃, ..., u_{m.m + 1}To give 2 Using a hex decoder, length 2^{m + 1}Element of the secondary Reed-Muller code That is, it is decoded as RM (2, m + 1). These coefficients are given by the following relations: It is converted to the required quaternary output coefficients. a '= 2u' + u₁ a_i= (2u_{i + 1}+ u₁,_{i + 1}(1 + 2u₁)) Mod4 (when 1 = <1 = <m) a_ik= (U_{i + 1, k + 1}+ u_{1, i + 1} u_{1, k + 1}) Mod2 (when 1 = <i <k = <m) Value a₁, A_Two,…, A_mAnd a ′ are 2 (m + 1) decoded data via the Gray code in Table 5. -Give a bit. The remaining bits are the coefficient a₁₂, A₁₃, ... a_{m-1, m}According to the value of It is restored with reference to the look-up table of the coset selected for use.   2 of each symbol in each pair of Gray code binary symbols_{m + 1}Bit Assignment of each binary code word to each half is not the only possibility. Coefficient u₁, U_Two, ..., u_{m + 1}, U ', u₁₂, U₁₃, ..., u_{m, m + 1}To a quaternary output coefficient To convert the binary symbols to the respective positions of the binary code word. An alternative method of assigning to the location is possible. For example, the quaternary code to be decoded De word is of the same order as the corresponding quaternary symbol in the quaternary code word. Can be extended by concatenating pairs of Gray code binary symbols You. Thus, the quaternary word 1322 0212 1300 2031 becomes the binary word 0110 11 11 0011 0111 0110 0000 1100 1001 In this case, the relational expression for coefficient conversion is as follows. a '= 2u' + u_{m + 1} a_i= ((2u_i+ u_{i, m + 1}) (1 + 2u_{m + 1})) Mod4 (when 1 = <1 = <m) aik = (u_{i, k}+ u_{i, m}u_{k, m + 1}) Mod2 (when 1 = <i <k = <m)   Also, using a decoder having the format of the 16Q / 13B decoder 200 in FIG. It is also possible to replace the hexadecimal processing with the quaternary processing. This is the In that case, the received quaternary codeword and coset storage 204 Both the output coset representative values are 3 in symbol units using the Gray code in Table 5. By modifying the decoder to be converted to a two-bit code word, It can be implemented. Correspondingly, subtractor 202 modulates instead of modulo 2 It is modified to perform b4 operation.   The 32-bit data output by the subtracter 202 for the current time coset representative value. The code S is sent to the vector generator 210. This vector generator is also 16 Two 32-bit input vector sequences D for each of those vectors To supply to a Green machine 212 that performs a fast Hadamard transformation You. That is, the correction vector generator 210 calculates the total number of possible combinations of 16 coefficients. Four coefficients b to cover all the₁, B_Two, B_ThreeAnd b_Four0 or 1 for each of Assign the value of For each such combination, the quaternary 16-symbol value is It is derived according to the relational expression. (B₁X₁+ b_TwoX_Two+ b_ThreeX_Three+ b_FourX_Four) Mod4 This value is also calculated in the same way as the input quaternary codeword and the current next coset representative. Is converted to a 32-bit word B. The above relationship is for the uncorrected decoder 200 But not identical to the original for the value B of In particular, the modified function The relation does not include the coefficient b ′.   Each vector D is a 32-bit word S and a respective 32-bit word B And its value in the vector D + 1 And -1 are replaced by the binary values 0 and 1, respectively. Next, the vector D is a Green machine 2 modified to perform a fast Hadamard transform of length 32 12 is supplied.   The output of the Green machine 212 is 32 conversion components for each vector D And they are the comparator 2 (modified to receive 32 components) 14 and latch 216, all of the values of B for each of the coset representative values. With maximum absolute value and corresponding z value (from 0 to 31) over Component E_z, Four coefficients b₁, B_Two, B_ThreeAnd b_Four, And coset indexes Is selected. Selected z value, E_zSign and coefficient b₁, B_Two, B_ThreeAnd b_FourIs As described below, the value of the lower 10 bits to be decoded is determined, while the coset index is determined. The mix forms the upper 3 bits output.   To determine the value of the lower 10 bits to be output, the coefficient C ′ is E_zTo 1 if is negative , Is set to 0 if positive, and the value of z is a 5-bit binary number c₁c_Twoc_Threec_Fourc_FiveExpressed as It is. 5 row coefficients a_Four, A_Three, A_Two, A₁And a ′ are determined by the following equation: a₁= (2c_Two+ b₁(1 + 2c₁)) Mod4 a_Two= (2c_Three+ b_Two(1 + 2c₁)) Mod4 a_Three= (2c_Four+ b_Three(1 + 2c₁)) Mod4 a_Four= (2c_Five+ b_Four(1 + 2c₁)) Mod4 a '= 2c' + c₁ Next, the data bits to be decoded are stored in these rows by reference to the Gray code in Table 5. It is derived in pairs from the coefficients.   2^mThis is the general case of a quaternary code word containing symbols. Codewords and cosets to use the modified decoder 200 in a simple way The representative value is symbolically converted to a corresponding binary code word. Binary number Number b₁, B_Two,…, B_mIs 2^mCover the possible combinations of values. The value B is (b₁X₁+ B_TwoX_Two+ ... + b_mX_m) Mod4 The quaternary value of 2^{m + 1}Derived by converting to a bit binary word . To determine the value of the 2 (m + 1) least significant output bits from the row coefficients, the value of z is Number c₁c_Two... c_{m + 1}And the row coefficient a to be decoded₁, A_Two, ..., a_mAnd a 'is Determined by the formula. a_i= (2c_{i + 1}+ b_i(1 + 2c₁)) Mod4 (when 1 = <1 = <m) a '= 2c' + c₁   The above description of the quaternary code is based on a quaternary code with possible values of 0, 1, 2, 3. It was done on a symbol. For actual transmission, for example, quadrature phase shift key-in By means of modulation (i.e., QPSK) modulation, these values will be Corresponding to One possibility is that the complex vectors 1, i, -1 and (as i = √-1) Phase shifts of 0, 90, 180 and 270 degrees (equivalent to You. However, other variants are possible. For example, complex vector (1 + i) / √2, (-1+ i) / √2, (-1-i) / √2 and (1-i) / √2 are equivalent to 45 ° It may be 315 degrees. This principle works for codes with many (j> 2) symbol values. And the actual symbol value is any 2^jIdentifiable signal modulation And their amplitudes need not be equal, For shift keying, the phase differences between adjacent pairs need not be equal.   The above-described method and apparatus for encoding and decoding a COFDM signal provides for error detection and And correction characteristics, low PMEPR, ease of implementation and useful data rate. Provides several benefits. Particularly long code words (i.e. multi-carrier systems) (Many carriers in the network) need to provide high data rates, Search for all code words (impossible and time-consuming) These properties can be obtained without performing. In certain situations, this The code word selection can be modified to compromise between these different characteristics. Sometimes it is necessary.   For example, increasing the data rate instead of accepting the relatively high maximum value of PMEPR If it is desired to increase the available codeword population There is. For example, consider a binary code of length 16 (ie, m = 4). Any of the base codes whose representative value is a linear combination of additional rows in the generator array Constructs a possible coset and results from the use of codewords within the coset It is possible to derive the maximum peak envelope power (PEP). this Is the remainder class in ascending order of the maximum PEP for each coset, as shown in Table 7 below. To be listed. In such a table, the first column contains the code Including six coefficients, including a multiplier for each additional row of the generator array, Coefficient is symbol unit modulo 2^jThe coset representative value is created by adding It is. Thus, for example, 1 under the heading '12' is added to the additional line (X₁* X_Two) Is multiplied by 1 so that is included in the coset representative value, and 0 is multiplied by 0 (effectively excluded) Means to be done. The middle column shows the coset representative value itself, and the last column shows , The largest PEP resulting from the use of code words in the coset.                          Table 7: Binary (length 16) The derivation of PEP for one coset follows the envelope in the time domain Complex waves generated by code words at consecutive sample points It should be noted that it is necessary to calculate the amplitude of the shape envelope . The interval between time values for successive samples affects the accuracy of the results. Blur. That is, the shorter the time interval, the higher the accuracy, but the overall waveform The number of required samples increases, and therefore the cost of computation increases. pay. Typically, the derivation of the PEP is based on the fact that the acquired PEP is a predetermined number. A relatively short continuous interval until stable for a significant number of digits or a decimal number Is repeated for In the case of Table 7, the second decimal place after rounding The calculation was repeated in this way until the location stabilized.   As described above, the first twelve remainders listed above the dotted line (1) in Table 7 The 384 code words associated with the class representative value are PEPR not exceeding 3 dB, That is, it gives a factor of 2 (average error for a 2 m long codeword). The envelope power is 2^mAnd therefore for a code word of length 16 Has an average envelope power of 16, up to 32 PEPs and up to 2 PMEPRs give).   The code word used is the upper half of Table 7 or the first half above dotted line (2). 27 coset representative values plus any of the next 25 coset representative values above the dotted line (3) If it is associated with any of these five, the minimum Hamming distance remains unchanged at 2 * 2 (= 4) , It is possible to encode 10 data bits. this is, The maximum PMEPR is simply increased from 2 to 4 (in any code word Only increase to 4 (as opposed to 16 if good) The number of data bits that can be represented by a single codeword from eight to ten Let These additional data bits are adapted by the selector 112 of FIG. However, in this case, the selector determines the most significant (3 Selected (2) according to the value of 5 bits (not bits)^Five) 32 coset representatives Modified to select from values.   Prepare a table similar to Table 7 for binary codewords of other lengths it can. 2^mFor the code word length of Any code, including the coset set, has a PMEPR that does not exceed 3dB (As in binary code) 2^m-2With a minimum Hamming distance of Any of the devices can be used.   Similar compromises between data rates and PMEPR have been made for quaternions, octals, and so on. It can be applied to codes of higher order. Table 8 below shows the 16 The coset representative value for the quaternary code of the symbol is represented for each coset in the same format as in Table 7. Are shown in the order of the largest PEP.Table 8: Quaternary (length 16) This table can be used as in Table 7. For example, the dotted line (1 The code words associated with the first twelve cosets above) are between the dotted lines (1) and (2). Together with the code word associated with any 20 of the next 40 cosets of Encode up to 15 data bits as a symbol length quaternary code word Can be used for A multi-carrier signal modulated according to these code words The large PMEPR is 4 (6 dB).   Prepare a table equivalent to Table 8 for any other length of quaternary code word Can be 2^mSelected from such a table for the codeword length of Any code that contains a coset set must have a PMEPR not exceeding 3 dB. (Similar to the case of the aforementioned quaternary code)^m-1Minimum Hamming distance and 2^m-1 With a minimum Lee distance. Use any of the decoders described above for such codes In this case, the above-mentioned characteristics are maintained.   Each of the coset representatives listed in Tables 7 and 8 is associated with a respective generator array. Is a linear combination of additional rows. Generate larger tables in a similar way It is possible to include other coset representatives for the base code, but Selecting a code word from such a table based solely on the PEP Is generally less desirable. Because the choice of such a code word is Because it may have low error detection and correction characteristics. Table 7 and Table The generator array used in deriving 8 is the binary and 4 The same nascent array used to derive the minimum PMEPR code in hexadecimal It is.   Table 9 shows the coset representative values for the octal code of length 16 symbols in each coset. It is shown in the order of maximum PEP. In this case, the first column is 2^mSymbol 2 (X_i* X_k) Contains six coefficients that make up a multiplier of the form The additional row of the octal generator array is 4 (X_i* X_k)), They are symbolic Unit modulo 2^jAre added to generate a coset representative value. Therefore, headings 3 below '12' is word 2 (X₁* X_Two) Are included in the coset representative value. Is multiplied by 3. Thus, in this case, the listed remainder Class representative values are not limited to linear combinations of additional rows of the octal generator array. this Because the table is large, only the selected rows are included in Table 9 below.Table 9-8 octal (length 16) In creating Table 9, the PEP calculation is based on that at least the first decimal place after rounding For relatively short intervals between envelope sample time values until stable And was repeated.   As can be seen from Table 9, in the case of octal code, if PMEPR is higher than 4, There are many cosets available that are not available.   For octal codewords other than 16 symbols long, see Table 9. You can prepare a simple table. Length 2^mSuch code words Any code containing the coset set selected from the table (3 dB as described above) Minimum hamming distance 2 (as in octal code for PMEPR not to be exceeded)^{m -2} And minimum Lee distance 2^m-1have. However, in the octal case, the list Coset representative value limited to linear combination of additional rows in octal generator array Not necessarily. This unexpected feature is associated with a very large number associated with low PMEPR. Contribute to the coset.   Each of the coset representatives listed in Table 9 is of the form 2 (X_i* X_k) 2^mSymbol W It is a linear combination of codes. Format (X_i* X_k) Is a linear combination of 2 "symbol words Accepting makes more octal cosets available . In this case, the minimum Lee distance may decrease, but the minimum Hamming distance of the code The separation remains unchanged. Actually, the selector 2 in the encoder 230 (FIG. 14) Adds a row to 52 so that the selector determines the choice of coset representative value By allowing additional input data to be accepted, Is implemented. The octal decoder described above can be used for such codes In this case, the above-described characteristics can be maintained.   More remainders on the underlying code to create larger tables in a similar way It is possible to include class representatives, but based only on the tabulated PEPs, It is generally less desirable to select code words from such tables. No. Because such a code word selection has low error detection and correction This is because they may have characteristics.   Each of the coset representative values is of format 2 (X_i* X_k) 2^mA linear combination of symbol words Thus, a table similar to Table 9 can be generated. Length 2^mCode word With respect to any code containing a coset set selected from such a table. Small hamming distance 2^m-2And minimum Lee distance 2^m-1have. Format (X_i* X_k) 2^mSymbol More cosets available by accepting linear combinations of Le words It can be. In this case, the minimum Lee distance may decrease, but the The minimum Hamming distance of the node remains unchanged. Proper use of a decoder as described above To decode such a code.   The ordered list of cosets as seen in Tables 7, 8 and 9 is It can be generated using the following programs in the programming language. Wear.   In addition to choosing a compromise between data rate and PMEPR, for example, It is also possible to make a compromise between data rate and minimum Lee distance. For example, in Table 7 Select 4 subsets from the coset defined by the first 12 rows To maintain a maximum PMEPR of 3 dB while reducing the number of data bits to eight. Instead of decreasing from 7 to 7, the minimum distance can be increased from 4 to 6. That One example of such a subset is the second, third, fourth and sixth Consists of cosets defined by rows. Similar compromises have been made with other code This is possible for code lengths and higher order codes. In extreme cases, the underlying code For any order code using a single coset of^m-1The smallest Hami Encode only j (m + 1) data bits, while still providing It is also possible.   To construct a list of coset representative values in the order of the largest PEP for each coset Thus, a possible compromise between data rate and PMEPR is as described above. , Especially for relatively long code words and higher order codes It has the disadvantage that. The following technique does not calculate the maximum PEP for each coset, In combination with the base code, a code related to PMEPR that does not exceed a known value Identify the coset representative value set that can generate the word (this technique uses Classes are not ordered by PMEPR). The first example of using this technique is Less than This is the case for a binary code of length 16 (m = 4) with a generator array as below. (0000 0000 1111 1111) (X₁) (0000 1111 0000 1111) (X_Two) (0011 0011 0011 0011) (X_Three) (0101 0101 0101 0101) (X_Four) (1111 1111 1111 1111) (X ') (0000 0000 0000 1111) (X₁* X_Two) (0000 0000 0011 0011) (X₁* X_Three) (0000 0000 0101 0101) (X₁* X_Four) (0000 0011 0000 0011) (X_Two* X_Three) (0000 0101 0000 0101) (X_Two* X_Four) (0001 0001 0001 0001) (X_Three* X_Four) As a first step in generating a set of coset representative values, eight (2)^m-1The first of the formula A term set is built. Each of these has the following form: (X₁* X_Two) (+) (X_Two* X_Three) (+) V₁(X₁* X_Four) (+) V_Two(X_Two* X_Four) (+) V_Three(X_Three* X_Four) Where v₁, V_TwoAnd v_ThreeIs a coefficient with the value 0 or 1 and the sign (+) is in bits 3 shows an addition modulo 2; Therefore, the initial set is as follows. (X₁* X_Two) (+) (X_Two* X_Three) (V₁= 0, v_Two= 0, v_Three= 0) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X₁* X_Four) (V₁= 1, v_Two= 0, v_Three= 0) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X_Two* X_Four) (V₁= 0, v_Two= 1, v_Three= 0) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X₁* X_Four) (+) (X_Two* X_Four) (V₁= 1, v_Two= 1, v_Three= 0) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X_Three* X_Four) (V₁= 0, v_Two= 0, v_Three= 1) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X₁* X_Four) (+) (X_Three* X_Four) (V₁= 1, v_Two= 0, v_Three= 1) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X_Two* X_Four) (+) (X_Three* X_Four) (V₁= 0, v_Two= 1, v_Three= 1) (X₁* X_Two) (+) (X_Two* X_Three) (+) (X₁* X_Four) (+) (X_Two* X_Four) * (X_Three* X_Four) (V₁= 1, v_Two= 1, v_Three= 1) This initial set then contains the row identifier X in each expression₁, X_Two, X_ThreeAnd X_FourSubscribe to Expanded by applying all possible permutations of the lip. The final set is Created by identifying dissimilar expressions within the extended set You. For this purpose, (X₁* X_TwoAnd X_Two* X₁Subscript pairs are the same But the items in the opposite order are considered to be the same, and Ordering of items such as is ignored.   Table 10 contains the final set of 49 expressions that define the required set of coset representative values. Is shown. That of the additional row of the code generator array modulo 2 added The table is constructed in terms of the six coefficients that make up the multiplier for each, and the corresponding remainder Class representative values have been created. For example, one under the heading '12' is added to the additional line (X₁* X_Two) Is multiplied by 1 so that it is included in the coset representative value, and 0 is multiplied by 0. That is, excluded.Table 10   One of these coset representatives is obtained by combining with the base code. The resulting code has the following characteristics: -The minimum Hamming distance of the code is 4 (2^m-2). A multi-carrier signal modulated according to these code words has a P No MEPR. The Hamming distance-based decoder used for such codes Can be operated using analysis techniques, such as at least partially combinatorial logic. Can be implemented. Code words available to achieve acceptable data rate . Length 2 with m> = 3^mIn the general case of a binary code of bits, 2^m-1Pieces An initial set of expressions of the form is constructed in the form (X₁* X_Two) (+) (X_Two* X_Three) (+) ... (+) (X_m-2* X_m-1) (+) V₁(X₁* X_m) (+) V_Two(X_Two* X_m) (+) ... (+) v_m-1(X_m-1* X_m) Where v₁, V_Two, ... v_m-1Is a coefficient with a value of 0 or 1 The permutation of the default and the identification of dissimilar expressions are performed as described above. (Parameter m Given by this procedure for various codeword lengths. The number of cosets and the total number of codewords are as follows:m Coset Code words 3 7 1792 4 49 50 176 5 552 2260992 6 7800 127795200 7 126 360 8281 128 960   A modification of this technique that can be applied to word lengths greater than 64 bits (m> 6) is The coset can be large without having to examine the set of expressions to find and exclude similar expressions. To identify the best subset. In other words, this modification technique Can guarantee to generate an extended set of expressions that are inherently similar to no other . Therefore, the coefficient v₁, V_Two, ... v_m-1The combination of values of the form v₁v_Twov_Three... v_m-2v_m-12 Binary words are limited to combinations that have the following two properties: sand That is, each word v₁v_Twov_Three... v_m-2v_m-1Have a Hamming weight of at least 4 ,and, It is. The initial set, given in the form above, which is a limited set of coefficient combinations Constructs the formulas of the set, and in every possible order of the subscripts of the line identifier in each formula By applying columns, an explicit definition of the required subset of coset representatives A set of expressions is created. The base code is The code word generated by the combination with the class representative value also does not exceed 6 dB. Produce PMEPR. at least m! .2 if m> 7^m-4The remainder class representative value of this Identified by the manager.   Using similar techniques, the quaternion and quota results in a known maximum PMEPR result. And the cosets of higher order code words can be identified. Length 2 when m> = 3^m For a quaternary code word of^m-1The initial set of expressions is of the form Be built. 2 (X₁* X_Two) (+) 2 (X_Two* X_Three) (+) ... (+) 2 (X_m-2* X_m-1) (+) 2v₁(X₁* X_m) (+) 2v_Two(X_Two* X_m) (+) ... (+) 2v_m-1(X_m-1* X_m) Where v₁, V_Two, ... v_m-1Is a coefficient having a value of 0 or 1, and the symbol (+) is a symbol 4 shows a unit addition modulo 4. This initial set is then the row identifier in each expression X₁, X_Two, X_Three... X_mBy applying all possible permutations of the subscript Expanded. Final set identifies dissimilar expressions within the expanded set Created by doing. (As far as parameter m is concerned) The number of cosets and code words given by this procedure for The total number of codes is as follows.m Coset Code words 3 37 9472 4 661 676864 5 14472 59277312 6 360600 5908070400 7> = 1290240> = 84557168640 In this case, there are even more cosets, and thus potentially higher data transmission rates. Available. What are the coset representatives identified by this modification technique and 4 The code word generated by the combination of the radix base codes also exceeds 6 dB. No PMEPR, but Lee minimum distance is 2^m-2To decrease.   Furthermore, the coefficient v₁And v_m-1Is fixed at 1 and 3, and other coefficients are from 0 to 3. By allowing any value of Utilizes a modification procedure that generates an extended set of dissimilar expressions in columns can do. Length 2 when m> = 3^m4 for the octal code word^m-1Initial set of Bits are constructed in the following format: 4 (X₁* X_Two) (+) 4 (X_Two* X_Three) (+) ... (+) 4 (X_m-2* X_m-1) (+) 2v₁(X₁* X_m) (+) 2v_Two(X_Two* X_m) (+) ... (+) 2v_m-1(X_m-1* X_m) Where v₁, v_Two, ... v_m-1Is a coefficient having a value of 0 or 1, and the symbol (+) is a symbol 9 shows a unit addition modulo 8. This initial set is then the row identifier in each expression X₁, X_Two, X_Three... X_mBy applying all possible permutations of the subscript Expanded. Final set identifies dissimilar expressions within the expanded set Created by doing.   One of these coset representatives is obtained by combining with the base code. The octal code has the following characteristics. ・ The minimum Hamming distance of the cord is 2^m-2And the minimum Lee distance is 2^m-1It is. A multi-carrier signal modulated according to these code words has a P No MEPR. The Hamming distance-based decoder used for such codes Can be operated using analysis techniques, such as at least partially combinatorial logic. Can be implemented. Code words available to achieve acceptable data rate . In general, this procedure is at least m! .4 when m => 4^m-3Dissimilar remainders Identify extra representative values.   This procedure for various codeword lengths (as far as the parameter m is concerned) The number of cosets and the total number of code words given by .m Coset Code words 3 37 151552 4 661 21659648 5 14472 3793747968 6 360600 756233011200 7> = 1290240> 2.16x10¹³   As with the quaternion, the word length of 16 or more (m => 4) symbols is Large subsets of these cosets can be identified without checking for similarity. Wear. This is the coefficient v₁And v_m-1Are fixed at 1 and 3, and the other coefficients are Achieved by allowing any value up to three. This procedure Exactly m! .4 by ja^m-3A number of dissimilar coset representatives is obtained.   Length 2 when m> = 3^mFor the octal code word of₁, V_Two, ... v_m-1But Allow any number from 0 to 7 and modify the format of the initial set of expressions as follows: Further corrections are possible by correction. 4 (X₁* X_Two) (+) 4 (X_Two* X_Three) (+) ... (+) 4 (X_m-2* X_m-1) (+) V₁(X₁* X_m) (+) V_Two(X_Two* X_m) (+) ... (+) v_m-1(X_m-1* X_m) This initial set then contains the row identifier X in each expression₁, X_Two, X_Three... X_mSubscription of It is extended by applying all possible permutations of the put. The final set is Identifies dissimilar expressions within the expanded set, as in the binary case above Created by doing. (As far as parameter m is concerned) The number of cosets and code words given by this procedure for The total number of codes is as follows.m Coset Code words 3 169 692224 4 5917 193888256 5 243912 63940067328 6> 5.52x10⁶        > 1.15x10¹³ 7> 3.09x10⁸        > 5.19x10¹⁵ In this case, there are even more cosets, and thus potentially higher data rates. Available. What are the coset representatives identified by this correction technique and 4 The code word generated by the combination of the radix base codes also exceeds 6 dB. No PMEPR, but Lee minimum distance is 2^m-2To decrease.   In addition, after creating a permutation of row identifier subscripts, make sure that dissimilar expressions Another modification procedure is provided that generates an extension set. This professional The sheather has a coefficient v₁And v_m-1Values to an ordered value pair as shown below. Set. That is, v₁, v_m-1Values are 1,2; 1,3; 1,5; 1,6,1,7,2,3; 2,5,2,6; 2,7; 3, 5,3,6,3,7; 5,6; 5,7,6,7; Other coefficients have any value from 0 to 7 be able to. With this approach, if m> = 4, at least 15.m! .8^m-3 Are obtained.   Identify the cosets that define the code words that generate PMEPRs that do not exceed 6 dB. These other techniques can be extended to higher order codes. Generalize further Then a large set of cosets would have 2^jWith possible values of (m> = 3) 2^mSymbol Identifiable to the code containing the file, where PMEPR is 1 = <p = <m-2 At most 2 for any fixed integer p^{P + 1}Is guaranteed. That is, the coefficient The p sets are defined as follows: (V_1,1, V_1,2, ... v_{1, mp}) (V_2,1, V_2,2, ... v_{2, mp}) ... (V_{p, 1}, V_{p, 2}, ... v_{p, mp}) However, each coefficient is from 0 to 2^jIt has an integer up to -1. 1 = <i <j = <p, each of which is 0 2^jAn additional p (p-1) / 2 coefficients with integers up to -1 are defined. Coefficient V_ij， O And W_ijFor the initial selection of, an expression of the form Two^j-1(X₁* X_Two) (+) 2^j-1(X_Two* X_Three) (+) ... (+) 2^j-1(X_mp-1* X_mp) (+) V_1,1(X₁* X_{m-p + 1}) (+) V_1,2(X_Two* X_{m-p + 1}) (+) ... (+) v_{1, mp}(Xm-p * X_{m-p + 1}) (+) V_2,1(X₁* X_{m-p + 2}) (+) V_2,2(X2 * X_{m-p + 2}) (+) ... (+) v_{2, mp}(X_{m -p} * X_{m-p + 2}) ... (+) V_{p, 1}(X₁* X_m) (+) V_{p, 2}(X_Two* X_m) (+) ... (+) v_{p, mp}(X_mp* X_m) (+) W_1,2(X_{m-P + 1}* X_{m-P + 2}) (+) W_1,3(X_{m-p + 1}* X_{m-p + 3}) (+) W_2,3(X_{m-p + 2}* X_{m-p + 3}) (+) W_1,4(X_{m-P + 1}* X_{m-P + 4}) (+) W_2,4(X_{m-P + 2}* X_{m-P + 4}) (+) W_3,4(X_{m-P + 3}* X_{m-p + 4}) ... (+) W_{1, p}(X_{m-p + 1}* X_m) (+) W_{2, p}(X_{m-p + 2}* X_m) (+) ... (+) w_{p-1, p}(X_m-1* X_m) However, the sign (+) is symbol unit addition modulo 2.^jIs shown. A further expression of this form is Coefficient v_ijAnd w_ijBuilt on the remaining possible combinations of the choice of 2^{jp (mp) + jp (p-1) / 2} An initial set of expressions is obtained. This initial set is And row identifier X₁, X_Two, X_Three, ... X_mApply all possible permutations of subscripts in Extended by doing. The end of the expression that defines the required set of coset representatives The set recognizes other dissimilar expressions without the extended set as described above for binary numbers. Generated by another. Any remainder identified by this procedure Extra value and 2^jCode generated by combining the following base codes ・ The word sets are both 2^m-2Minimum minimum distance and minimum hamming distance One. Coefficient V_ijAnd W_ijAre further modified to be limited to have the above range values It is also possible to add. In this case, fewer cosets are identified, but as a result The resulting code is 2^m-1With a minimum Lee distance.   Data transmission rate, PMEPR, and error detection / correction capability using methods other than those described above. A compromise between the two. For example, as described above, 11 data bits Instead of using 32-bit code words for concatenating, derive from Table 3. Contiguous as a 16-bit code word using the 8B / 16B code issued Encode an 8-bit block and then concatenate such codeword pairs To generate a hybrid 32-bit word for transmission. like this The concatenated code has the following characteristics compared to the direct use of the 11B / 32B code. One. Better data transfer rates (16 data represented by 32 bits transmitted) Data bits). PMEPR is at worst 4 (= 6 dB), and the component code word Twice the maximum PMEPR for As described above, the connection method requires an excellent data transmission rate, and is not necessarily required. It may not be optimal but can be useful in situations where PMEPR is controlled. Wear.   When decoding the concatenated code, the code word is first divided into two consecutive 1s It is split into 6-bit code words and then the 16B / 8B decoder 120 or 1 40, each of which is individually processed to retrieve an 8-bit data word. You.   Short code lengths other than by concatenating pairs of code words for transmission It is possible to combine words into longer hybrid code words for transmission. An example For example, to concatenate four code words and decode the received hybrid code word A method of dividing into four parts is also possible. As an alternative, individual The bits or symbols of the components of the short codeword one by one or Interleaving can also be done in pairs or quadruplets. Configure mixed code words The resulting code words need not be members of the same coset. Constituent features Make the choice of coset representative value for a given codeword span multiple values You can also. In this case, for example, a form similar to the function of the selector of the decoder 60 in FIG. By defining one or more additional bits of the data word in the form The identity of the coset to which each of the selected constituent codewords belongs Can be used for transmission.   Hybrid transmission codewords created by such concatenation or interleaving Mode is not derived from a generator matrix with the same length of rows as described above (example It is derived from a generator matrix with, for example, half) rows. However, that The code that is actually composed of such hybrid code words is Code with one row of generator matrices of the same length as An embodiment in which selection is made from a set is also possible.   Another modification is to use a 16-bit code word and a 32-bit code word. Selecting words for transmission, each selected from a code derived respectively as described above It is also possible to concatenate them into a hybrid 48-bit word. Furthermore, this 4 By concatenating the additional 16 zero bits into the 8-bit word, the transmission IFFT It is also possible to generate 64-bit words for processing in a manner. With 16 pieces The 0 bits provide the best choice of a power of two with respect to the number of samples on receipt. Enables the use of oversampling. Another possibility is that each has 16 bits The three 48-bit code words selected from the code of Generate a bit code word.   Note that binary, quaternary and octal numbers of the given length as described above Are derived from a single common generator matrix. Real Such generator matrices are used to derive similar higher-order codes be able to. Change the code order used according to the general characteristics of the communication path These characteristics are valuable in implementing an adaptive system. in this way, Use a higher order code, such as octal, when a high quality path is in use A high data transmission rate can be obtained. Bad route quality and increased error rate In addition, it is possible to switch to using lower order codes such as quaternary or binary numbers. There are advantages that can be done. This is a complete duplicate of the circuit for each different order code Can be achieved with the present invention without the need to use. Ma Similar circuits incorporating or based on a common generator matrix For example, a switchable parameter corresponding to the value in the additional row of the generator array for each code It can also be used with a meter. Selecting appropriate values for those parameters Depending on the choice, the same basic circuit can be used to generate the required specific order code, Can be used.   Coset representatives derived as described above provide particularly advantageous properties. In the most general case, coset representatives shift their values in a systematic way Symbol unit addition modulo 2^jCombined with code words by Can be any value.   The code word in the code defined as above also It has properties useful for establishing timing synchronization with the received signal. Obedience Therefore, the above-described circuit for decoding these codes is also synchronized with the received data. Can be used to identify a sequence. These codes are It is observed that it has advantageous properties for communication with technologies other than FDM. For example, the codes are related to code division multiple access (or CDMA) technology. For modem transmission of digital data as a card or on an analog communication path Can be used as code.

────────────────────────────────────────────────── ─── Continuation of front page    (31) Priority claim number 9625710.0 (32) Priority date December 11, 1996 (December 11, 1996) (33) Priority claim country United Kingdom (GB) (31) Priority claim number 97157721.8 (32) Priority date July 26, 1997 (July 26, 1997) (33) Priority claim country United Kingdom (GB) (81) Designated countries EP (AT, BE, CH, DE, DK, ES, FI, FR, GB, GR, IE, IT, L U, MC, NL, PT, SE), JP, US

Claims (1)

[Claims] 1. A method of encoding a data word as a 2 ^m symbol code word for transmission using a multi-carrier frequency division multiplexing system such as orthogonal frequency division multiplexing (ie, COFDM), comprising: And z has an integer value from 0 to 2 ^m -1, y has an integer value from m to 0, and x is a bit-wise multiplication of a factor expressed as a (m + 1) -bit binary number. And the result of the division is represented as a single digit value, A data word according to a code word selected in full coset form with respect to a desired peak average envelope power ratio of said transmission from one or more coset sets of linear subcodes of the code having a generator matrix of Is encoded, the data word encoding method. 2. 2. The method of claim 1, wherein m is at least 3. 3. 3. The method of claim 2, wherein m is at least 4. 4. 2. The method according to claim 1, wherein m is 4, and the generator matrix is (0000 0000 1111 1111) (0000 1111 0000 1111) (0011 0011 0011 0011) (0101 0101 0101 0101) (1111 1111 1111 1111). the method of. 5. m is 5 and the generator matrix is (0000 0000 0000 0000 1111 1111 1111 1111) (0000 0000 1111 1111 0000 0000 1111 1111) (0000 1111 0000 1111 0000 1111 0000 1111) (0011 0011 0011 0011 0011 0011 0011 0011) 0011) The method according to claim 1, wherein (0101 0101 0101 0101 0101 0101 0101 0101) (1111 1111 1111 1111 1111 1111 1111 1111). 6. A method according to any of the preceding claims, wherein the number of code words is an integer power of two. 7. The method according to any of the preceding claims, wherein each symbol has 2 ^j different possible values, where j is a positive integer. 8. The method of claim 7, wherein the value of j is 1. 9. The method of claim 7, wherein the value of j is 2. 10. The method of claim 7, wherein the value of j is 3. 11. All of the code words are derived by symbolic multiplication of the generator matrix and all possible pairs of the first m rows of the generator matrix, and s, which is a power of 2 less than 2 ^j , is The method of claim 7, comprising additional rows to be multiplied. 12. The method of claim 11, wherein s has a value of one. 13. The method of claim 11, wherein 13.1 is at least 2 and s has a value of 2. 14． 12. The method of claim 11, wherein j is at least 3 and s has a value of ^2j-1 . 15. m is 4 and the generator array is (0000 0000 1111 1111) (0000 1111 0000 1111) (0011 0011 0011 0011) (0101 0101 0101 0101) (1111 1111 1111 1111) (0000 0000 0000 ssss) (0000) The method according to any one of claims 11 to 14, wherein 0000 00ss 00ss) (0000 0000 0s0s 0s0s) (0000 00ss 0000 00ss) (0000 0s0s 0000 0s0s) (000s 000s 000s 000s). 16. m is 5 and the generator array is (0000 0000 0000 0000 1111 1111 1111 1111) (0000 0000 1111 1111 0000 0000 1111 1111) (0000 1111 0000 1111 0000 1111 0000 1111) (0011 0011 0011 0011 0011 0011 0011 0011 0011) (0011) (0101 0101 0101 0101 0101 0101 0101 0101) (1111 1111 1111 1111 1111 1111 1111 1111) (0000 0000 0000 0000 0000 0000 0000 sssss) (0000 0000 0000 0000 0000 0000 sssss 0000 s ssss 0000 s s 00ss) (0000 0000 0000 0000 s0s 0s0s 0s0s 0s0s) (0000 0000 0000 ssss 0000 0000 0000 ssss) (0000 0000 00ss 00ss 0000 0000 00ss 00ss) (0000 0000 0s0s 0s0s 0000 0000 0s0s 0s 0000 s 0s 0000 s The method according to any one of claims 11 to 14, wherein (00ss) (0000 0s0s 0000 0s0s 0000 0s0s 0000 0s0s) (000s 000s 000s 000s 000s 000s 000s 000s). 17． Let X _{1 to} X _m be the row labels of the first m rows in the generator matrix, * denote symbol-wise multiplication, and (+) denote symbol-wise addition modulo 2 ^j , All of the class representative values are 2 ^j-1 (X ₁ * X ₂ ) (+) 2 ^j-1 (X ₂ * X ₃ ) (+) .. (+) 2 ^j-1 (X _m-1 * X _m) or they have the form of subscript permutation method according to claim 11. 18. The code word is selected from a list of cosets of a linear subcode of the code, the list comprising a maximum peak envelope power ratio for all cod words in one coset with respect to the other coset. 17. The method according to any of the preceding claims, wherein the method is arranged in the order of comparison. 19. 19. The method of claim 18, wherein at least a predetermined number of cosets are selected such that a maximum peak-to-average envelope power ratio for all codewords in the selected coset is minimized. 20. 19. The method of claim 18, wherein the codeword is selected from a coset such that the maximum peak-to-average envelope power ratio does not exceed a predetermined value. 21. The code word is binary or quaternary and the predetermined value is 4, or the code word is an octal code word and the predetermined value is 3; The method according to claim 20. 22. Its representative value is derived by symbol-wise multiplication of all possible pairs of the first m rows of the generator matrix and further multiplied by a positive integer s, which is a power of two less than the number of different possible values for each symbol. 19. The method of claim 18, wherein the codeword is selected from a coset defined by a linear combination of the rows performed. 23. Modulating a multi-carrier signal according to a code word encoding the data word; combining the modulated carrier signal to reduce the peak average envelope as compared to a hybrid signal directly modulated according to the data word. A method according to any of the preceding claims, comprising: generating a hybrid signal having a power ratio. 24. A method of transmitting a data word as a 2 ^m symbol code word for transmission using a multi-carrier frequency division multiplexing scheme such as orthogonal frequency division multiplexing (ie, COFDM), comprising: And z has an integer value from 0 to 2 ^m -1, y has an integer value from m to 0, and x is a bit-wise multiplication of a factor expressed as a (m + 1) -bit binary number. And the result of the division is represented as a single digit value, Each data according to a code word selected in one or more cosets with respect to a desired peak-to-average envelope power ratio of the transmission from one or more cosets of a linear subcode of the code having a generator matrix of A method of transmitting data words, comprising: encoding a word; and decoding each code word using an analytic decoder. 25. A method of encoding a data word as a 2 ^m symbol code word for transmission using a multi-carrier frequency division multiplexing scheme such as orthogonal frequency division multiplexing (ie, COFDM), wherein m is At least 4, the value of the data word and z has an integer value from 0 to 2 ^m -1, y has an integer value from m to 0, and x is represented as a (m + 1) bit binary number Indicates the bitwise multiplication of the factor by which the result of the division is represented as a single digit value, A data word according to a code word selected in full coset form with respect to a desired peak average envelope power ratio of said transmission from one or more coset sets of linear subcodes of the code having a generator matrix of Is encoded, the data word encoding method. 26. For transmission using a multi-carrier frequency division multiplexing system such as orthogonal frequency division multiplexing (ie, COFDM), the data words are represented as 2 ^m symbol code words, each having 2 ^j different possible values. A method for encoding, wherein m is at least 4, j is at least 3, the value of a data word and z have integer values from 0 to 2 ^m -1 and y is from m to 0. Has integer values up to and x indicates a bitwise multiplication of a factor expressed as a (m + 1) -bit binary number, and assuming that the result of the division is expressed as a one-digit value, A data word according to a code word selected in full coset form with respect to a desired peak average envelope power ratio of said transmission from one or more coset sets of linear subcodes of the code having a generator matrix of Is encoded, the data word encoding method. 27. A method for encoding a data word for transmission over a communication path as code words of 2 ^m symbols each having 2 ^m different possible values, wherein j is a positive integer variable, A word value and a bit unit of a factor in which z has an integer value of 0 to 2 ^m -1, y has an integer value of m to 0, and x is expressed as (m + 1) -bit binary number Indicates multiplication and the result of the division is represented as a single digit value. A data word is encoded according to a code having a code word of value j contained in a set of one or more cosets of linear subcodes of a code having a generator matrix of . 28. A method according to any of the preceding claims, wherein the coset set comprises a plurality of cosets, and wherein the linear subcode is the same for each coset. 29. 29. The method of claim 28, wherein said linear subcode comprises all of said codes. 30. z has an integer value from 0 to 2 ^m -1; y has an integer value from m to 0; x indicates a bit-wise multiplication of a factor expressed as a (m + 1) -bit binary number; Is represented as a single digit value, each code word symbol has 2 ^j different possible values with J> 2, A method for identifying code words of 2 ^m symbols within a set of code words having a generator matrix, wherein the coefficients for the rows of the matrix comprise a plurality of fast Hadamard transforms or equivalent transform schemes. A code word identification method wherein the coefficients for the rows of the matrix are derived by applying to the 2 ^m symbol input words and combining the results of the conversion to identify the code word. 31. 31. The method of claim 30, wherein a total of 2 ^{(j-1) (m + 1)} fast Hadamard or equivalent transforms are applied to the input word. 32. X _{1 to} X _m are the row labels of the first m rows of the matrix, X ′ is the row label of the last row of the matrix, and (+) indicates the symbol unit addition modulo 2 ^j. For all possible combinations of coefficients b ₁ , b ₂ ,..., B _m and b ′ having integer values from to 2 ^j−1 , B = b ₁ X ₁ (+) b ₂ X ₂ (+ ) ... (+) b _m X _m (+) b′X ′, generating 2 ^{(j−1) (m + 1)} values B according to the relation: s is the input word, Assuming that the subtraction and modulo operations | ··· | with the values X ′, s and B are performed symbol-by-symbol, D = | 2 ^j−1 X ′ − ((SB) mod 2 ^j ) | −2 ^j− Generating a vector D that obeys the relation ² X ′ for each value B; applying a fast Hadamard transform or an equivalent transformation scheme to each vector D to derive 2 ^m transform components; value Identifying a transform component; and deriving a row coefficient corresponding to the value of B and the result of a rapid Hadamard or equivalent transform associated with the identified now-comparing absolute magnitude transform component. 32. The method of claim 31, wherein coefficients for rows of the matrix are derived. 33. 33. The method according to any of claims 30 to 32, wherein the symbols of the input word have integer values. 34. An encoder for encoding a data word as a 2 ^m symbol code word for transmission using a multi-carrier frequency division multiplexing scheme such as orthogonal frequency division multiplexing (ie, COFDM), comprising: Receive a word and the value of that data word, and z has an integer value from 0 to 2 ^m -1, y has an integer value from m to 0, and x is represented as a (m + 1) bit binary number Indicates the bitwise multiplication of the factor by which the result of the division is represented as a single digit value, A code received from one or more coset sets of linear subcodes of a code having a generator matrix according to a code word selected in the form of a full coset with respect to the desired peak average envelope power ratio of the transmission. A data word encoder comprising means for encoding words.