WO2019022334A1

WO2019022334A1 - Parity-check-concatenated polar encoder

Info

Publication number: WO2019022334A1
Application number: PCT/KR2018/003298
Authority: WO
Inventors: 송홍엽; 박진수; 김인선
Original assignee: 연세대학교 산학협력단
Priority date: 2017-07-25
Filing date: 2018-03-21
Publication date: 2019-01-31
Also published as: KR101891093B1

Abstract

Provided is a parity-check-concatenated polar encoder. The parity-check-concatenated polar encoder may comprise an outer encoder and a polar encoder concatenated to the outer encoder. The outer encoder may be configured to receive m information bits and provide the m information bits and r parity check bits as output. The polar encoder may be configured to receive, as input, the m information bits and the r parity check bits provided from the outer encoder and n bits including [n-(m+r)] frozen bits provided from selected bit indices and provide n output bits. Bit indices and bit values for the r parity check bits, which are output together with the m information bits by means of the outer encoder, may be set such that corresponding input vector(s) are not input into the polar encoder as possible, so that any encoded vector(s) having a minimum hamming weight (MHW) among encoded vectors that can theoretically be output from the polar encoder is(are) not actually output.

Description

Parity check concatenated coder

More particularly, the present invention relates to an outer encoder for a parity-check encoder.

Along with the rapid development of telecommunication services and the continuous increase of Internet traffic, wired / wireless communication systems are rapidly evolving to enable high-speed data transmission and high-performance data error correction. As a result, the polar codes, soft-decision error detection codes, are attracting attention as new error correction codes. The polarity code is a code suggested by Arikan in 2008. It has a low complexity in terms of encoding / decoding, and has a channel capacity of infinite code length in a binary input discrete memoryless channel (BI-DMC) ), Which is theoretically proven to be possible. In recent years, the drawbacks of the polar codes - that is, when the code length is finite and the code length is the same and the polarity is lower than that of the LDPC Codes and Turbo Codes in the same channel environment, A parity-check-concatenated polar encoder which is constructed by concatenating a parity-check encoder and a polar encoder has been proposed in order to compensate for the performance of the apparatus. Researches are being actively conducted. However, most studies on parity check concatenated encoders are focused on developing efficient and good decoders, and there are relatively few studies on such encoders.

SUMMARY OF THE INVENTION It is an object of the present invention to provide a parity check concatenated polarizer capable of providing polarized codes having improved characteristics in terms of a minimum Hamming distance.

Another object of the present invention is to provide an outer encoder for a parity check concatenated polarizer capable of providing polarized codes having improved characteristics in terms of minimum hamming distance.

A further object of the present invention is to provide a method for designing an outer encoder for a parity check concatenated polarizer capable of providing polarized codes with improved characteristics in terms of minimum hamming distance.

The problems to be solved by the present invention are not limited to the above-mentioned problems, and other matters not mentioned can be clearly understood by those skilled in the art from the following description.

In one aspect, a parity check concatenated polarizer is provided. The parity check concatenated polar coder includes an outer encoder and a polarizer connected to the outer encoder. Wherein the external encoder is configured to output 20 parity check bits and 64 information bits, and in response to the input of 20 parity check bits, 64 information bits, and 172 frozen bits, And the 172 frozen bits are set to the frozen bit index set F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17 , 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, , 50, 51, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, , 84, 85, 86, 87, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 113, 114, , 125, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155 , 157, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181, 183, 185, 187, 210, 211, 213, 215, 217, 218, 219, 221, 225, 226, 227, 219, 193, 194, 195, 196, 197, 198, 199, 223, 234, 235, 237, 241, 242, 243, 245, 249}, and the outer encoder inputs the 20 parity check bits Parity check bit index set P = {159,191,122,182,101,30,62,52,76,44,28,124,72,40,24,184,16,121,164,64} Lt; RTI ID = 0.0 > defined bit indices.

According to embodiments of the present invention, there is a technical effect of being able to provide polarized codes with improved characteristics in terms of minimum hamming distance.

FIG. 1 is a diagram illustrating the configuration of a parity-check-concatenated (PCC) Polar Code Generator according to an embodiment of the present invention.

2 is a diagram illustrating an encoding operation by an external encoder according to an embodiment of the present invention.

3 is a diagram illustrating a configuration of a polarizer according to an embodiment of the present invention.

4A to 4C are diagrams illustrating an embodiment of a flowchart for illustrating bit indexes for r parity check bits and a method for determining the bit values thereof, which are output together with m information bits by an outer encoder .

5 shows the case where the length n of the coded vector output by the polarizer 120 is 256, the number m of information bits is 64, and the numbers r of parity check bits are 18, 20 and 22, respectively, The parity check concatenated coders are designed according to the method illustrated in FIG. 4C and the frame error rates (FERs) of the output codewords output from the designed parity check concatenated coders are simulated and compared with each other .

6 shows the case where the length n of the encoded vector output by the polarizer 120 is 256, the number m of information bits is 85, and the numbers r of parity check bits are 16, 18, 20, 22 and 24, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

7 shows a case where the length n of the encoded vector output by the polarizer 120 is 256, the number of information bits m is 128, and the numbers r of parity check bits are 16, 18, 20, 22 and 24, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

8 shows the case where the length n of the encoded vector output by the polarizer 120 is 256, the number of information bits m is 171, and the numbers r of parity check bits are 12, 14, 16, 18 and 20, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

9 shows a case where the length n of the encoded vector output by the polarizer 120 is 512, the number of information bits m is 128, and the numbers r of parity check bits are 18, 20, 22, 24 and 26, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

10 shows a case where the length n of the encoded vector output by the polarizer 120 is 512, the number of information bits m is 171, and the numbers r of parity check bits are 16, 18, 20, 22 and 24, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

11 shows a case where the length n of the encoded vector output by the polarizer 120 is 512, the number of information bits m is 256, and the numbers r of parity check bits are 18, 20, 22, 24 and 26, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

12 shows the case where the length n of the encoded vector output by the polarizer 120 is 512, the number of information bits m is 341, and the numbers r of parity check bits are 28, 30 and 32, respectively, 4A and 4B are diagrams illustrating a comparison of the frame error rates of the output codes output from the designed parity check concatenated coders according to the parity check concatenated coders according to the exemplary embodiment of the present invention.

13 shows the case where the length n of the coded vector output by the polarizer 120 is 1,024, the number of information bits m is 256, and the numbers r of parity check bits are 30, 32 and 34, respectively, 4A and 4B are diagrams illustrating a comparison of the frame error rates of the output codes output from the designed parity check concatenated coders according to the parity check concatenated coders according to the exemplary embodiment of the present invention.

14 shows a case where the length n of the coded vector output by the polarizer 120 is 1,024, the number of information bits m is 341, and the numbers r of parity check bits are 22, 24, 26, 28 and 30, respectively 4A to 4C, and compares the frame error rates of the output codes output from the designed parity check concatenated coders with each other to simulate the parity check concatenated coders according to the method illustrated in FIGS.

15 shows the case where the length n of the coded vector outputted by the polarizer 120 is 1,024, the number m of information bits is 512, and the numbers r of parity check bits are 24, 26 and 28, respectively, 4A and 4B are diagrams illustrating a comparison of the frame error rates of the output codes output from the designed parity check concatenated coders according to the parity check concatenated coders according to the exemplary embodiment of the present invention.

16 shows the case where the length n of the coded vector output by the polarizer 120 is 1,024, the number of information bits m is 683, and the numbers r of parity check bits are 26, 28 and 30, respectively, 4A and 4B are diagrams illustrating a comparison of the frame error rates of the output codes output from the designed parity check concatenated coders according to the parity check concatenated coders according to the exemplary embodiment of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS The advantages and features of the present invention and the manner of attaining them will become apparent with reference to the embodiments described in detail below with reference to the accompanying drawings. The present invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of the invention to those skilled in the art. To fully disclose the scope of the invention to a person skilled in the art, and the invention is only defined by the scope of the claims.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the present invention. For example, an element expressed in singular < Desc / Clms Page number 5 > terms should be understood to include a plurality of elements unless the context clearly dictates a singular value. In addition, in the specification of the present invention, it is to be understood that terms such as "include" or "have" are intended to specify the presence of stated features, integers, steps, operations, components, The use of the term does not exclude the presence or addition of one or more other features, numbers, steps, operations, elements, parts or combinations thereof. Further, in the embodiments described herein, 'module' or 'sub-unit' may mean at least one function or a functional part performing an operation.

In addition, all terms used herein, including technical or scientific terms, unless otherwise defined, have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Terms such as those defined in commonly used dictionaries should be construed as meaning consistent with meaning in the context of the related art and may be interpreted in an ideal or overly formal sense unless explicitly defined in the specification of the present invention It does not.

Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the following description, well-known functions or constructions will not be described in detail if they obscure the subject matter of the present invention.

1, a PCC Polar Code Generator 100 according to an exemplary embodiment of the present invention includes an outer encoder 110 and an inner encoder, which are parity check encoders, And may include a Polar Code Generator (POR) 120. Since the outer encoder 110 and the polarizer 120 are concatenated, the output of the outer encoder 110 is provided as an input to the polarizer 120 as shown. The outer encoder 110 receives m information bits (m is a natural number) and receives m information bits and r (r is a natural number) parity check bits (SPC bits) As an output. The polarizer 120 includes m pieces of information bits and r pieces of parity check bits supplied from the outer encoder 110 and n pieces of n pieces of [n - (m + r)] frozen bits provided from predetermined bit indices Lt; RTI ID = 0.0 > n, < / RTI > where n is a natural number. The bit indices to which the frozen bits are provided may be determined by any predetermined criteria, such as, for example, a Gaussian Approximation method, and may be identified by the frozen bit index set (F). The bit indices (p _j , j = 1 to r) and the values (q _j ) for which the parity check bits are provided can be determined according to embodiments of the present invention described below.

2 is a diagram illustrating an encoding operation by an external encoder according to an embodiment of the present invention. In the illustrated embodiment, when n = 16, m = 8, r = 1 and the frozen bit index set F is set to F = {1, 2, 3, 5, 6, 7, 9}, the outer encoder 110 8 information bits and one parity check bit _{_{(a 4, a 8, a}} 10, a 11, a 12, a 13, a 14, a 15, a 16) of the

bit indexes

4, 8, 10, 11, 12, 13, 14, 15 and 16, and outputs them. The outer encoder 110 determines a parity check bit in the bit index 11 (p ₁ ) as a bit index into which the parity check bits among the

bit indices

4, 8, 10, 11, 12, 13, 14, The value can be determined as q ₁ (= a ₄ + a ₈ (mod 2)). The outer encoder 110 adds eight bits to the remaining bit indices except for the bit indexes 11 for the parity check bits and the

bit indexes

1, 2, 3, 5, 6, 7 and 9 included in the frozen bit index set F Arranges and outputs information bits. In this disclosure, the support for the jth parity check bit is denoted by T _j , where T ₁ = {4, 8} is established in the above example. In the embodiment illustrated in FIG. 2, n = 16, m = 8 and r = 1 are exemplified, but it should be noted that the configuration of the outer encoder 110 is not limited to this case.

3 illustrates a polarizer having a 16-bit length, it should be appreciated that the polarizer according to embodiments of the present invention is not limited to a specific bit length encoder. The illustrated 16-bit-length polarizer 120 may be configured by recursive Plotkin concatenation of the 2-bit length polarizers 312-319 in a Plotkeen manner. A 4-bit length polarizer 322 can be constructed by concatenating a 2-bit length polarizer 312 and a 2-bit length polarizer 313. A 2-bit-length polarizer 314 and a 2-bit-length polarizer 315 are concatenated to form a 4-bit-length polarizer 324. A 2-bit-length polar encoder 316 and a 2-bit-length polar encoder 317 are concatenated to form a 4-bit length polarizer 326. A 4-bit length polarizer 328 can be constructed by concatenating a 2-bit-length polarizer 318 and a 2-bit-length polarizer 319. A 8-bit-length polarizer 335 can be constructed by concatenating a 4-bit-length polarizer 322 and a 4-bit-length polarizer 324. The 8-bit length polarizer 337 can be constructed by concatenating the 4-bit length polarizer 326 and the 4-bit length polarizer 328. Finally, the 16-bit-length polarizer 120 can be constructed by concatenating the 8-bit-length polarizer 335 and the 8-bit-length polarizer 337.

In the illustrated embodiment, the polar coder 120 generates a 16-bit output code c (m + r) in response to [16- (m + r)] frozen bits provided by the outer encoder 110 and ₁ , c ₂ , c ₃ , ..., c ₁₆ . For example, the bit value (c ₁ ) of the output code can be calculated as shown in Equation ( ₁ ) below.

[Equation 1]

Here, mod 2 means modulo 2 addition operation.

As another example, the bit value (c ₄ ) of the output code can be calculated as shown in the following equation (2).

&Quot; (2) "

As another example, the bit number of the output code (c ₈₎ can be calculated as the following equation (3).

&Quot; (3) "

The relationship between the output code (coded vector) c output from the polar coder 120 and the input vector a is defined as Equation (4) below.

&Quot; (4) "

Where c and a represent an encoded vector of length n and an input vector, and G represents a generator matrix defined by the structure of the polarizer 120 as illustrated in FIG.

The generator matrix G can be expressed by the following equation (5).

&Quot; (5) "

here

Represents the Kronecker power of the t-th wheel. For example, if the polarizer 120 provides a pole code (output code) of length 16, then t can be set to 4 according to the relationship n = 16 = 2 ^t .

The parity check concatenation code encoder 100 including the external decorrelator 110 and the polarizer 120 described above can be integrated into one or more modules. These integrated modules can be implemented in hardware, such as application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs) , Controllers, micro-controllers, and microprocessors, for example. In addition, such an integration module may be implemented as a firmware / software module that is executable on a hardware platform to perform at least one function or operation. Such a firmware / software module may be implemented by software code written in a suitable programming language. In this case, the software code may be stored in a separate system memory module (not shown). The system memory module for storing the software codes may be a flash memory type, a hard disk type, a multimedia card micro type, a card type memory (for example, SD or XD A random access memory (SRAM), a read-only memory (ROM), an electrically erasable programmable read-only memory (EEPROM), a programmable read-only memory (PROM) , A magnetic memory, a magnetic disk, or an optical disk.

The bit indexes and the bit values for r parity check bits output by the outer encoder 110 together with the m information bits are output from the parity checker 120 to the minimum Hamming The input vector (s) corresponding thereto so that the encoded vector (s) having a minimum Hamming Weight (MHW) is not actually outputted can be set so as not to be input to the polarizer 120 if possible.

The method includes the steps of inputting the input vector to the polar encoder 120, length n of the encoded vector output by the polarizer 120, number m of information bits, number r of parity check bits, generator matrix G, And the frozen bit index set F (S403). In step S405, the bit indices b ₁ , b ₂ , b ₃ , ..., b _r are calculated. The bit indices b ₁ , b ₂ , b ₃ , ..., b _r can be calculated by the following equations (6) to (8).

&Quot; (6) "

here

The

Represents an integer that is not less than.

&Quot; (7) "

&Quot; (8) "

In step S407, the variable j is set to one. In step S409, it is checked whether j is less than or equal to r. If it is determined in step S409 that the check result j is less than or equal to r, the parity check bit indexes p _{j '} , j' = 1 to j - 1 (Q _{j '} , j' = 1 to j -1) based on the corresponding support sets (T _{j ',} j' = 1 to j - 1) (PCC pole codes) by constructing input vectors while changing the information bits while arranging the frozen bits according to the vector F, and applying the generator matrix G to the constructed input vectors, respectively . When step S411 is first performed, it should be noted that parity check bits are not arranged in each of the input vectors for generating encoded vectors. In step S413, the minimum Hamming distance (d _min ) between the generated PCC polar codes is calculated. The minimum Hamming distance (d _min ) may be a Hamming weight of the PCC polar codes having the smallest Hamming distance in relation to a null vector among generated PCC polar codes, and the corresponding PCC polar sign Is denoted by A _dmin . This step also generates a set of input vectors (V ' _dmin ) of the polarizer 120 to generate the PCC pole codes belonging to the set (A _dmin ) as the output vectors of the polarizer 120. This step also excludes frozen bits from each of the constituent vectors of the set of input vectors (V ' _dmin ) and takes only information bits and parity check bits to generate the set (V _dmin ). The constituent vectors of the set (V _dmin ) may have a length of (m + r).

In step S415, null vectors x and z having a length of (m + r) are generated. In step S417, a value of true is assigned to the variable continue. In step S419, it is checked whether continue is true. If it is determined in step S419 that continue is true, the procedure proceeds to step S421 to check whether x is a null vector. If it is determined that the check result x in step S421 is a null vector, a set W is constructed according to the following equation (9) in step S423.

&Quot; (9) "

Where s sets the vector that belongs to _{(V dmin), wt (s} , j) denotes the Hamming weight of the vector s up to b _j-th bit wt (s, j - 1) is a vector s of b _j-1 th Indicates the Hamming weight up to the bit.

According to Equation (9) at least one vector consisting of a set (W) s are set back bekteoyi belonging to (V _dmin) the Hamming weight of up to that b _j-th bit is greater than the Hamming weight of up to that b _j-1 th bit It is a vector satisfying the condition. On the other hand, if it is determined in step S421 that x is not a null vector, the set W is constructed according to the following equation (10) in step S425.

&Quot; (10) "

Where s is a vector belonging to the set (V _dmin ), and f (s, x, j) is a bit index equal to or smaller than b _j and the number of bit indexes .

According to Equation (10), at least one vector s constituting the set W is a vector belonging to the set V _dmin and the condition of f (s, x, j) = 0 and the condition of wt Lt; / RTI > Step S427 is performed following step S423 or step S425. In step S427, it is checked whether W is an empty set and x is not a null vector. If it is determined in step S427 that the check result W is an empty set and x is not a null vector, the process proceeds to step S429 to assign a false value to the variable continue, and the procedure returns to step S419 . On the other hand, if it is determined in step S427 that the check result W is an empty set and x is not a null vector, it is determined in step S431 whether W is an empty set and x is a null vector. If it is determined that the check result W in step S431 is an empty set and x is a null vector, it is determined in step S433 that a value of 1 is added to the bits at bit indices of b _j or less of the vector z, . Here, a may have a value between 0 and 1. In one embodiment, the value of alpha may be set to 0.5. Following step S433, step S435 assigns the bit values of vector z to the bits of x. Subsequently, in step S437, a value of false is assigned to the variable continue, and the procedure returns to step S419.

On the other hand, if it is determined that the check result W in step S431 is an empty set and x is a null vector, the vector S is determined according to the following equation (11) in step S439.

&Quot; (11) "

According to Equation (11), the vector S is a vector s having the minimum Hamming weight from the vectors s, which are elements of W, up to the b _j- th bit index. Next, in step S441, the value of the bit index of the non-zero most right bit among the bits up to the _jth bit index of the vector S is assigned to the variable i. In step S443, a value of 1 is assigned to the bit at the ith bit index of the vector z.

In step S445, the vector S is accumulated in the vector x

The process returns to step S419.

On the other hand, if it is determined in step S419 that the continue value is not true, the procedure proceeds to step S447 so that the Hamming weight wt (z) of the vector z is 1 and the Hamming weight wt (x) of the vector x is 2 It is checked whether the number of holes is larger. If it is determined in step S447 that the Hamming weight of the vector z is 1 and the condition that the Hamming weight of the vector x is a number of holes larger than 2 is not satisfied, the procedure proceeds to step S449, Check whether the weight is 1 or not. If it is determined in step S449 that the Hamming weight of the vector z is 1, it is determined in step S451 that the bits of the vector z at the bit indices of b _j or less, where the bit values of the vector x are 0, 1 " In one embodiment, the value of alpha may be set to 0.5. After step S451, the procedure goes to step S453. On the other hand, if it is determined in step S447 that the Hamming weight of the vector z is 1 and the Hamming weight of the vector x is an odd number greater than 2, in step S455, the bit values of the vector x are added to the bits of the vector z And the procedure goes to step S453. If it is determined in step S449 that the hamming weight of the vector z is not 1, the procedure proceeds to step S453. Step (S453), we assign the last (rightmost) bit value of the index of a bit of non-zero bits in the vector z to the p-bit index _j. In step S457 subsequent to step S453, the support set Tj is constructed according to the following equation (12).

&Quot; (12) "

According to Equation (12) above, the support set Tj is composed of bit indexes that are not p _j and bit indexes of b _j or less in which the bit values of the vector z are 1. Subsequent to step S457, step S459 adds 1 to the variable j, and the procedure returns to step S409. On the other hand, if it is determined in step S409 that the check result j is less than or equal to r, it is determined in step S461 whether bit indices p ₁ to _{r r for} parity check bits and parity check bits And outputs the support sets T ₁ to T _r . The bit indices (p ₁ to p _r ) for the parity check bits and the support sets (T ₁ to T _r ) for the corresponding parity check bits are determined for each of the constituent vectors of the set of input vectors (V ' _dmin ) (V _dmin ) is generated by taking only the information bits and the parity check bits excluding the frozen bits (refer to step S413), and it is determined based on this. After that, in this step, the bit indexes p ₁ to p _r ) and the indexes included in the support sets (T ₁ to T _r ) for the parity check bits can be changed accordingly. In the following description, the bit indices (p ₁ to p _r ) for the parity check bits are indicated by the parity check bit index set P = {p ₁ , p ₂ , p ₃ , ..., p _r } .

Referring back to FIG. 1, the outer encoder 110 arranges and outputs r parity check bits from the bit indices p ₁ to p _r determined by the procedure illustrated in FIG. 4, and outputs a parity check bit index set P and the bit indexes included in the frozen bit index set F, and output the m information bits. The outer encoder 110 may be configured to determine the values of the r parity check bits q ₁ to q _r by the support sets T ₁ to T _r determined by the procedure illustrated in FIG. For example, _{T 3 = {2, 4,} 8} in case the outer code encoder 110 the second, fourth, and eighth bit of the second add operation, the value of the bit value to the module in the index bit index 3 (p ₃₎ It can be configured to determine a bit value (q ₃₎ of the bit (a ₃₎ of the. Not only m information bits and r parity check bits are input to the polarizer 120 but also [n - (m + r)] frozen bits are input in the bit indices included in the frozen bit index set F .

In designing the parity check concatenated coders used in the simulation of FIG. 5, the frozen bit indices are obtained by a Gaussian approximation method. As can be seen from FIG. 5, when the number r of parity check bits is 20, the best performance is shown. A frozen bit index set F including 172 bit indices is shown in Equation (13) below, and a parity check bit index set P including 20 bit indices is represented by 14, and 20 support sets (T ₁ to T ₂₀ ) for the parity check bits are shown in the following equations (15) to (34).

&Quot; (13) "

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59 , 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, , 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 123, 125, 129, 130, 131, 132, 133, 134, 135 , 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 161, 162, 163, 164, 165, 166, 167 , 169, 170, 171, 173, 177, 178, 179, 181, 183, 185, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 207, 209 , 210, 211, 213, 215, 217, 218, 219, 221, 225, 226, 227, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243,

&Quot; (14) "

P = {159, 191, 122, 182,110, 30,62, 52,76, 44,28,124,72,40,24,184,16,112,160,64}

&Quot; (15) "

T ₁ = {119, 239, 247, 251}

&Quot; (16) "

T ₂ = {63, 95, 111, 119, 175}

&Quot; (17) "

T ₃ = {111, 127, 175, 186, 191, 239, 251, 253, 255}

&Quot; (18) "

T ₄ = {63, 95, 111, 119, 127, 159, 186, 223, 239, 250, 253}

&Quot; (19) "

T ₅ = {111, 159, 174, 182, 186, 191, 206, 214, 223, 239, 246, 247, 250, 251, 253, 255}

&Quot; (20) "

T ₆ = {63, 95, 119, 122, 127, 159, 175, 247, 251, 255}

&Quot; (21) "

T ₇ = {119, 159, 174, 182, 186, 206, 214, 238, 239, 247, 253, 255}

&Quot; (22) "

T ₈ = {111, 118, 119, 122, 127, 159, 214, 246, 253, 255}

&Quot; (23) "

T ₉ = {52, 110, 118, 119, 175, 186, 190, 191, 212, 222, 223, 244, 250, 253, 254, 255}

&Quot; (24) "

T ₁₀ = {63, 95, 159, 175, 191, 223, 251, 253, 255}

&Quot; (25) "

T ₁₁ = {52, 62, 76, 118, 119, 122, 126, 158, 172, 174, 175, 182, 190, 204, 214, 222, 223, 228, 238, 246,

&Quot; (26) "

T ₁₂ = {28,52,62,95,108,126,127,156,172,174,175,180,182,186,190,204,206,222,228,236,238,246, 253, 255}

&Quot; (27) "

T ₁₃ = {136}

&Quot; (28) "

T ₁₄ = {72}

&Quot; (29) "

T ₁₅ = {168}

&Quot; (30) "

T ₁₆ = {16, 44, 52, 76, 88, 94, 95, 104, 108, 110, 116, 120, 159, 175, 182, 186, 191, 200, 204, 206, 216, 232, 244, 246, 252, 254}

&Quot; (31) "

T ₁₇ = {120}

(32)

T ₁₈ = {122, 127, 174, 175, 186, 191, 214, 238, 239, 247, 255}

&Quot; (33) "

T ₁₉ = {24, 28, 48, 52, 56, 60, 62, 64, 76, 80, 88, 92, 94, 110, 111, 112, 116, 124, 127, 144, 180, 190, 204, 206, 216, 223, 228, 232, 240, 246, 247, 248, 251, 252, 255}

&Quot; (34) "

T ₂₀ = {24, 28, 56, 60, 88, 124, 144, 152, 208}

In designing the parity check concatenated coders used in the simulation of FIG. 6, frozen bit indices are obtained by using a Gaussian approximation method. As can be seen from FIG. 6, the best performance is shown when the number r of parity check bits is 20. A frozen bit index set F including 151 bit indexes is shown in Equation 35 below and a parity check bit index set P including 20 bit indexes is set to 36, and 20 support sets (T ₁ to T ₂₀ ) for parity check bits are shown in the following equations (37) to (56).

&Quot; (35) "

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 29, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 55, 57, 58, 59, , 67, 68, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89, 90, 91, 93, 97, 98, 99, 101, 102, 103 , 105, 106, 107, 109, 113, 114, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 143, , 150, 151, 153, 154, 155, 157, 161, 162, 163, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181, 185, 189, 193, 194, 195 , 196, 197, 198, 199, 201, 202, 203, 205, 209, 210, 211, 213, 217, 219, 221, 225, 226, 227, 229, 233, 235, 237, 241, 242, 243 , 245, 249}

&Quot; (36) "

P = {247, 31, 191, 186, 86, 142, 46, 30, 164, 84, 140, 44, 28, 136, 40, 24,

&Quot; (37) "

T ₁ = {123, 187, 253}

&Quot; (38) "

T ₂ = {111, 183, 215, 231, 247, 251}

[Equation 39]

T ₃ = {31, 63, 95, 111, 123, 175, 187, 231, 247, 253}

[Equation 40]

T ₄ = {63, 119, 123, 127, 175, 187, 191, 218, 223, 251, 253, 255}

(41)

T ₅ = {63, 111, 119, 123, 125, 159, 175, 187, 215, 223, 234, 239, 247, 250, 255}

(42)

T ₆ = {54}

Equation (43)

T ₇ = {78}

&Quot; (44) "

T ₈ = {46}

&Quot; (45) "

T ₉ = {30}

&Quot; (46) "

T ₁₀ = {100, 148}

&Quot; (47) "

T ₁₁ = {52, 84}

&Quot; (48) "

T ₁₂ = {52, 76}

&Quot; (49) "

T ₁₃ = {44}

(50)

T ₁₄ = {92}

&Quot; (51) "

T ₁₅ = {60, 72}

(52)

T ₁₆ = {28, 40}

&Quot; (53) "

T ₁₇ = {24}

(54)

T ₁₈ = {30, 31, 44, 52, 62, 63, 76, 86, 94, 100, 110, 116, 118, 119, 123, 126, 127, 142, 156, 158, 164, 182, 186, 230, 231, 234, 238, 239, 253}, < / RTI >

(55)

T ₁₉ = {16, 24, 40, 48, 56, 72}

&Quot; (56) "

T ₂₀ = {30, 40, 46, 54, 62, 63, 76, 78, 80, 88, 92, 94, 95, 96, 108, 110, 111, 112, 124, 127, 142, 156, 159, 164, 168, 172, 180, 183, 186, 188, 190, 200, 208, 215, 223, 228, 234, 240, 246, 247, 250, 251,

In designing the parity check concatenated coders used in the simulation of FIG. 7, the frozen bit indexes are obtained using a Gaussian approximation method. As can be seen from FIG. 7, when the number r of parity check bits is 20, the best performance is shown. A frozen bit index set F including 108 bit indices is shown in Equation (57) below, and a parity check bit index set P including 20 bit indices is represented by The following 20 sets of support sets (T ₁ to T ₂₀ ) for parity check bits are shown in Equation (58) and Equation (59) to Equation (78) below.

&Quot; (57) "

F = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 25, 26, 27, 29, , 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 57, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, , 85, 89, 93, 97, 98, 99, 101, 105, 109, 113, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, , 146, 147, 149, 153, 157, 161, 162, 163, 165, 169, 171, 173, 177, 179, 181, 185, 193, 194, 195, 197, 201, 203, 205, , 211, 213, 217, 225, 226, 227, 229, 233, 241}

(58)

P = {91, 199, 247, 239, 191, 106, 186, 22, 14, 238, 190, 36, 20, 12, 28, 8, 24,

(59)

T ₁ = {107, 221, 235, 237}

(60)

T ₂ = {59, 61, 123, 155, 187, 221, 235, 243, 253}

&Quot; (61) "

T ₃ = {55, 61, 87, 91, 103, 123, 125, 155, 167, 183, 187, 189, 215, 219, 231, 235, 245, 253}

(62)

T ₄ = {61, 87, 103, 119, 125, 151, 155, 167, 183, 187, 189, 199, 207, 215, 219, 235, 247}

Equation (63)

_{T 5 = {31, 55,} 79, 87, 91, 95, 107, 125, 151, 159, 167, 183, 187, 189, 207, 223, 235, 237, 239, 243, 245, 247}

Equation (64)

T ₆ = {47, 55, 91, 95, 111, 119, 125, 143, 151, 155, 159, 167, 170, 175, 178, 183, 187, 202, 215, 231, 242, 245, 249}

Equation (65)

T ₇ = {55, 58, 59, 63, 79, 90, 95, 103, 106, 111, 123, 125, 127, 151, 155, 221, 237, 239, 243, 249, 255}

[Equation 66]

_{T 8 = {31, 47,} 55, 58, 59, 61, 79, 87, 90, 95, 107, 111, 119, 125, 127, 143, 151, 155, 167, 178, 183, 187, 207, 215, 218, 221, 223, 230, 231, 235, 239, 243, 245, 249, 253, 255}

Equation (67)

T ₉ = {22, 31, 54, 55, 58, 59, 86, 87, 90, 95, 107, 111, 114, 118, 119, 123, 125, 143, 151, 154, 170, 175, 178, 183, 186, 187, 191, 199, 219, 221, 231, 242, 245, 250, 251, 253, 255}

Equation (68)

T ₁₀ = {54, 55, 58, 59, 78, 86, 87, 91, 95, 106, 107, 110, 111, 119, 125, 127, 151, 154, 175, 178, 206, 207, 214, 218, 219, 221, 231, 242, 243, 245, 247, 249, 250, 255}

[Equation 69]

_{T 11 = {30, 46,} 58, 59, 62, 94, 102, 107, 111, 119, 122, 125, 151, 159, 182, 183, 186, 187, 189, 191, 207, 215, 218, 221, 230, 234, 238, 239, 242, 243, 247, 250, 253}

[Equation 70]

T ₁₂ = {68}

&Quot; (71) "

T ₁₃ = {36}

(72)

T ₁₄ = {20}

Equation (73)

_{T 15 = {22, 44,} 47, 52, 54, 55, 61, 62, 76, 79, 87, 90, 91, 94, 95, 108, 110, 111, 114, 116, 122, 125, 126, 228, 230, 231, 238, 239, 244, 250, 238, 234, 234, 251, 255}

[Equation 74]

T ₁₆ = {76}

[Equation 75]

T ₁₇ = {8, 12, 20, 36, 40, 52, 68, 76}

[Equation 76]

T ₁₈ = {8, 24, 30, 72, 108, 136}

[Equation 77]

T ₁₉ = {28, 104, 136, 254}

[Equation 78]

T ₂₀ = {16, 208}

In designing the parity check concatenated coders used in the simulation of FIG. 8, the frozen bit indices are obtained by a Gaussian approximation method. As can be seen from FIG. 8, when the number r of parity check bits is 16, the best performance is shown. A frozen bit index set F including 69 bit indices is shown in Equation 79 below and a parity check bit index set P including 16 bit indices is represented by The following sixteen sets of support (T ₁ to T ₁₆ ) for the parity check bits are shown in the following Equation (81) to Equation (96).

[Equation 79]

F = {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21, 25, 33, 34, 35, 37, 41, 45, 49, 51, 53 , 57, 65, 66, 67, 69, 73, 77, 81, 83, 85, 89, 97, 98, 99, 101, 105, 113, 129, 130, 131, 133, 193, 194, 195, 197, 201, 209, 217, 225, 229, 233, 241, 147, 149, 153, 161, 162, 163, 165, 169, 177,

[Equation 80]

P = {179, 155, 39, 247, 111, 50, 90, 230, 142, 190, 4, 12, 28, 8, 16,

Equation (81)

T ₁ = {29, 93, 109, 117, 157, 181, 185, 205, 211, 213, 245, 249}

Equation (82)

T ₂ = {27, 61, 75, 93, 121, 157, 171, 173, 203, 205, 211, 213, 235, 237, 243}

[Equation 83]

T ₃ = {27, 29, 43, 59, 71, 91, 107, 109, 117, 121, 125, 135, 171, 173, 179, 203, 205, 219, 235, 245,

Equation (84)

T ₄ = {27, 39, 115, 119, 121, 123, 155, 167, 173, 179, 187, 189, 199, 205, 211, 215, 221, 227, 235, 237, 243, 249}

Equation (85)

T ₅ = {29, 39, 43, 55, 79, 93, 107, 109, 119, 143, 151, 155, 157, 171, 173, 175, 183, 185, 189, 199, 211, 215, 219, 221, 231, 235, 249, 251, 253}

[Equation (86)

T ₆ = {29, 31, 39, 43, 63, 75, 82, 87, 91, 103, 121, 125, 151, 173, 187, 199, 207, 211, 215, 219, 223, 226, 227, 231, 235, 243, 247, 249, 251, 253, 255}

&Quot; (87) "

T ₇ = {23,26,27,31,43,95,61,74,75,79,82,91,93,103,106,109,111,114,117,145,155,159,171,171,171,171,171,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,16 234, 235, 239, 243, 247, 249, 231, 235,

(88)

T ₈ = {23,26,27,29,38,39,42,43,55,59,63,70,71,74,79,82,87,90,95,109,117,117,119,119,12,13,14,15,16,16,16,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,15,16,14,14,14,15,16,14,14,15,16,14,16,14,15,16,14,15,16,14,15,16,14,16,16 181, 199, 203, 205, 210, 215, 234, 235, 242, 250, 251}, 121, 127, 138, 143, 166, 170, 173, 178, 179, 186,

[Equation 89]

T ₉ = {15, 26, 29, 39, 42, 47, 50, 54, 55, 58, 59, 61, 63, 70, 71, 79, 82, 86, 87, 91, 93, 109, 111, 114, 121, 123, 125, 134, 135, 146, 150, 151, 159, 178, 179, 182, 185, 186, 187, 189, 202, 207, 213, 214, 223, 226, 230, 231, 234, 235, 237, 242, 243, 245, 246, 249, 251, 253, 255}

[Equation 90]

T ₁₀ = {14, 15, 22, 23, 27, 30, 47, 50, 54, 55, 58, 61, 62, 71, 75, 78, 82, 86, 111, 114, 115, 117, 119, 121, 122, 142, 146, 150, 155, 158, 159, 167, 171, 175, 179, 181, 183, 185, 187, 211, 214, 215, 221, 222, 230, 234, 239, 242, 245, 247, 250, 255}

&Quot; (91) "

_{T 11 = {14, 22,} 23, 26, 29, 30, 38, 39, 42, 47, 50, 54, 55, 58, 62, 63, 71, 74, 86, 87, 90, 91, 102, 171, 171, 173, 179, 181, 182, 183, 187, 189, 191, 202, 206, 206, 106, 110, 111, 114, 115, 121, 125, 225, 211, 213, 214, 218, 221, 222, 223, 226, 227, 238, 239, 247, 250, 253, 254, 255}

(92)

T ₁₂ = {20, 30, 36, 68, 132, 182}

(93)

T ₁₃ = {14, 36}

Equation (94)

T ₁₄ = {12, 118}

[Equation (95)

T ₁₅ = {4, 14, 22, 23, 24, 26, 28, 29, 31, 42, 43, 44, 52, 54, 56, 59, 60, 62, 70, 76, 79, 184, 174, 175, 181, 183, 185, 186, 174, 148, 154, 159, 166, 173, 174, 175, 118, 121, 122, 123, 124, 186, 187, 189, 191, 196, 200, 207, 210, 211, 212, 215, 218, 219, 221, 228, 230, 231, 232, 234, 235, 236, 239, 243, 246, 247, 248, 249, 251, 252, 254}

[Equation 96]

T ₁₆ = {20}

In designing the parity check concatenated coders used in the simulation of FIG. 9, frozen bit indexes are obtained using a Gaussian approximation method. As can be seen from FIG. 9, the best performance is shown when the number r of parity check bits is 22. A frozen bit index set F including 362 bit indices is shown in Equation 97 below and a parity check bit index set P including 22 bit indices is represented by Equation (98), and the 22 support sets (T ₁ to T ₂₂ ) for the parity check bits are shown in the following equations (99) to (120).

(97)

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, , 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, , 85, 86, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 111, 113, , 119, 121, 122, 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149 , 150, 151, 153, 154, 155, 157, 159, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 175, 177, 178, 179, 181, 182, 183 , 185, 186, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 206, 207, 209, 210, 211, 213, 214, 215, 217, 218 , 219, 221, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243, 245, 249, 251, 253, 257, 258, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274, 275, 276, 277, 278, 279, 281, 282, 313, 314, 315, 315, 315, 315, 315, 318, 315, 342, 343, 345, 346, 347, 349, 342, 333, 334, 335, 337, 338, 339, 340, 373, 377, 379, 381, 385, 386, 387, 388, 389, 390, 391, 361, 362, 363, 365, 369, 370, 393, 395, 397, 398, 399, 401, 402, 403, 404, 405, 406, 407, 409, 410, 411, 413, 417, 418, 419, 420, 421, 422, 423, 425, 426, 427, 429, 433, 434, 435, 437, 441, 443, 445, 449, 450, 451, 452, 453, 454, 455, 457, 458, 459, 461, 465, 466, 467, 469, 471, 473, 474, 475, 477, 481, 482, 483, 485, 487, 489, 490, 491,

Equation (98)

P = {255, 438, 430, 286, 62, 254, 180, 396, 428, 236, 92, 380, 296, 488, 24, 184, 16, 80, 48, 32,

Equation (99)

T ₁ = {63, 239, 247, 319, 351, 431, 463, 495, 503}

(100)

T ₂ = {63, 191, 239, 250, 255, 319, 367, 375, 383, 470, 479, 486, 495, 506, 507, 509, 511}

&Quot; (101) "

T ₃ = {127, 174, 247, 319, 375, 378, 383, 439, 442, 447, 462, 463, 479, 502, 503, 506, 507, 511}

Equation (102)

T ₄ = {63, 191, 223, 319, 366, 374, 415, 430, 431, 438, 439, 442, 447, 463, 470, 494,

&Quot; (103) "

T ₅ = {63, 94, 127, 158, 191, 239, 246, 247, 250, 255, 319, 350, 351, 366, 374, 375, 378, 383, 414, 430, 431, 439, 442, 462, 463, 470, 478, 479, 486, 494, 495, 502, 503, 511}

Equation (104)

_{T 6 = {63, 110,} 127, 158, 191, 239, 318, 366, 378, 383, 415, 431, 438, 439, 442, 462, 463, 478, 479, 486, 494, 502, 506, 511}

Equation (105)

_{T 7 = {62, 63,} 94, 110, 127, 158, 191, 212, 222, 238, 239, 246, 247, 250, 255, 308, 318, 351, 367, 374, 375, 378, 382, 415, 438, 439, 442, 446, 447, 462, 463, 478, 506, 507, 510}

&Quot; (106) "

_{T 8 = {62, 110,} 126, 158, 212, 222, 223, 238, 247, 254, 255, 286, 308, 318, 319, 374, 383, 414, 430, 431, 436, 442, 447, 495, 500, 502, 503, 506}

&Quot; (107) "

T ₉ = {62,144,110,127,158,180,190,191,204,222,223,239,247,250,300,308,318,350,351,366,367,374,383, 496, 495, 500, 507, 509, 510, 511}, < / RTI >

(108)

T ₁₀ = 110, 116, 126, 174, 180, 191, 204, 212, 222, 238, 239, 246, 250, 254, 300, 318, 332, 350, 364, 366, 375, 383, 430, 438, 439, 446, 447, 460, 468, 470, 484, 486, 500, 506, 509, 510}

(109)

_{T 11 = {62, 110,} 116, 126, 127, 190, 191, 212, 222, 236, 308, 318, 332, 364, 367, 378, 382, 383, 396, 415, 428, 438, 439, 446, 447, 460, 462, 463, 468, 478, 479, 484, 486, 492, 495, 500, 502, 507, 509, 510, 511}

(110)

T ₁₂ = {62, 92, 94, 116, 124, 188, 212, 236, 238, 239, 300, 308, 318, 332, 348, 350, 367, 372, 374, 375, 383, 414, 436, 438, 444, 447, 468, 470, 476, 479, 484, 495, 500, 502, 503, 507, 511}

(111)

T ₁₃ = {60, 62, 110, 124, 127, 158, 172, 174, 188, 191, 200, 212, 239, 244, 247, 284, 286, 300, 308, 319, 332, 367, 372, 508, 509, 510, 511}, < / RTI >

(112)

T ₁₄ = {62, 63, 92, 94, 124, 158, 172, 204, 220, 222, 223, 238, 239, 244, 247, 250, 254, 286, 296, 300, 318, 364, 367, 375, 382, 414, 415, 424, 428, 431, 436, 438, 439, 442, 446, 456, 462, 468, 470, 479, 484, 492, 494, 495, 506, 507, 509, 510, 511}

&Quot; (113) "

T ₁₅ = {104, 116, 127, 168, 172, 180, 190, 191, 200, 212, 220, 223, 232, 238, 239, 250, 252, 254, 284, 286, 296, 300, 308, 316, 318, 319, 328, 348, 350, 351, 364, 366, 374, 375, 378, 380, 382, 383, 392, 415, 424, 428, 431, 436, 442, 470, 476, 478, 479, 484, 502, 506, 510}

(114)

T ₁₆ = {24, 56, 60, 104, 110, 116, 124, 156, 168, 172, 190, 200, 204, 216, 220, 222, 223, 244, 246, 250, 252, 254, 284, 286, 316, 318, 328, 332, 344, 348, 351, 360, 364, 367, 372, 374, 375, 412, 430, 436, 442, 446, 456, 460, 463, 468, 470, 472, 478, 479, 486, 488, 500, 502, 507, 508, 509, 510, 511}

(115)

T ₁₇ = {56, 60, 94, 110, 120, 124, 156, 158, 168, 180, 190, 191, 204, 212, 216, 220, 222, 238, 244, 247, 248, 462, 463, 468, 438, 434, 436, 438, 439, 447, 456, 462, 464, 464, 470, 472, 476, 478, 479, 486, 488, 492, 494, 495, 500, 503, 506, 507, 509}

&Quot; (116) "

T ₁₈ = {24, 88, 144, 152, 248, 272}

(117)

T ₁₉ = {80, 144, 184}

&Quot; (118) "

T ₂₀ = {80, 88, 112}

[Formula 119]

T ₂₁ = {16}

(120)

T ₂₂ = {32}

In designing the parity check concatenated coders used in the simulation of FIG. 10, the frozen bit indices are obtained using a Gaussian approximation method. As can be seen from FIG. 10, when the number r of parity check bits is 20, the best performance is shown. A frozen bit index set F including 321 bit indices is shown in Equation 121 below and a parity check bit index set P including 20 bit indices is represented by 122, and 20 support sets (T ₁ to T ₂₀ ) for the parity check bits are shown in the following equations (123) to (142).

(121)

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58 , 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86, , 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 123, 125, 129, 130, 131, 132, 133, 134 , 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 161, 162, 163, 164, 165, 166 , 167, 169, 170, 171, 173, 177, 178, 179, 181, 183, 185, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 207 , 209, 210, 211, 213, 215, 217, 218, 219, 221, 225, 226, 227, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243, 245, 249, 257 278, 275, 276, 277, 278, 279, 281, 282, 283, 285, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274, , 289, 290, 291, 292, 293, 294, 295, 297, 298, 299, 301, 305, 306, 307, 309, 311, 313, 31 343, 345, 346, 347, 349, 353, 334, 333, 335, 337, 338, 339, 341, 343, 324, 325, 326, 327, 329, 374, 377, 385, 386, 387, 388, 389, 390, 391, 393, 394, 395, 397, 375, 362, 363, 365, 369, 370, 419, 421, 422, 423, 425, 426, 427, 429, 433, 434, 435, 437, 419, 419, 419, 419, 419, 401, 402, 403, 405, 406, 441, 445, 449, 450, 451, 452, 453, 454, 455, 457, 458, 459, 461, 465, 466, 467, 469, 473, 477, 481, 482, 483, 485, 493, 497, 499, 501, 505}

(122)

P = {175, 223, 223, 506, 118, 430, 30, 382, 340, 396, 44, 28, 60, 40, 24, 88, 16,

[Formula 123]

T ₁ = {119, 247, 251, 443, 471, 487, 503, 507}

Equation (124)

T ₂ = {95, 159, 175, 253, 287, 303, 351, 367, 379, 381, 439, 443, 463, 475, 487, 495, 507}

[Formula 125]

T ₃ = {95, 159, 175, 287, 381, 415, 439, 463, 471, 487, 507, 509}

[Equation 126]

_{T 4 = {63, 95,} 122, 127, 159, 191, 251, 319, 351, 367, 378, 379, 381, 439, 447, 463, 471, 474, 479, 495, 498, 503, 507}

[Equation 127]

T ₅ = 95, 122, 127, 159, 175, 182, 191, 247, 250, 251, 253, 255, 287, 303, 342, 439, 442, 443, 470, 471, 475, 498, 503, 506, 507}

(128)

_{T 6 = {63, 95,} 119, 122, 159, 174, 175, 186, 250, 251, 253, 287, 302, 310, 319, 342, 367, 378, 379, 383, 398, 431, 442, 447, 463, 470, 474, 479, 486, 490, 498, 503}

(129)

_{T 7 = {95, 110,} 122, 159, 174, 175, 182, 186, 206, 223, 247, 255, 287, 303, 310, 319, 366, 367, 374, 381, 383, 430, 431, 438, 442, 447, 463, 471, 486, 487, 490, 498, 503, 506, 507, 509, 511}

(130)

_{T 8 = {62, 63,} 94, 110, 111, 122, 127, 159, 174, 175, 190, 206, 214, 222, 223, 246, 250, 251, 286, 287, 302, 334, 350, 466, 374, 375, 378, 398, 415, 430, 439, 443, 462, 463, 470, 471, 474, 475, 479, 486, 490, 494, 498,

[Formula 131]

T ₉ = {110, 119, 127, 175, 182, 186, 191, 206, 222, 223, 238, 239, 246, 247, 250, 251, 286, 287, 302, 318, 319, 356, 366, 375, 378, 379, 381, 383, 398, 414, 415, 431, 442, 443, 446, 447, 462, 471, 479, 486, 490, 494, 495, 506,

(132)

T ₁₀ = 30, 62, 63, 94, 111, 116, 127, 159, 174, 175, 206, 214, 222, 223, 244, 250, 253, 462, 463, 468, 479, 484, 487, 490, 494, 495, 500, 372, 372, 375, 378, 398, 404, 430, 436, 438, 442, 446, 502, 507, 509}

(133)

T ₁₁ = {76}

(134)

T ₁₂ = {44}

Equation (135)

T ₁₃ = {63, 94, 108, 110, 111, 116, 119, 126, 158, 159, 174, 175, 182, 186, 191, 204, 214, 239, 388, 404, 414, 415, 428, 430, 431, 436, 447, 462, 463, 468, 471, 475, 478, 487, 490, 495, 500, 502, 507, 510, 511}

[Equation (136)

T ₁₄ = {72, 92, 136}

(137)

T ₁₅ = {40, 156}

(138)

T ₁₆ = {316}

[Formula 139]

T ₁₇ = {72}

(140)

T ₁₈ = {24, 28, 32, 40, 44, 48, 56, 60, 80, 92, 95, 104, 111, 122, 124, 186, 188, 191, 200, 204, 214, 222, 228, 236, 239, 244, 247, 248, 251, 253, 272, 286, 287, 300, 316, 318, 332, 342, 350, 356, 440, 442, 456, 463, 468, 471, 475, 476, 484, 487, 490, 492, 495, 504, 507, 510, 511}

Equation (141)

T ₁₉ = {24, 40, 48, 56, 136, 144}

[Equation 142]

T ₂₀ = {24,28,30,32,44,60,62,72,80,104,108,111,116,118,119,122,124,126,127,168,180,182,184, 248, 251, 252, 253, 255, 272, 280, 287, 288, 296, 382, 383, 392, 396, 400, 404, 408, 370, 370, 370, 370, 412, 414, 416, 420, 424, 428, 430, 432, 436, 438, 439, 442, 462, 468, 475, 476, 478, 480, 486, 488, 490, 494, 508}

In designing parity check concatenated coders used in the simulation of FIG. 11, frozen bit indices are obtained by using a Gaussian approximation method. As can be seen from FIG. 11, when the number r of parity check bits is 22, the best performance is shown. A frozen bit index set F including 234 bit indexes is shown in Equation (143) below, and a parity check bit index set P including 22 bit indices is represented by The following set of 22 support sets (T ₁ to T ₂₂ ) for the parity check bits are shown in the following mathematical expression (144) and the following mathematical expressions (145) to (166).

[Formula 143]

F = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, , 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71 , 73, 74, 75, 77, 81, 82, 83, 85, 89, 91, 93, 97, 98, 99, 101, 103, 105, 107, 109, 113, 114, , 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 145, 146, 147, 149, 151, 153, 155, 157, 161, 162, 163, 165, 167, 169, 170 , 171, 173, 177, 178, 179, 181, 185, 193, 194, 195, 197, 198, 199, 201, 202, 203, 205, 209, 210, 211, 213, 217, 225, 226, 227 , 229, 233, 237, 241, 245, 249, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 269, 273, 274, 275, 277, 279, 281, 282, 283 , 285, 289, 290, 291, 293, 295, 297, 298, 299, 301, 305, 306, 307, 309, 313, 321, 322, 323, 325, 326, 327, 329, 330, 331, 333 , 337, 338, 339, 341, 345, 349, 353, 354, 355, 357, 361, 365, 369, 371, 373, 377, 385, 386, 387, 389, 390, 391, 393, 394, 395 , 397, 401, 402, 403, 405, 409, 413, 417, 418, 419, 421, 425, 427, 429, 4 467, 469, 473, 481, 482, 483, 485, 489, 497, 435, 437, 441, 449, 450, 451, 453, 457, 459, 461, 465,

(144)

P = {379, 343, 271, 31, 127, 362, 442, 230, 438, 78, 414, 324, 404, 500, 12, 476, 8, 40, 24,

[Equation 145]

T ₁ = {219, 221, 235, 347, 363, 411, 443, 445, 493, 499, 501, 505}

[Equation 146]

T ₂ = {87, 125, 187, 231, 235, 243, 315, 347, 359, 379, 381, 411, 423, 455, 475, 501, 507}

&Quot; (147) "

T ₃ = {87, 123, 189, 219, 231, 235, 247, 251, 253, 311, 343, 347, 375, 381, 407, 475, 477, 487, 501, 505, 507, 509}

(148)

T ₄ = {55, 87, 123, 143, 183, 207, 235, 239, 253, 271, 311, 315, 359, 363, 375, 381, 407, 423, 431, 443, 445, 455, 463, 487, 491, 499, 501, 503}

[Equation 149]

_{T 5 = {31, 79,} 111, 123, 125, 143, 159, 183, 207, 219, 235, 247, 251, 253, 287, 311, 335, 343, 347, 351, 359, 367, 399, 407, 411, 439, 445, 455, 463, 471, 477, 479, 491, 501}

[Equation 150]

T ₆ = {31, 55, 63, 79, 95, 106, 119, 123, 143, 159, 175, 183, 187, 189, 191, 219, 239, 243, 251, 271, 287, 317, 319, 343, 347, 363, 367, 370, 381, 383, 407, 431, 434, 439, 443, 445, 455, 458, 463, 466, 475, 487, 493, 495, 505, 509, 511}

(151)

_{T 7 = {47, 63,} 127, 154, 183, 186, 221, 243, 247, 251, 255, 271, 303, 311, 315, 347, 351, 362, 363, 367, 375, 411, 415, 423, 426, 431, 434, 439, 443, 447, 466, 471, 474, 477, 491, 493, 495, 498, 501, 503, 507}

(152)

_{T 8 = {31, 63,} 79, 87, 90, 95, 102, 111, 119, 127, 143, 166, 186, 187, 189, 207, 218, 219, 221, 223, 231, 242, 251, 431, 434, 439, 442, 445, 447, 455, 458, 466, 381, 415, 422, 423, 471, 474, 477, 479, 490, 493, 499, 506, 507}

[Formula 153]

T ₉ = {47, 54, 58, 79, 87, 95, 111, 119, 122, 123, 125, 150, 166, 175, 187, 214, 215, 218, 221, 223, 230, 235, 243, 361, 363, 378, 381, 383, 406, 407, 410, 411, 423, 426, 431, 434, 439, 442, 443, 447, 454, 458, 466, 471, 475, 477, 479, 491, 495, 505, 507}

(154)

T ₁₀ = {47, 58, 63, 79, 90, 102, 106, 111, 118, 142, 143, 166, 182, 183, 186, 187, 207, 214, 219, 230, 373, 407, 415, 422, 431, 443, 373, 374, 377, 372, 445, 454, 455, 458, 471, 491, 493, 495, 498, 501, 503, 506, 507}

(155)

_{T 11 = {30, 46,} 79, 87, 90, 95, 119, 125, 142, 154, 158, 166, 174, 182, 183, 186, 206, 214, 215, 218, 221, 230, 234, 235, 238, 239, 242, 246, 247, 250, 251, 253, 302, 303, 310, 311, 314, 317, 334, 342, 343, 346, 351, 366, 367, 370, 375, 378, 381, 398, 407, 410, 415, 423, 426, 431, 439, 442, 445, 454, 455, 458, 462, 466, 471, 475, 486, 487, 491, 493, 495, 506, 507}

(156)

T ₁₂ = {46, 54, 63, 79, 90, 94, 95, 122, 125, 126, 127, 142, 143, 174, 175, 183, 189, 190, 206, 207, 215, 218, 219, 314, 315, 318, 319, 359, 363, 374, 375, 370, 370, 378, 381, 383, 388, 398, 399, 407, 410, 414, 415, 423, 431, 439, 442, 445, 447, 454, 463, 466, 470, 471, 475, 477, 478, 493, 494, 503, 509}

[Formula 157]

T ₁₃ = {31, 46, 54, 62, 63, 78, 87, 94, 95, 102, 118, 119, 127, 148, 154, 166, 175, 186, 189, 190, 191, 292, 294, 302, 317, 319, 324, 347, 350, 351, 359, 373, 470, 474, 479, 484, 490, 440, 440, 423, 430, 438, 439, 442, 491, 493, 494, 495, 505, 507, 510, 511}

[Formula 158]

T ₁₄ = {46, 47, 52, 54, 58, 63, 78, 86, 90, 94, 95, 100, 102, 110, 111, 116, 118, 119, 154, 186, 314, 315, 318, 319, 324, 346, 350, 358, 362, 274, 363, 366, 370, 374, 378, 381, 388, 399, 406, 407, 411, 420, 422, 426, 430, 431, 434, 438, 439, 443, 445, 455, 463, 466, 468, 470, 471, 474, 479, 487, 490, 491, 495, 499, 501, 505}

[Equation 159]

T ₁₅ = 30, 31, 47, 52, 55, 58, 62, 78, 84, 90, 94, 95, 102, 111, 116, 123, 126, 142, 143, 148, 150, 158, 159, 234, 239, 246, 251, 253, 255, 276, 278, 292, 303, 308, 310, 311, 315, 318, 218, 221, 228, 230, 439, 443, 443, 431, 432, 424, 414, 415, 422, 426, 430, 431, 438, 494, 498, 501, 502, 506, 510}, < / RTI > 447, 452, 454, 463, 471, 475, 478, 479, 484, 487, 490, 491,

[Formula 160]

T ₁₆ = {12, 30, 31, 44, 46, 52, 62, 78, 84, 86, 92, 94, 95, 102, 108, 110, 119, 140, 148, 150, 158, 270, 180, 183, 187, 189, 190, 196, 207, 214, 220, 221, 230, 231, 234, 235, 238, 244, 247, 251, 253, 254, 268, 271, 358, 359, 363, 366, 367, 370, 375, 378, 379, 381, 399, 404, 407, 411, 415, 422, 423, 431, 438, 439, 442, 443, 445, 446, 447, 452, 460, 462, 466, 468, 470, 471, 474, 475, 478, 479, 486, 487, 490, 492, 494, 495, 499, 500, 503, 505, 506, 507, 509, 510, 511}

(161)

T ₁₇ = {28, 31, 44, 47, 52, 54, 55, 62, 63, 76, 78, 87, 92, 94, 100, 102, 111, 116, 118, 119, 123, 126, 140, 142, 143, 156, 159, 172, 174, 180, 183, 188, 190, 191, 196, 212, 219, 223, 228, 231, 234, 239, 242, 246, 247, 268, 366, 374, 362, 364, 364, 362, 364, 368, 375, 379, 380, 381, 382, 383, 396, 399, 404, 406, 407, 410, 411, 412, 414, 415, 420, 426, 428, 430, 431, 436, 438, 445, 446, 458, 468, 477, 478, 479, 484, 492, 495, 498, 499, 508, 509, 510, 511}

(162)

T ₁₈ = {8, 12, 72, 136, 158, 172, 264, 284, 380, 460}

[Formula 163]

T ₁₉ = {40, 44, 136, 156, 364}

[Equation 164]

T ₂₀ = {28, 40, 72, 124}

[Equation 165]

T ₂₁ = {24, 184}

[Equation 166]

T ₂₂ = {16, 220}

In designing the parity check concatenated coders used in the simulation of FIG. 12, frozen bit indexes are obtained using a Gaussian approximation method. As can be seen from FIG. 12, when the number r of parity check bits is 30, the best performance is shown. A frozen bit index set F including 141 bit indices is shown in Equation 167 below and a parity check bit index set P including 30 bit indices is set to The following set of 30 support sets (T ₁ to T ₃₀ ) for the parity check bits are shown in Equation (168) below and in Equations (169) to (198) below.

[Equation 167]

F = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21, 25, 29, 33, 34, 35, , 49, 51, 53, 57, 65, 66, 67, 69, 71, 73, 75, 77, 81, 82, 83, 85, 89, 97, 98, 99, 101, , 130, 131, 133, 135, 137, 138, 139, 141, 145, 146, 147, 149, 153, 161, 162, 163, 165, 169, 177, 181, 185, 193, 194, 195, 197 , 201, 209, 213, 217, 225, 227, 229, 233, 241, 257, 258, 259, 261, 263, 265, 266, 267, 269, 273, 274, 275, 277, , 291, 293, 297, 305, 309, 313, 321, 322, 323, 325, 329, 333, 337, 341, 345, 353, 355, 357, 361, 369, 385, 386, 387, , 397, 401, 403, 405, 409, 417, 419, 421, 425, 433, 449, 450, 451, 453, 457, 465, 481}

(168)

P = {493, 125, 435, 235, 27, 39, 23, 15, 367, 95, 210, 330, 26, 58, 38, 22, 14, 46, 190, 68, 20, 52, 12, 28 , 8, 40, 24, 144, 240, 256}

[Equation 169]

T ₁ = {109, 117, 173, 237, 301, 437, 441, 461, 489, 501}

(170)

T ₂ = {61, 117, 173, 205, 221, 237, 317, 349, 377, 441, 461, 473, 493, 497}

[Equation 171]

T ₃ = {61,93,125,179,205,221,249,301,307,317,349,365,373,437,441,445,496,473,485,489}

[Equation 172]

T ₄ = {93, 109, 125, 171, 205, 211, 221, 245, 249, 285, 299, 349, 363, 377, 395, 413, 427, 429, 435, 459, 469, 485, 497, 499, 505, 509}

[Formula 173]

T ₅ = {61, 93, 107, 115, 117, 157, 171, 173, 203, 205, 211, 237, 245, 301, 317, 349, 365, 377, 381, 427, 429, 441, 473, 477, 483, 493, 499, 501, 505, 509}

[Equation 174]

T ₆ = {59, 115, 155, 171, 179, 189, 205, 221, 237, 253, 283, 299, 301, 307, 315, 339, 347, 365, 379, 391, 395, 427, 443, 445, 455, 459, 467, 473, 475, 477, 483, 485, 489, 491, 493, 497, 499, 507}

[175]

T ₇ = {27, 91, 109, 115, 117, 123, 125, 155, 157, 171, 189, 211, 219, 231, 235, 243, 249, 283, 285, 295, 299, 315, 317, 363, 365, 373, 411, 413, 423, 427, 441, 443, 455, 459, 461, 469, 475, 477, 485, 491, 497, 499,

[Equation 176]

T ₈ = {27, 39, 55, 59, 87, 91, 107, 109, 117, 119, 151, 157, 167, 171, 173, 179, 189, 203, 205, 221, 231, 235, 243, 375, 377, 379, 407, 413, 429, 435, 461, 467, 469, 473, 477, 375, 371, 483, 485, 487, 491, 493, 497, 501, 503}

[Equation 177]

T ₉ = {15, 27, 39, 47, 59, 61, 87, 91, 93, 103, 111, 117, 125, 157, 175, 203, 207, 215, 221, 235, 237, 243, 245, 371, 373, 377, 379, 381, 395, 423, 427, 429, 435, 439, 441, 443, 463, 467, 471, 473, 483, 491, 497, 499, 501, 503, 507}

[Equation 178]

T ₁₀ = {27, 39, 55, 61, 79, 93, 103, 107, 111, 117, 119, 125, 155, 157, 173, 175, 187, 199, 203, 205, 211, 435, 375, 377, 379, 399, 407, 411, 415, 429, 435, 437, 439, 443, 445, 455, 461, 471, 475, 477, 485, 489, 491, 497, 501, 505, 507}

[179]

_{T 11 = {15, 23,} 27, 31, 47, 61, 63, 103, 109, 115, 117, 119, 143, 151, 155, 159, 167, 175, 179, 191, 203, 207, 215, 391, 331, 338, 339, 343, 347, 349, 359, 365, 371, 373, 375, 381, 383, 395, 402, 411, 493, 497, 499, 501, 503, 507, 509}, < / RTI > 413, 415, 418, 441, 447, 459, 469, 473, 477, 479, 483, 485,

(180)

T ₁₂ = {50}

[Equation 181]

T ₁₃ = {42}

&Quot; (182) "

T ₁₄ = {26, 42}

[Formula 183]

T ₁₅ = {26, 70}

[Equation 184]

T ₁₆ = {38}

[Formula 185]

T ₁₇ = {54}

[Formula 186]

T ₁₈ = {22, 26, 38, 42}

[Equation 187]

T ₁₉ = {15, 22, 23, 26, 27, 30, 31, 39, 42, 55, 59, 62, 70, 74, 78, 79, 86, 87, 91, 93, 94, 106, 110, 111, 114, 115, 127, 143, 150, 151, 166, 167, 171, 174, 175, 179, 182, 186, 198, 202, 206, 207, 221, 222, 315, 295, 298, 299, 301, 302, 303, 306, 310, 314, 315, 318, 330, 331, 334, 342, 346, 347, 349, 351, 363, 365, 371, 373, 374, 375, 377, 381, 383, 399, 402, 410, 413, 418, 422, 423, 426, 427, 429, 430, 431, 435, 437, 438, 439, 447, 454, 458, 459, 461, 462, 466, 467, 471, 474, 479, 486, 489, 491, 501, 506, 511}

[Formula 188]

T ₂₀ = {132, 278}

[Equation 189]

T ₂₁ = {36, 68, 86}

(190)

T ₂₂ = {206}

(191)

T ₂₃ = {20}

[Equation 192]

T ₂₄ = {15, 22, 26, 27, 30, 38, 39, 46, 47, 50, 52, 54, 55, 58, 61, 63, 74, 76, 78, 79, 186, 189, 190, 196, 198, 199, 202, 203, 204, 205, 210, 211, 219, 288, 292, 294, 298, 299, 300, 228, 231, 235, 236, 238, 242, 244, 245, 246, 247, 249, 250, 251, 254, 262, 268, 270, 278, 368, 362, 365, 366, 371, 377, 378, 381, 383, 388, 390, 394, 395, 396, 402, 404, 407, 410, 414, 418, 420, 423, 426, 427, 430, 437, 445, 452, 455, 463, 471, 473, 475, 477, 479, 483, 485, 486, 487, 489, 492, 495, 500, 506, 507, 510, 511}

(193)

T ₂₅ = {44}

(194)

T ₂₆ = {14, 22, 23, 27, 28, 31, 38, 39, 42, 46, 50, 54, 55, 60, 61, 62, 86, 87, 92, 94, 100, 102, 109, 115, 117, 118, 119, 124, 127, 132, 134, 136, 140, 143, 148, 150, 151, 154, 156, 164, 166, 187, 188, 190, 196, 198, 202, 203, 204, 210, 214, 218, 220, 222, 223, 228, 230, 231, 234, 237, 238, 242, 244, 245, 247, 250, 253, 254, 262, 264, 268, 270, 278, 279, 282, 283, 292, 302, 306, 308, 311, 314, 316, 378, 380, 382, 383, 391, 392, 396, 398, 402, 364, 366, 440, 412, 413, 418, 423, 429, 431, 435, 437, 438, 444, 445, 452, 459, 461, 468, 469, 470, 474, 475, 476, 477, 482, 483, 484, 485, 486, 490, 491, 494, 495, 500, 503, 505, 508, 510, 511}

(195)

T ₂₇ = {22, 27, 38, 40, 42, 47, 58, 61, 63, 76, 86, 87, 91, 92, 93, 102, 103, 108, 109, 186, 189, 191, 196, 198, 200, 204, 207, 210, 211, 212, 214, 215, 223, 228, 230, 232, 234, 243, 249, 250, 251, 252, 253, 254, 255, 262, 283, 285, 286, 292, 298, 299, 301, 302, 303, 307, 308, 374, 372, 373, 344, 342, 343, 349, 354, 359, 360, 363, 365, 366, 370, 371, 372, 434, 438, 439, 441, 442, 447, 452, 423, 424, 427, 428, 430, 434, 435, 436, 434, 379, 380, 383, 391, 399, 414, 415, 454, 455, 456, 459, 461, 462, 463, 467, 469, 470, 471, 473, 483, 486, 487, 488, 489, 490, 491, 493, 495, 497, 499, 502, 503, 507, 508, 509, 510, 511}

(196)

T ₂₈ = {12, 20, 22, 27, 28, 30, 42, 46, 58, 61, 62, 72, 86, 87, 88, 109, 111, 118, 122, 123, 124, 125, 126, 142, 143, 148, 152, 154, 155, 156, 158, 159, 166, 173, 174, 175, 180, 184, 188, 190, 231, 237, 242, 247, 248, 251, 252, 254, 260, 262, 268, 311, 312, 314, 315, 318, 326, 327, 332, 338, 340, 342, 343, 350, 351, 271, 276, 279, 282, 283, 286, 295, 299, 374, 377, 378, 379, 390, 391, 392, 404, 411, 414, 420, 422, 424, 428, 429, 431, 434, 475, 476, 482, 483, 484, 485, 475, 475, 475, 475, 486, 487, 489, 491, 492, 495, 500, 501, 506, 510, 511}

(197)

T ₂₉ = {12, 14, 15, 20, 23, 24, 26, 27, 28, 30, 38, 39, 42, 46, 47, 52, 54, Wherein the first and second optical elements are arranged at a plurality of positions spaced apart from one another at a distance from the first optical element to the second optical element. 216, 218, 220, 222, 223, 226, 230, 231, 242, 246, 248, 249, 191, 198, 199, 206, 214, 216, 318, 314, 315, 316, 318, 324, 328, and 328, 373, 374, 375, 377, 378, 379, 381, 383, 392, 396, 364, 362, 364, 367, 368, 371, 372, 398, 400, 402, 404, 408, 411, 412, 415, 420, 422, 423, 426, 427, 429, 430, 434, 435, 437, 438, 442, 443, 444, 446, 447, 455, 489, 490, 491, 492, 493, 494, 495, 498, 499, 478, 482, 483, 485, 500, 501, 502, 503, 505}

(198)

T ₃₀ = {8,12,14,23,26,27,31,32,38,39,40,44,46,47,52,56,95,63,64,70,78,80,84. 108, 91, 93, 94, 95, 96, 102, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 123, 124, 125, 134, 143, 148, 150, 151, 152, 154, 155, 156, 157, 159, 160, 164, 168, 170, 178, 183, 184, 189, 191, 196, 199, 200, 204, 206, 212, 278, 283, 285, 286, 294, 295, 296, 306, 308, 309, 358, 359, 362, 363, 346, 348, 349, 351, 354, 356, 358, 359, 394, 394, 396, 398, 399, 400, 402, 404, 408, 411, 412, 413, 414, 415, 416, 420, 423, 426, 427, 429, 432, 434, 435, 438, 440, 444, 445, 446, 447, 448, 454, 466, 468, 469, 471, 475, 477, 478, 495, 497, 500, 501, 503, 504, 505, 506, 510,

In designing the parity check concatenated coders used in the simulation of FIG. 13, frozen bit indices are obtained by using a Gaussian approximation method. As can be seen from FIG. 13, when the number r of parity check bits is 32, the best performance is shown. A frozen bit index set F including 736 bit indexes is shown in Equation 199 below and a parity check bit index set P including 32 bit indexes is represented by The number of support sets (T ₁ to T ₃₆ ) for the parity check bits is shown in Equation (201) to Equation (232) below.

[Formula 199]

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 , 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51 , 52, 53, 54, 55, 57, 58, 59, 61, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81 , 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, , 114, 115, 116, 117, 118, 119, 121, 122, 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, , 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 158, 159, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 174 , 175, 177, 178, 179, 180, 181, 182, 183, 185, 186, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 204, 205, 206 , 207, 209, 210, 211, 212, 213, 214, 215, 217, 218, 219, 221, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 241, 242 , 243, 245, 247, 249, 251, 253, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 292, 293, 294, 295, 297, 298, 299, 295, 286, 287, 323, 324, 325, 326, 327, 329, 330, 331, 311, 315, 317, 321, 322, 323, 342, 343, 345, 346, 347, 349, 353, 354, 355, 356, 357, 358, 359, 361, 362, 363, 333, 334, 335, 337, 399, 395, 396, 397, 398, 399, 396, 390, 391, 395, 414, 414, 420, 421, 422, 423, 425, 426, 427, 429, 431, 433, 434, 435, 437, 439, 441, 442, 443, 445, 449, 450, 451, 452, 453, 454, 455, 457, 458, 459, 461, 463, 465, 466, 467, 469, 473, 474, 475, 477, 481, 482, 483, 485, 486, 487, 489, 490, 491, 493, 497, 498, 499, 501, 505, 513, 514, 515, 516, 525, 526, 527, 529, 530, 531, 532, 533, 534, 535, 537, 538, 539, 541, 542, 543, 545, 546, 547, 54 568, 569, 565, 566, 567, 569, 570, 571, 573, 577, 578, 579, 595, 596, 597, 598, 599, 601, 602, 603, 605, 609, 610, 591, 592, 583, 584, 585, 586, 587, 588, 611, 612, 613, 614, 615, 617, 618, 619, 621, 623, 625, 626, 627, 629, 631, 633, 634, 635, 637, 641, 642, 643, 644, 645, 646, 667, 669, 673, 674, 675, 676, 677, 647, 648, 649, 650, 651, 652, 653, 654, 655, 657, 658, 659, 660, 661, 662, 663, 698, 699, 701, 705, 706, 707, 708, 709, 710, 711, 713, 714, 612, 698, 698, 698, 679, 681, 682, 683, 685, 715, 717, 719, 721, 722, 723, 725, 726, 727, 729, 730, 731, 733, 737, 738, 739, 741, 742, 743, 745, 746, 747, 749, 753, 754, 778, 779, 780, 781, 782, 783, 785, 786, 787, 788, 789, 790, 770, 771, 772, 773, 801, 802, 803, 804, 805, 806, 807, 809, 810, 811, 813, 815, 817, 818, 819, 821, 822, 823, 825, 82 854, 855, 857, 858, 859, 858, 859, 858, 858, 859, 861, 866, 867, 868, 869, 870, 871, 873, 874, 875, 877, 881, 882, 883, 885, 889, 893, 897, 898, 899, 900, 901, 902, 903, 905, 906, 907, 909, 910, 911, 913, 914, 915, 916, 917, 918, 919, 921, 922, 923, 925, 929, 930, 931, 932, 933, 934, 938, 939, 941, 945, 946, 947, 949, 953, 955, 957, 961, 962, 963, 964, 965, 966, 967, 969, 970, 971, 973, 977, 978, 983, 985, 987, 989, 993, 994, 995, 997, 999, 1001, 1002, 1003, 1005, 1009, 1010, 1011, 1013,

[Formula (200)

P = {887, 495, 735, 959, 1023, 762, 1014, 366, 926, 62, 446, 484, 948, 908, 108, 796, 92, 60, 252, 968, 104, 152, 88, 56 , 1016, 80, 48, 176, 32, 96, 64, 128}

[Formula 201]

T ₁ = {507, 891, 1019, 1021}

&Quot; (202) "

T ₂ = {239, 367, 507, 751, 759, 887, 943, 975, 1019, 1021}

[Formula 203]

T ₃ = {223, 367, 495, 507, 509, 759, 879, 887, 891, 943, 975, 1007, 1015, 1019}

[Equation 204]

T ₄ = {191, 223, 239, 319, 367, 479, 503, 509, 575, 671, 703, 831, 863, 879, 887, 951, 1007, 1015, 1021}

&Quot; (205) "

T ₅ = {127, 191, 255, 351, 367, 383, 447, 479, 503, 507, 509, 511, 639, 703, 735, 751, 763, 831, 863, 943, 951,

(206)

T ₆ = {223, 250, 351, 367, 383, 415, 503, 509, 575, 639, 759, 763, 863, 890, 1021, 1023}

[Equation 207]

_{T 7 = {127, 191,} 223, 239, 246, 374, 383, 479, 495, 511, 607, 694, 735, 758, 762, 763, 767, 886, 887, 895, 943, 951, 954, 975, 982, 986, 1018, 1021}

[Formula 208]

_{T 8 = {127, 191,} 223, 250, 319, 351, 367, 383, 415, 430, 438, 447, 479, 502, 506, 509, 511, 575, 607, 630, 671, 718, 735, 982, 986, 998, 1007, 1018, 1019, 1021}, < / RTI >

[Formula 209]

T ₉ = 127, 191, 223, 238, 250, 255, 366, 367, 374, 383, 415, 430, 479, 494, 495, 607, 622, 630, 639, 671, 694, 703, 957, 959, 982, 991, 998, 1007, 1014, 1019,

[Formula 210]

_{T 10 = {127, 191,} 222, 223, 239, 250, 255, 350, 351, 415, 430, 438, 462, 478, 479, 494, 495, 506, 509, 606, 607, 630, 686, 694, 703, 750, 751, 758, 762, 763, 798, 831, 879, 890, 927, 950, 954, 959, 975, 982, 991, 1006,

[Formula 211]

_{T 11 = {127, 190,} 222, 223, 238, 246, 318, 319, 350, 366, 415, 430, 462, 503, 506, 511, 575, 630, 639, 670, 694, 702, 703, 978, 943, 950, 959, 975, 990, 991, 998, 1006, 1007, 1021, 1023,

[Formula 212]

T ₁₂ = {62, 126, 127, 223, 239, 254, 319, 350, 374, 382, 430, 438, 462, 479, 495, 502, 506, 507, 575, 606, 638, 639, 670, 686, 702, 703, 735, 740, 759, 762, 798, 814, 862, 878, 886, 890, 891, 926, 927, 942, 958, 959, 975, 986, 991, 998, 1015, 1019, 1022}

[Formula 213]

T ₁₃ = {62, 94, 126, 222, 246, 250, 255, 318, 319, 366, 383, 430, 436, 446, 468, 479, 503, 506, 970, 980, 982, 990, 996, 991, 992, 994, 992, 994, 998, 1006, 1014, 1015, 1021, 1022, 1023}

[Formula 214]

T ₁₄ = {62, 126, 127, 222, 246, 318, 350, 351, 382, 383, 414, 415, 430, 462, 478, 494, 502, 506, 575, 628, 638, 670, 671, 884, 887, 890, 891, 927, 942, 943, 958, 959, 956, 974, 982, 990, 991, 996, 998, 1006, 1014, 1021, 1022}

[Formula 215]

_{T 15 = {126, 222,} 223, 239, 246, 319, 350, 366, 372, 382, 415, 447, 460, 494, 500, 502, 506, 507, 509, 511, 574, 575, 606, 880, 820, 830, 831, 844, 862, 863, 878, 879, 884, 890, 891, 908, 926, 943, 950, 951, 958, 972, 975, 982, 986, 990, 991, 1006, 1014, 1015, 1021,

[Formula 216]

T ₁₆ = {62, 94, 172, 222, 238, 246, 250, 255, 284, 319, 364, 374, 382, 414, 415, 446, 460, 462, 479, 484, 495, 503, 506, 507, 511, 540, 574, 606, 607, 638, 639, 670, 684, 686, 694, 703, 716, 718, 724, 748, 756, 758, 759, 763, 766, 767, 820, 830, 921, 991, 998, 1004, 1015, 1019, 1021, 1022, 1023,

[Equation 217]

T ₁₇ = 94, 108, 126, 172, 190, 191, 222, 236, 238, 246, 254, 318, 319, 350, 351, 372, 382, 412, 430, 436, 446, 460, 462, 468, 478, 479, 484, 492, 494, 495, 500, 502, 506, 509, 510, 606, 628, 668, 670, 684, 692, 694, 703, 718, 735, 740, 942, 943, 948, 951, 954, 942, 944, 864, 866, 879, 884, 891, 895, 908, 926, 927, 942, 943, 958, 972, 980, 986, 991, 1004, 1006, 1012, 1015, 1019, 1021, 1022, 1023}

[Formula 218]

T ₁₈ = {88, 94, 126, 127, 152, 168, 172, 190, 191, 200, 222, 238, 239, 244, 246, 252, 254, 255, 318, 319, 328, 344, 360, 366, 367, 374, 380, 436, 438, 446, 447, 460, 488, 494, 507, 536, 540, 552, 607, 616, 628, 636, 668, 670, 680, 684, 686, 692, 840, 840, 844, 852, 863, 872, 879, 884, 850, 820, 982, 982, 991, 1000, 1012, 1014, 1015, 1018, 1022}, < / RTI >

[Equation 219]

T ₁₉ = {222, 246, 250, 622, 670, 694, 750, 798, 814, 890, 926, 954, 982, 1006,

[Equation 220]

T ₂₀ = {62, 94, 222, 246, 250, 414, 430, 438, 478, 502, 506, 622, 670, 686, 694, 734, 750, 758, 798, 814, 878, 926, 942, 950, 954, 974, 990, 1014, 1018}

[Equation 221]

T ₂₁ = {255, 351, 415, 479, 507, 509, 511, 575, 607, 671, 703, 763, 767, 887, 895, 951, 975, 991, 1007, 1015, 1023}

[Equation 222]

T ₂₂ = {62,80,144,104,126,127,144,156,172,90,191,223,239,244,254,272,296,318,319,503,366,372, 504, 506, 507, 509, 511, 536, 540, 572, 490, 490, 714, 735, 744, 748, 758, 762, 766, 767, 792, 740, 712, 716, 718, 724, 734, 820, 830, 860, 863, 872, 878, 879, 887, 891, 892, 895, 904, 924, 940, 942, 948, 950, 958, 959, 968, 972, 990, 1012, 1014, 1015, 1022}

Equation (223)

T ₂₃ = 127, 191, 223, 239, 319, 367, 495, 503, 507, 511, 639, 671, 735, 767, 863, 879, 887, 951, 975, 991, 1007,

[Equation 224]

T ₂₄ = {62, 88, 92, 108, 126, 127, 152, 168, 172, 191, 200, 220, 238, 244, 252, 328, 344, 351, 360, 367, 508, 507, 510, 511, 536, 540, 552, 574, 616, 620, 622, 628, 636, 638, 412, 415, 447, 462, 468, 488, 492, 494, 840, 840, 844, 860, 872, 876, 878, 710, 720, 740, 987, 980, 984, 988, 991, 998, 1000, 1004, 1006, 1021, 1022,

[Equation 225]

T ₂₅ = {60, 62, 94, 104, 108, 120, 124, 126, 127, 152, 156, 168, 188, 190, 191, 200, 216, 244, 246, 254, 284, 296, 316, 460, 462, 472, 476, 478, 484, 488, 492, 495, 500, 440, 444, 446, 434, 424, 424, 740, 732, 740, 744, 748, 750, 758, 759, 760, 612, 873, 764, 766, 767, 808, 812, 820, 828, 831, 862, 863, 872, 876, 878, 884, 886, 887, 891, 892, 894, 895, 904, 920, 927, 943, 950, 951, 952, 954, 956, 958, 968, 982, 986, 991, 998, 1012, 1022, 1023}

[Equation 226]

T ₂₆ = {144, 272}

[Equation 227]

T ₂₇ = {80}

[Equation 228]

T ₂₈ = {60,62,88,104,120,168,200,222,223,232,236,246,248,250,254,280,284,312,318,31,336,344,351, 382, 408, 412, 415, 424, 428, 446, 447, 456, 460, 464, 472, 476, 478, 479, 488, 492, 495, 500, 507, 509, 511, 528, 568, 574, 604, 616, 620, 622, 628, 638, 656, 670, 671, 680, 694, 696, 700, 703, 712, 716, 720, 724, 734, 744, 748, 750, 759, 763, 764, 876, 879, 887, 888, 890, 894, 904, 924, 926, 814, 820, 828, 831, 840, 844, 848, 860, 927, 940, 951, 954, 956, 959, 968, 980, 982, 988, 991, 1000, 1004, 1006, 1007, 1018, 1019, 1021, 1022,

[Equation 229]

T ₂₉ = {210}

[Equation 230]

T ₃₀ = {60, 62, 92, 94, 127, 152, 156, 160, 168, 184, 188, 190, 191, 220, 222, 232, 236, 240, 244, 246, 248, 444, 456, 460, 464, 468, 476, 478, 484, 500, 502, 434, 436, 438, 444, 508, 509, 511, 536, 540, 560, 568, 574, 604, 607, 622, 628, 636, 638, 639, 656, 668, 670, 671, 686, 688, 814, 816, 820, 830, 831, 812, 814, 812, 814, 816, 840, 852, 860, 876, 878, 879, 880, 891, 895, 908, 920, 926, 927, 936, 940, 942, 948, 950, 952, 954, 958, 968, 972, 990, 1000, 1007, 1012, 1014, 1015, 1018, 1019, 1021, 1023}

[Equation 231]

T ₃₁ = {56, 88, 94, 104, 108, 112, 120, 127, 152, 156, 190, 191, 200, 216, 220, 223, 224, 232, 236, 239, 240, 248, 250, 272, 280, 284, 318, 328, 336, 348, 350, 351, 366, 383, 408, 412, 415, 428, 444, 447, 460, 472, 476, 479, 480, 502, 503, 504, 735, 732, 732, 734, 735, 632, 636, 664, 670, 671, 680, 684, 686, 692, 703, 716, 724, 728, 832, 816, 820, 830, 840, 844, 848, 852, 878, 884, 887, 895, 904, 912, 920, 924, 926, 927, 936, 942, 944, 950, 952, 958, 974, 975, 984, 986, 988, 991, 1004, 1006, 1007, 1008, 1012, 1016, 1020, 1021, 1022}

(232)

T ₃₂ = {32, 60, 62, 88, 120, 127, 156, 176, 188, 191, 220, 222, 236, 239, 240, 246, 250, 272, 280, 284, 296, 304, 316, 468, 476, 479, 480, 484, 488, 492, 496, 448, 446, 444, 446, 447, 456, 562, 692, 694, 568, 572, 574, 576, 592, 604, 620, 630, 636, 638, 672, 688, 692, 694, 508, 509, 510, 511, 536, 540, 544, 696, 700, 702, 703, 712, 716, 718, 720, 724, 728, 736, 740, 744, 751, 756, 759, 762, 763, 764, 766, 767, 792, 798, 800, 812, 820, 831, 840, 860, 862, 863, 864, 880, 884, 890, 891, 892, 894, 895, 912, 920, 924, 926, 942, 948, 950, 952, 954, 956, 958, 960, 972, 975, 992, 996, 1000, 1007, 1023}

In designing the parity check concatenated coders used in the simulation of FIG. 14, the frozen bit indexes are obtained using the Gaussian approximation method. As can be seen from FIG. 14, when the number r of parity check bits is 26, the best performance is shown. A frozen bit index set F including 657 bit indices is shown in Equation 233 below and a parity check bit index set P including 26 bit indices is represented by 234 and 26 support sets (T ₁ to T ₂₆ ) for the parity check bits are shown in the following equations (235) to (260).

[Equation 233]

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54 , 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, , 86, 87, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114, 115, , 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153 , 154, 155, 157, 159, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 175, 177, 178, 179, 181, 183, 185, 186, 187, 189 , 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 207, 209, 210, 211, 213, 214, 215, 217, 218, 219, 221, 225, 226, 227 , 228, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243, 245, 249, 253, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267 , 268, 269, 270, 271, 273, 274, 275, 276, 277, 278, 279, 281, 282, 283, 285, 287, 289, 290, 29 313, 314, 315, 317, 321, 322, 323, 324, 325, 308, 309, 346, 347, 349, 353, 354, 355, 356, 357, 358, 359, 341, 342, 343, 345, 391, 395, 397, 398, 399, 401, 402, 361, 362, 363, 365, 369, 370, 371, 373, 377, 381, 385, 386, 387, 388, 389, 390, 391, 393, 403, 404, 405, 406, 407, 409, 410, 411, 413, 417, 418, 419, 420, 421, 422, 423, 425, 426, 427, 429, 433, 434, 435, 443, 445, 449, 450, 451, 452, 453, 454, 455, 457, 458, 459, 461, 465, 466, 467, 469, 471, 473, 475, 477, 481, 482, 483, 485, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 517, 515, 516, 517, 517, 529, 530, 531, 532, 533, 534, 535, 537, 538, 539, 541, 543, 545, 546, 547, 548, 549, 550, 551, 553, 554, 555, 557, 559, 561, 562, 563, 565, 566, 567, 569, 570, 571, 573, 577, 578, 579, 580, 581, 582, 583, 585, 586, 587, 589, 590, 591, 593, 594, 595, 59 612, 613, 614, 615, 617, 618, 619, 621, 625, 626, 627, 629, 633, 635, 637, 641, 642, 643, 665, 666, 667, 669, 673, 643, 644, 645, 646, 647, 649, 650, 651, 653, 654, 655, 657, 658, 659, 660, 661, 662, 674, 675, 676, 677, 678, 679, 681, 682, 683, 685, 689, 690, 691, 693, 697, 699, 701, 705, 706, 707, 708, 709, 710, 711, 713, 714, 715, 717, 721, 722, 723, 725, 727, 729, 731, 733, 737, 738, 739, 741, 743, 745, 746, 747, 749, 753, 754, 755, 757, 761, 769, 770, 771, 772, 773, 774, 775, 777, 778, 779, 780, 781, 782, 783, 785, 786, 787, 788, 789, 790, 791, 793, 801, 802, 803, 804, 805, 806, 807, 809, 810, 811, 813, 817, 818, 819, 821, 823, 825, 827, 829, 833, 834, 835, 836, 837, 838, 869, 871, 873, 874, 875, 877, 881, 882, 853, 851, 858, 859, 861, 865, 866, 903, 905, 906, 907, 909, 911, 913, 914, 915, 917, 919, 921, 922, 923, 925, 929, 93 963, 967, 969, 970, 971, 973, 977, 924, 933, 934, 935, 937, 938, 939, 941, 945, 946, 947, 949, 953, 961, 978, 979, 981, 985, 989, 993, 994, 995, 997, 1001, 1005, 1009, 1011, 1013, 1017}

[Equation 234]

P = {695, 847, 927, 63, 474, 742, 630, 302, 366, 990, 254, 852, 884, 556, 1004, 476, 636, 200, 40, , 96, 192}

[Equation 235]

T ₁ = {251, 379, 507, 509, 765, 893, 955, 957, 999, 1003, 1019}

[Equation 236]

T ₂ = {379, 631, 695, 763, 765, 891, 951, 955, 983, 987, 999, 1019, 1021}

[Equation 237]

T ₃ = {247, 251, 375, 415, 463, 503, 509, 623, 631, 671, 695, 765, 799, 847, 879, 893, 943, 983, 987, 999, 1015, 1019,

[Equation 238]

T ₄ = {223, 247, 375, 379, 415, 495, 503, 507, 687, 695, 759, 763, 799, 847, 879, 893, 943, 951, 955, 983, 991, 999, 1007, 1015, 1019, 1021}

[Equation 239]

_{T 5 = {63, 247,} 255, 319, 351, 383, 415, 431, 439, 447, 503, 507, 509, 575, 631, 639, 671, 687, 719, 735, 759, 763, 765, 987, 991, 1003, 1007, 1015, 1019, 1021, 1023}

[Equation 240]

_{T 6 = {63, 95,} 191, 247, 250, 251, 375, 379, 431, 439, 447, 463, 495, 506, 511, 575, 631, 634, 639, 698, 703, 719, 730, 735, 751, 759, 762, 815, 863, 887, 891, 893, 943, 954, 957, 975, 987, 999, 1007, 1010,

[Equation 241]

_{T 7 = {63, 182,} 191, 319, 351, 375, 378, 438, 439, 447, 470, 479, 486, 507, 623, 634, 639, 703, 735, 751, 765, 767, 799, 847, 863, 870, 879, 887, 891, 893, 943, 959, 983, 986, 987, 991, 999, 1002, 1007}

[Equation 242]

_{T 8 = {118, 239,} 247, 250, 251, 255, 319, 379, 439, 442, 447, 462, 463, 486, 495, 502, 503, 506, 507, 558, 623, 630, 631, 975, 943, 955, 959, 975, 982, 935, 935, 935, 987, 998, 999, 1007, 1010}

[Equation 243]

_{T 9 = {127, 206,} 223, 319, 351, 367, 375, 378, 383, 430, 431, 439, 447, 470, 495, 502, 507, 511, 622, 623, 634, 639, 671, 686, 687, 718, 726, 730, 735, 751, 758, 762, 767, 799, 815, 822, 831, 846, 854, 870, 879, 890, 891, 895, 918, 950, 954, 957, 959, 975, 982, 983, 991, 999, 1002, 1010, 1014, 1019, 1023}

[Equation 244]

T ₁₀ = {63, 118, 127, 191, 222, 238, 239, 250, 251, 351, 374, 383, 415, 430, 439, 442, 447, 462, 463, 470, 474, 478, 486, 494, 495, 502, 503, 506, 507, 509, 558, 575, 606, 607, 623, 686, 695, 698, 703, 718, 719, 726, 734, 735, 742, 762, 765, 767, 999, 950, 954, 955, 957, 974, 986, 987, 1002, 1003, 1018, 1021,

[Equation 245]

T ₁₁ = {63, 94, 95, 110, 126, 158, 191, 223, 238, 246, 247, 255, 318, 462, 463, 470, 486, 495, 503, 558, 607, 622, 623, 638, 670, 671, 686, 687, 694, 698, 718, 719, 726, 742, 750, 759, 762, 763, 942, 943, 950, 951, 957, 974, 982, 983, 832, 830, 854, 862, 863, 879, 886, 890, 891, 894, 895, 927, 942, 991, 998, 999, 1002, 1003, 1007, 1010, 1014, 1018}

[Equation 246]

_{T 12 = {63, 94,} 95, 110, 118, 126, 174, 191, 222, 223, 238, 247, 251, 254, 302, 318, 340, 374, 378, 382, 414, 415, 438, 508, 509, 511, 542, 574, 575, 596, 606, 607, 623, 630, 634, 694, 698, 702, 703, 718, 730, 734, 742, 758, 763, 765, 766, 814, 822, 846, 847, 854, 862, 863, 870, 878, 879, 886, 887, 890, 895, 918, 943, 951, 954, 957, 974, 987, 996, 999, 1003, 1007, 1019, 1021, 1022, 1023}

[Equation 247]

T ₁₃ = 94, 110, 116, 158, 174, 190, 222, 223, 239, 246, 255, 286, 308, 351, 367, 372, 374, 379, 383, 414, 415, 431, 436, 468, 470, 479, 486, 494, 502, 506, 558, 564, 596, 606, 623, 634, 638, 686, 687, 692, 698, 702, 703, 718, 724, 734, 740, 742, 875, 910, 918, 942, 950, 951, 952, 954, 862, 878, 873, 766, 767, 815, 822, 826, 830, 831, 846, 852, 854, 954, 955, 982, 983, 986, 987, 990, 996, 998, 1002, 1003, 1006, 1010, 1014, 1021}

[248]

T ₁₄ = {62, 63, 116, 127, 180, 182, 190, 191, 206, 222, 223, 239, 244, 250, 251, 254, 255, 286, 308, 318, 332, 366, 375, 378, 379, 382, 383, 415, 431, 438, 446, 462, 479, 486, 494, 495, 500, 503, 506, 509, 510, 542, 558, 575, 588, 606, 766, 767, 798, 758, 759, 763, 765, 766, 767, 798, 724, 726, 734, 735, 740, 742, 751, 752, 628, 638, 671, 686, 687, 702, 703, 999, 932, 942, 943, 950, 951, 955, 957, 964, 832, 863, 870, 884, 890, 893, 908, 916, 974, 983, 991, 996, 1006, 1012, 1014, 1015, 1022}

[Equation 249]

_{T 15 = {62, 95,} 110, 116, 126, 172, 180, 182, 206, 212, 222, 223, 244, 251, 254, 302, 332, 351, 366, 367, 372, 375, 378, 479, 484, 486, 495, 502, 503, 506, 507, and 478, 689, 682, 692, 694, 698, 702, 716, 719, 724, 730, 734, 740, 742, 748, 750, 758, 759, 762, 763, 765, 767, 798, 814, 815, 822, 826, 831, 846, 862, 868, 879, 884, 886, 893, 895, 908, 910, 918, 932, 942, 943, 958, 959, 964, 972, 983, 986, 987, 996, 999, 1003, 1006, 1010, 1014,

[Equation 250]

T ₁₆ = {158, 180, 182, 190, 191, 206, 220, 223, 236, 239, 246, 250, 251, 255, 286, 302, 319, 332, 348, 350, 351, 374, 382, 503, 507, 509, 540, 556, 558, 564, 574, 588, 607, 622, 623, 474, 478, 490, 628, 630, 639, 652, 668, 670, 671, 684, 686, 687, 694, 695, 698, 716, 719, 726, 730, 735, 748, 751, 756, 758, 762, 763, 765, 876, 796, 798, 799, 814, 815, 826, 830, 844, 846, 847, 852, 854, 862, 863, 868, 870, 879, 887, 891, 893, 910, 916, 918, 927, 932, 940, 943, 950, 957, 975, 980, 982, 983, 986, 987, 990, 991, 996, 998, 1002, 1006, 1007, 1010, 1012, 1015, 1018, 1019, 1021, 1022, 1023}

[Equation 251]

T ₁₇ = 94, 95, 110, 124, 156, 172, 188, 204, 220, 236, 238, 239, 244, 246, 250, 286, 302, 318, 332, 340, 350, 351, 364, 366, 367, 374, 378, 382, 396, 414, 415, 428, 438, 439, 442, 444, 446, 447, 463, 468, 470, 474, 478, 495, 507, 510, 511, 542, 688, 574, 588, 604, 606, 607, 620, 622, 623, 628, 630, 634, 638, 639, 668, 671, 684, 686, 687, 694, 695, 700, 702, 716, 718, 879, 886, 846, 848, 844, 846, 847, 854, 870, 879, 886, 890, 891, 893, 894, 895, 908, 910, 916, 918, 924, 926, 940, 943, 950, 951, 955, 956, 957, 964, 975, 980, 988, 991, 996, 998, 999, 1003, 1004, 1007, 1015, 1023}

[Equation 252]

T ₁₈ = {108, 118, 126, 127, 156, 158, 174, 180, 191, 238, 239, 244, 246, 247, 250, 286, 300, 316, 319, 340, 348, 350, 351, 468, 470, 474, 476, 478, 486, 492, 494, 502, 508, 509, 684, 687, 692, 698, 700, 702, 719, 726, 730, 732, 632, 638, 638, 639, 648, 862, 863, 876, 878, 890, 892, 732, 740, 742, 750, 751, 756, 762, 763, 764, 766, 767, 776, 799, 812, 822, 828, 831, 893, 894, 895, 904, 910, 924, 927, 940, 942, 948, 955, 957, 958, 964, 972, 974, 980, 982, 983, 990, 999, 1003, 1004, 1007, 1010, 1015, 1019, 1022}

[Equation 253]

T ₁₉ = 94, 110, 118, 127, 206, 238, 239, 250, 251, 319, 351, 374, 379, 431, 438, 439, 442, 446, 463, 470, 478, 494, 506, 702, 703, 718, 719, 734, 751, 758, 759, 762, 763, 767, 826, 831, 854, 863, 870, 987, 990, 991, 998, 1007, 1015}, < / RTI >

[Equation 254]

T ₂₀ = {40, 48, 60, 62, 80, 92, 104, 108, 116, 124, 126, 127, 144, 172, 180, 182, 188, 190, 191, 200, 436, 438, 446, 432, 434, 424, 432, 434, 424, 426, 508, 509, 510, 542, 552, 556, 564, 584, 588, 596, 606, 620, 623, 628, 634, 486, 494, 495, 503, 506, 507, 636, 638, 639, 648, 652, 668, 670, 671, 686, 687, 695, 698, 700, 712, 716, 719, 724, 726, 732, 735, 740, 742, 744, 748, 751, 876, 879, 884, 886, 890, 892, 904, 850, 832, 824, 862, 763, 765, 766, 767, 776, 798, 799, 812, 924, 932, 936, 940, 942, 950, 951, 982, 986, 988, 991, 996, 998, 1000, 1002, 1004, 1007, 1012, 1015,

[Equation 255]

T ₂₁ = {24, 56, 62, 94, 95, 110, 118, 152, 156, 158, 168, 172, 174, 180, 188, 191, 206, 222, 232, 247, 254, 296, 312, 328, 340, 348, 350, 360, 366, 367, 374, 375, 378, 379, 380, 392, 412, 415, 424, 430, 431, 438, 439, 442, 462, 463, 470, 472, 476, 479, 486, 494, 500, 502, 506, 507, 510, 540, 542, 552, 572, 584, 588, 600, 607, 620, 622, 623, 628, 752, 763, 764, 765, 766, 776, 792, 796, 732, 735, 730, 732, 735, 740, 742, 744, 748, 878, 884, 887, 890, 893, 894, 908, 920, 927, 940, 942, 840, 812, 814, 820, 824, 826, 828, 840, 846, 854, 863, 948, 951, 954, 957, 958, 964, 972, 974, 975, 980, 987, 990, 996, 998, 999, 1003, 1012, 1015, 1020, 1022,

[Equation 256]

T ₂₂ = {40, 56, 80, 88, 92, 94, 96, 118, 120, 126, 127, 152, 158, 168, 172, 180, 182, 188, 190, 191, 328, 328, 350, 351, 360, 364, 366, 380, 396, 408, 424, 506, 508, 507, 508, 509, 510, 511, 536, 540, 552, 556, 564, 574, 540, 440, 446, 463, 468, 470, 479, 486, 488, 495, 588, 596, 604, 607, 616, 623, 628, 630, 631, 632, 648, 671, 680, 684, 687, 692, 696, 712, 718, 732, 751, 756, 759, 760, 776, 924, 924, 932, 940, 943, 832, 863, 868, 870, 879, 894, 904, 916, 918, 924, 951, 954, 956, 958, 964, 974, 980, 990, 999, 1000, 1004, 1006, 1010, 1012, 1014, 1016, 1020, 1022,

[Equation 257]

T ₂₃ = {60, 80, 144, 216, 312, 872}

[Equation 258]

T ₂₄ = {16, 48, 120, 664}

[Equation 259]

T ₂₅ = {80, 600}

(260)

T ₂₆ = {32}

In designing the parity check concatenated coders used in the simulation of FIG. 15, frozen bit indices are obtained using a Gaussian approximation method. As can be seen from FIG. 15, when the number r of parity check bits is 26, the best performance is shown. When the number r of parity check bits is 26, a frozen bit index set F including 486 bit indexes is shown in Equation (261) below, and a parity check bit index set P including 26 bit indexes is set to 262, and 26 sets of support (T ₁ to T ₂₆ ) for the parity check bits are shown in the following equations (263) to (288).

[Equation 261]

F = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, , 27, 29, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 55, 57, 58, 59, , 68, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89, 90, 91, 93, 97, 98, 99, 101, 102, 103, 105 , 106, 107, 109, 113, 114, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 143, 145, 146, 147, 149, 151 , 153, 154, 155, 157, 161, 162, 163, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181, 185, 189, 193, 194, 195, 196, 197 , 198, 199, 201, 202, 203, 205, 209, 210, 211, 213, 217, 221, 225, 226, 227, 229, 233, 235, 237, 241, 243, 245, 249, 257, 258 , 259, 260, 261, 262, 263, 265, 266, 267, 269, 271, 273, 274, 275, 277, 278, 279, 281, 282, 283, 285, 289, 290, 291, 293, 294 , 295, 297, 298, 299, 301, 305, 306, 307, 309, 313, 317, 321, 322, 323, 324, 325, 326, 327, 329, 330, 331, 333, 337, , 341, 345, 349, 353, 354, 355, 357, 361, 363, 365, 369, 371, 373, 377, 385, 386, 387, 388, 389, 390, 391, 393, 394, 395, 397, 401, 402, 403, 405, 409, 411, 413, 417, 418, 419, 421, 423, 425, 427, 429, 433, 434, 435, 437, 441, 449, 450, 451, 453, 455, 457, 458, 459, 461, 465, 466, 467, 469, 473, 481, 482, 483, 485, 489, 505, 513, 514, 515, 516, 517, 518, 519, 521, 522, 523, 525, 526, 527, 529, 530, 531, 533, 534, 535, 537, 538, 539, 541, 545, 546, 547, 548, 549, 550, 551, 553, 554, 555, 557, 561, 562, 563, 565, 569, 573, 577, 578, 579, 580, 581, 582, 583, 585, 586, 627, 629, 633, 641, 642, 643, 644, 608, 603, 604, 675, 677, 679, 681, 683, 685, 689, 690, 691, 665, 667, 669, 673, 693, 697, 705, 706, 707, 709, 711, 713, 714, 715, 717, 721, 722, 723, 725, 729, 737, 738, 739, 741, 745, 753, 757, 761, 769, 770, 795, 797, 801, 802, 803, 805, 807, 809, 810, 811, 813, 817, 818, 819, 821, 825, 833, 834, 835, 837, 839, 841, 842, 843, 845, 849, 850, 851, 853, 857, 865, 866, 867, 869, 873, 877, 881, 885, 889, 897, 898, 899, 901, 902, 903, 905, 906, 907, 909, 913, 914, 915, 917, 921, 929, 930, 931, 961, 963, 965, 969, 971, 973, 977, 979, 981, 985, 993, 995, 997, 1001, 1009,

[Equation 262]

P = {987, 871, 471, 207, 543, 191, 426, 858, 678, 214, 782, 46, 30, 638, 532, 52, 76, 44, 28, 904, , 32, 256}

[263]

T ₁ = {219, 253, 445, 475, 491, 493, 509, 859, 861, 923, 957, 1011, 1021}

[264]

T ₂ = {123, 125, 219, 253, 381, 443, 507, 571, 615, 635, 699, 701, 731, 733, 747, 763, 861, 883, 893, 925, 935, 939, 967, 989, 1011, 1013}

[265]

T ₃ = {123, 125, 187, 219, 231, 251, 253, 315, 347, 359, 477, 499, 571, 615, 635, 663, 699, 743, 749, 755, 763, 829, 855, 859, 891, 919, 923, 939, 987, 989, 1003, 1005, 1011, 1017, 1021}

[266]

T ₄ = 119, 123, 125, 187, 215, 231, 247, 399, 407, 443, 445, 475, 477, 507, 509, 591, 615, 635, 637, 695, 727, 731, 749, 963, 783, 827, 859, 871, 875, 883, 893, 923, 925, 935, 939, 951, 955, 957, 967, 987, 1005, 1015,

[Equation 267]

T ₅ = {111, 125, 215, 231, 247, 251, 253, 315, 375, 379, 381, 399, 407, 439, 445, 487, 491, 493, 507, 509, 567, 571, 599, 935, 939, 943, 951, 955, 957, 951, 952, 953, 975, 987, 1007, 1013, 1015, 1017, 1021}

[268]

T ₆ = {111, 159, 175, 207, 219, 223, 231, 247, 253, 287, 311, 319, 343, 347, 367, 381, 407, 415, 431, 439, 445, 471, 475, 477, 479, 487, 491, 495, 503, 559, 571, 591, 599, 615, 631, 655, 663, 671, 695, 719, 727, 733, 735, 743, 763, 827, 847, 871, 875, 879, 911, 923, 927, 935, 939, 943, 951, 957, 967, 975, 983, 987, 991,

[269]

T ₇ = {31, 95, 125, 183, 191, 207, 223, 247, 251, 253, 255, 439, 443, 445, 463, 471, 479, 493, 499, 503, 507, 509, 559, 571, 599, 615, 631, 635, 639, 671, 695, 701, 703, 727, 747, 749, 751, 763, 823, 829, 831, 847, 855, 861, 863, 879, 882, 887, 911, 923, 925, 927, 939, 946, 955, 957, 959, 983, 987, 994, 999, 1010, 1013, 1015, 1019, 1021, 1023}

[Equation 270]

_{T 8 = {123, 125,} 183, 187, 191, 207, 242, 251, 253, 287, 303, 319, 335, 343, 346, 347, 359, 362, 370, 375, 381, 399, 410, 415, 426, 431, 439, 443, 447, 463, 475, 490, 493, 499, 511, 543, 559, 607, 615, 635, 639, 655, 666, 671, 682, 703, 719, 731, 733, 743, 746, 749, 754, 759, 767, 794, 823, 829, 831, 847, 855, 859, 861, 875, 879, 882, 887, 893, 919, 923, 927, 955, 957, 959, 967, 975, 989, 991, 999, 1003, 1005, 1010, 1015, 1017, 1019, 1023}

[Equation 271]

_{T 9 = {63, 111,} 122, 123, 159, 183, 186, 187, 191, 215, 218, 223, 231, 234, 239, 242, 253, 255, 287, 303, 319, 335, 343, 347, 475, 493, 507, 509, 511, 602, 615, 631, 663, 671, 698, 823, 826, 827, 847, 858, 861, 871, 874, 875, 883, 890, 895, 911, 922, 923, 925, 938, 939, 946, 955, 966, 970, 991, 994, 999, 1002, 1003, 1005, 1007, 1011,

[Equation 272]

T ₁₀ = {31, 63, 86, 119, 122, 123, 125, 127, 150, 159, 183, 186, 207, 215, 223, 491, 499, 503, and 493, respectively, and 319, 346, 351, 359, 367, 375, 378, 379, 381, 383, 406, 415, 422, 442, 443, 445, 447, 471, 477, 479, 487, 706, 710, 719, 727, 735, 742, 746, 747, 749, 754, 925, 934, 946, 955, 957, 925, 924, 924, 934, 946, 955, 957, 959, 967, 986, 987, 989, 998, 999, 1007, 1011, 1013, 1015, 1017, 1019, 1021,

[Equation 273]

T ₁₁ = {54, 63, 86, 95, 111, 118, 123, 125, 159, 182, 183, 186, 187, 214, 242, 495, 498, 491, 492, 490, 490, 490, 490, 490, 499, 503, 509, 511, 543, 566, 567, 598, 602, 615, 618, 635, 637, 639, 655, 662, 663, 666, 671, 678, 694, 695, 699, 701, 703, 710, 727, 730, 733, 735, 742, 743, 746, 747, 755, 758, 759, 783, 794, 806, 822, 823, 826, 838, 854, 870, 879, 882, 886, 891, 911, 919, 923, 925, 927, 934, 935, 938, 943, 946, 950, 951, 954, 959, 966, 967, 982, 983, 986, 987, 991, 994, 998, 1005, 1007, 1010, 1013, 1014, 1015, 1017, 1018, 1021, 1023}

[274]

T ₁₂ = {54, 63, 78, 86, 95, 111, 118, 119, 123, 125, 127, 142, 175, 186, 187, 191, 214, 219, 223, 230, 247, 250, 253, 375, 377, 370, 375, 378, 381, 383, 398, 399, 407, 415, 426, 431, 439, 445, 454, 463, 470, 471, 475, 634, 635, 637, 639, 666, 634, 634, 635, 635, 878, 885, 863, 879, 886, 887, 732, 732, 990, 1013, 1014, 1019, 1021, 1023}, < / RTI >

[275]

T ₁₃ = {302}

[276]

T ₁₄ = {31, 54, 94, 95, 119, 127, 174, 182, 183, 186, 187, 190, 206, 207, 218, 219, 222, 230, 238, 242, 246, 431, 438, 439, 443, 446, 470, 478, 479, 331, 334, 335, 347, 350, 362, 367, 381, 415, 430, 431, 490, 493, 495, 498, 499, 502, 506, 507, 511, 542, 543, 558, 559, 566, 575, 590, 591, 606, 618, 622, 623, 631, 634, 654, 655, 663, 670, 671, 682, 694, 710, 727, 731, 734, 735, 743, 746, 747, 754, 758, 759, 763, 765, 782, 799, 806, 814, 815, 822, 826, 827, 829, 830, 831, 846, 855, 858, 861, 862, 863, 874, 883, 886, 887, 893, 918, 925, 934, 955, 957, 959, 966, 982, 983, 987, 994, 999, 1002, 1005, 1006, 1011, 1015, 1017, 1019, 1021}

[Equation 277]

_{T 15 = {47, 63,} 94, 95, 100, 118, 123, 142, 150, 158, 164, 174, 182, 183, 187, 190, 207, 214, 238, 242, 246, 250, 251, 434, 439, 443, 343, 359, 362, 382, 399, 406, 407, 410, 420, 422, 431, 438, 434, 570, 590, 599, 602, 612, 614, 615, 560, 560, 748, 754, 755, 762, 763, 743, 743, 744, 751, 754, 755, 762, 763, 630, 639, 662, 670, 676, 678, 682, 686, 687, 698, 702, 703, 718, 719, 872, 879, 882, 886, 890, 895, 911, 812, 861, 865, 858, 859, 861, 862, 874, 875, 876, 767, 790, 794, 799, 822, 827, 836, 919, 922, 934, 938, 939, 942, 950, 955, 958, 970, 975, 978, 982, 986, 987, 989, 990, 991, 994, 999, 1006, 1010, 1011, 1014,

[278]

T ₁₆ = {31, 47, 62, 94, 111, 118, 119, 127, 142, 148, 159, 186, 187, 191, 206, 215, 218, 219, 230, 239, 247, 366, 367, 374, 378, 379, 381, 399, 404, 406, 407, 410, 414, 415, 420, 430, 431, 438, 442, 446, 447, 462, 463, 474, 486, 491, 494, 495, 498, 499, 510, 511, 558, 559, 566, 571, 634, 635, 638, 639, 655, 660, 662, 663, 666, 670, 678, 682, 686, 612, 612, 614, 615, 618, 623, 687, 694, 702, 726, 734, 735, 740, 742, 746, 749, 750, 754, 762, 766, 767, 783, 788, 799, 804, 806, 815, 823, 830, 831, 838, 846, 852, 855, 859, 861, 868, 870, 874, 878, 879, 883, 887, 890, 891, 893, 894, 895, 900, 910, 916, 918, 922, 923, 925, 926, 927, 932, 934, 938, 939, 950, 951, 957, 958, 959, 964, 967, 974, 978, 980, 983, 987, 989, 990, 996, 998, 1002, 1003, 1006, 1015, 1019, 1022}

[Equation 279]

T ₁₇ = 30, 31, 52, 63, 84, 100, 110, 111, 116, 119, 123, 125, 126, 140, 142, 148, 159, 175, 183, 191, 207, 214, 286, 287, 302, 303, 310, 314, 319, 340, 346, 347, 351, 362, 366, 372, 374, 379, 383, 399, 404, 410, 415, 426, 438, 439, 442, 446, 462, 474, 475, 478, 484, 486, 491, 493, 499, 502, 507, 509, 524, 638, 639, 655, 662, 670, 678, 682, 687, 692, 699, 634, 635, 637, 638, 639, 655, 668, 564, 566, 567, 571, 702, 708, 710, 719, 726, 727, 731, 743, 756, 759, 762, 765, 766, 767, 780, 783, 794, 798, 804, 815, 823, 827, 829, 830, 831, 836, 838, 852, 854, 855, 859, 861, 863, 871, 874, 882, 883, 884, 886, 887, 893, 894, 895, 911, 918, 922, 923, 925, 932, 934, 935, 938, 939, 943, 950, 951, 954, 957, 958, 970, 986, 987, 989, 990, 991, 998, 999, 1005, 1010, 1011, 1012, 1014,

[Equation 280]

T ₁₈ = {30, 47, 52, 62, 76, 84, 86, 100, 110, 111, 116, 118, 119, 123, 125, 127, 158, 175, 182, 183, 190, 362, 366, 367, 375, 379, 383, 361, 362, 363, 498, 399, 404, 420, 422, 426, 430, 431, 443, 452, 454, 468, 474, 475, 479, 486, 491, 493, 495, 500, 503, 509, 510, 564, 571, 588, 606, 607, 612, 614, 618, 622, 623, 628, 630, 634, 652, 654, 655, 671, 676, 682, 687, 692, 699, 726, 730, 733, 734, 742, 747, 749, 751, 755, 756, 758, 759, 763, 767, 782, 804, 806, 815, 829, 836, 844, 847, 852, 854, 855, 858, 862, 870, 871, 874, 875, 878, 879, 883, 886, 890, 895, 900, 910, 911, 919, 922, 923, 925, 926, 934, 942, 946, 948, 951, 954, 966, 970, 972, 974, 975, 978, 980, 982, 986, 989, 991, 994, 999, 1003, 1006, 1010, 1011, 1012, 1014, 1015, 1019,

[Equation 281]

T ₁₉ = {31, 46, 52, 54, 95, 108, 110, 111, 122, 123, 125, 127, 158, 164, 174, 175, 180, 182, 183, 186, 187, 236, 238, 239, 242, 244, 246, 247, 250, 270, 287, 292, 300, 302, 303, 310, 314, 318, 319, 332, 334, 335, 340, 347, 351, 358, 362, 364, 370, 372, 375, 378, 379, 382, 383, 404, 406, 407, 495, 498, 499, 500, 502, 503, 507, 412, 412, 511, 524, 564, 571, 574, 575, 590, 591, 596, 598, 602, 607, 612, 620, 623, 631, 635, 655, 660, 662, 663, 670, 671, 676, 687, 692, 695, 699, 701, 708, 716, 719, 724, 726, 727, 733, 734, 746, 747, 748, 749, 751, 758, 759, 762, 765, 766, 767, 780, 790, 924, 922, 923, 926, 927, 926, 927, 928, 934, 935, 939, 940, 942, 943, 946, 948, 951, 954, 958, 964, 966, 967, 974, 975, 987, 989, 990, 991, 998, 1007, 1011, 1012, 1013, 101 4, 1015, 1018, 1021, 1022, 1023}

[Equation 282]

T ₂₀ = {31, 46, 47, 52, 60, 84, 95, 108, 110, 111, 122, 125, 126, 127, 136, 140, 148, 150, 159, 188, 190, 191, 204, 206, 212, 214, 215, 218, 220, 223, 230, 231, 234, 236, 238, 244, 247, 250, 254, 255, 270, 292, 372, 374, 380, 381, 382, 383, 398, 406, 407, 410, 412, 361, 362, 367, 370, 371, 314, 315, 334, 335, 340, 342, 348, 351, 356, 358, 414, 415, 420, 426, 431, 436, 438, 439, 443, 445, 447, 462, 463, 470, 475, 476, 477, 479, 484, 486, 491, 493, 494, 499, 502, 503, 511, 520, 540, 543, 558, 566, 567, 574, 590, 591, 598, 599, 607, 615, 620, 622, 628, 630, 631, 635, 636, 637, 638, 639, 652, 660, 662, 670, 676, 682, 686, 687, 694, 695, 699, 701, 716, 719, 724, 726, 727, 731, 732, 734, 743, 747, 748, 845, 820, 826, 836, 844, 846, 847, 852, 854, 740, 740, 860, 861, 862, 870, 874, 884, 890, 894, 895, 919, 922, 923, 924, 927, 932, 934, 935, 938, 939, 940, 942, 943, 948, 950, 956, 958, 959, 964, 966, 967, 970, 972, 978, 987, 988, 994, 1004, 1005, 1006, 1010, 1015, 1019, 1020, 1021,

[Equation 283]

T ₂₁ = {72}

[Equation 284]

T ₂₂ = {40}

[Equation 285]

T ₂₃ = {24}

[Equation 286]

T ₂₄ = 30, 31, 46, 60, 76, 94, 95, 100, 111, 116, 126, 127, 136, 144, 148, 150, 152, 238, 239, 244, 248, 250, 252, 253, 264, 268, 276, 280, 284, 286, 292, 296, 300, 302, 310, 311, 312, 316, 318, 319, 332, 343, 347, 348, 358, 359, 362, 364, 367, 378, 381, 396, 398, 477, 479, 488, 491, 492, 493, 495, 472, 474, 475, 472, 560, 606, 607, 616, 622, 532, 536, 543, 552, 556, 558, 559, 567, 572, 574, 584, 588, 590, 596, 702, 703, 708, 712, 718, 708, 701, 702, 704, 708, 782, 783, 788, 790, 794, 798, 804, 806, 814, 823, 824, 742, 743, 744, 750, 754, 755, 758, 760, 762, 763, 766, 780, 827, 828, 829, 830, 836, 838, 844, 847, 852, 854, 855, 856, 858, 859, 860, 861, 862, 863, 868, 875, 882, 884, 886, 891, 894, 895, 900, 904, 912, 916, 918, 919, 922, 924, 926, 932, 935, 936, 938, 939, 940, 942, 943, 946, 948, 950, 951, 952, 955, 959, 1003, 1004, 1005, 1006, 1011, 1012, 1013, 1015, 1017, 1021, 1022, 1023, 968, 970, 974, 975, 978, 980, 989, 991, 994, 996, 998,

[Equation 287]

T ₂₅ = {28, 30, 31, 47, 62, 76, 88, 94, 108, 110, 116, 118, 119, 120, 124, 126, 148, 152, 156, 158, 172, 174, 176, 280, 184, 187, 191, 200, 204, 207, 208, 212, 214, 219, 223, 228, 230, 232, 239, 240, 242, 247, 251, 252, 255, 264, 368, 372, 378, 383, 392, 396, 398, 407, 410, 414, 440, 442, 443, 445, 452, 462, 470, 477, 487, 490, 494, 499, 500, 502, 504, 506, 509, 602, 606, 607, 614, 620, 624, 630, 635, 638, 639, 562, 556, 559, 564, 568, 575, 584, 590, 591, 654, 655, 656, 660, 663, 664, 666, 671, 676, 678, 680, 686, 687, 694, 699, 700, 701, 702, 716, 719, 724, 726, 732, 733, 734, 784, 825, 815, 816, 822, 823, 824, 832, 835, 826, 827, 828, 830, 836, 838, 840, 844, 847, 848, 854, 856, 859, 861, 862, 874, 876, 880, 882, 883, 892, 895, 904, 910, 912, 920, 923, 926, 927, 932, 934, 935, 936, 939, 940, 942, 943, 946, 948, 951, 952, 956, 957, 958, 966, 970, 972, 974, 978, 986, 987, 990, 991, 994, 996, 998, 1000, 1006, 1008, 1012, 1013, 1019, 1020}

[Equation 288]

T ₂₆ = {52, 63, 76, 78, 92, 94, 100, 108, 110, 123, 124, 125, 126, 127, 148, 150, 156, 159, 164, 172, 174, 175, 182, 183, 188, 190, 191, 204, 207, 212, 214, 215, 219, 220, 222, 223, 230, 238, 250, 251, 254, 255, 264, 270, 276, 284, 287, 288, 386, 376, 378, 380, 381, 383, 343, 344, 346, 348, 351, 366, 374, 376, 378, 396, 399, 404, 406, 408, 415, 420, 432, 436, 438, 440, 443, 446, 456, 460, 472, 475, 476, 477, 478, 479, 480, 484, 486, 487, 492, 496, 498, 500, 502, 508, 510, 511, 528, 536, 543, 544, 556, 558, 560, 564, 572, 574, 575, 584, 588, 590, 591, 592, 596, 640, 640, 655, 660, 670, 672, 676, 678, 692, 694, 695, 699, 630, 630, 631, 632, 634, 638, 740, 732, 735, 736, 740, 742, 743, 744, 747, 752, 755, 756, 758, 759, 763, 765, 766, 790, 792, 794, 856, 860, 861, 862, 863, 832, 836, 838, 840, 846, 847, 852, 855, 856, 860, 864, 868 , 871, 875, 876, 878, 880, 886, 887, 891, 892, 894, 895, 904, 908, 911, 912, 920, 922, 923, 925, 926, 934, 935, 936, 938, 939 , 942, 943, 946, 950, 955, 957, 959, 960, 964, 970, 974, 976, 983, 989, 990, 991, 992, 994, 998, 999, 1000, 1002, 1003, 1005, 1011 , 1014, 1016, 1017, 1018, 1020, 1022, 1023}

In designing the parity check concatenated coders used in the simulation of FIG. 16, frozen bit indices are obtained by using a Gaussian approximation method. As can be seen from FIG. 16, when the number r of parity check bits is 28, the best performance is shown. The frozen bit index set F including 313 bit indices is shown in Equation (289) below, and a parity check bit index set P including 28 bit indices is represented by (281), and 28 support sets (T ₁ to T ₂₈ ) for the parity check bits are shown in the following Equations (291) to (318).

[Equation 289]

F = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 33, , 41, 42, 43, 45, 49, 50, 51, 53, 57, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 81, 82, , 98, 99, 101, 105, 109, 113, 117, 121, 129, 130, 131, 133, 134, 135, 137, 138, 139, 141, 145, 146, 147, 149, 153, 157, 161 , 162, 163, 165, 169, 173, 177, 179, 181, 185, 193, 194, 195, 197, 201, 203, 205, 209, 211, 213, 217, 225, 226, 227, 229, 233 , 241, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 269, 273, 274, 275, 277, 281, 285, 289, 290, 291, 293, , 307, 309, 313, 321, 322, 323, 325, 329, 331, 333, 337, 339, 341, 345, 353, 354, 355, 357, 361, 369, 377, 385, 386, , 391, 393, 395, 397, 401, 402, 403, 405, 409, 417, 418, 419, 421, 425, 433, 441, 449, 450, 451, 453, 457, 465, 469, 473, 481 , 485, 489, 497, 513, 514, 515, 516, 517, 518, 519, 521, 522, 523, 525, 529, 530, 531, 533, 537, 541, 545, 546, 547, 549, 553 , 555, 557, 561, 563, 565, 569, 577, 578, 579, 581, 585, 587, 641, 642, 643, 645, 647, 649, 651, 653, 657, 658, 659, 661, 775, 713, 721, 725, 729, 737, 741, 745, 753, 769, 770, 771, 773, 775, 705, 706, 707, 705, 835, 833, 834, 835, 837, 841, 845, 849, 853, 803, 807, 935, 931, 933, 937, 945, 961, 963, 965, 969, 961, 977, 993}

[Equation 290]

P = {797, 893, 459, 603, 327, 407, 1015, 15, 575, 210, 202, 26, 1018, 486, 22, 14, 94, 132, 36, 20, , 24, 16, 48, 96}

[Equation 291]

T ₁ = {249, 365, 373, 493, 501, 505, 685, 717, 757, 761, 813, 877, 885, 889, 973, 1001, 1005,

[Equation 292]

T ₂ = 125, 189, 245, 249, 349, 373, 413, 437, 621, 669, 693, 701, 757, 797, 813, 889, 953, 957, 997, 1001, 1009,

[Equation 293]

T ₃ = {61, 115, 125, 189, 221, 243, 245, 249, 253, 317, 349, 365, 371, 429, 435, 445, 467, 499, 501, 505, 509, 573, 605, 813, 829, 843, 861, 877, 885, 889, 941, 949, 957, 973, 985, 1001, 1005, 1009, 1013,

[Equation 294]

T ₄ = {61, 91, 93, 107, 115, 155, 221, 235, 237, 245, 253, 349, 365, 371, 381, 411, 427, 429, 461, 477, 483, 491, 493, 881, 861, 885, 881, 861, 885, 891, 892, 993, 994, 889, 907, 941, 949, 953, 957, 971, 985, 989, 1001, 1003, 1011, 1013, 1017, 1021}

[Equation 295]

_{T 5 = {59, 61,} 107, 115, 125, 155, 189, 243, 245, 299, 347, 365, 373, 379, 381, 427, 445, 475, 477, 483, 493, 571, 583, 813, 913, 953, 813, 813, 813, 813, 813, 813, 957, 971, 973, 987, 989, 1003, 1017, 1019}

[Equation 296]

T ₆ = {61, 91, 103, 107, 125, 151, 167, 171, 187, 199, 219, 231, 235, 237, 243, 245, 251, 295, 299, 315, 317, 327, 347, 603, 605, 615, 619, 621, 627, 635, 637, 669, 879, 683, 885, 889, 891, 893, 891, 891, 892, 903, 907, 923, 925, 941, 957, 967, 971, 973, 979, 985, 987, 989, 999, 1001, 1003, 1005, 1009,

[Equation 297]

T ₇ = {55, 61, 107, 115, 123, 151, 155, 167, 189, 219, 231, 237, 279, 295, 299, 327, 343, 349, 373, 407, 413, 427, 429, 437, 439, 467, 477, 483, 503, 505, 535, 551, 573, 583, 599, 605, 627, 629, 631, 663, 669, 679, 691, 695, 701, 711, 715, 717, 893, 903, 907, 923, 939, 951, 957, 813, 813, 819, 823, 827, 851, 871, 885, 889, 891, 893, 979, 985, 989, 995, 999, 1001, 1005, 1011, 1013, 1017, 1019}

[Equation 298]

T ₈ = {55, 59, 93, 107, 123, 151, 155, 167, 199, 219, 221, 237, 249, 311, 327, 347, 349, 359, 365, 373, 375, 381, 407, 413, 423, 435, 443, 445, 467, 475, 477, 483, 487, 491, 503, 535, 539, 567, 571, 599, 615, 619, 621, 627, 631, 663, 667, 669, 839, 843, 851, 855, 851, 852, 863, 871, 683, 691, 695, 699, 717, 731, 733, 755, 757, 761, 763, 765, 791, 795, 797, 811, 813, 819, 859, 861, 885, 891, 903, 919, 925, 941, 949, 953, 955, 957, 967, 971, 973, 981, 983, 997, 1001, 1003,

[Equation 299]

_{T 9 = {15, 31,} 55, 59, 63, 87, 103, 111, 115, 119, 155, 159, 167, 171, 183, 189, 199, 221, 231, 235, 237, 239, 245, 495, 499, 535, 461, 462, 474, 483, 491, 495, 429, 435, 439, 443, 445, 695, 699, 711, 715, 723, 727, 612, 638, 637, 671, 679, 685, 687, 693, 559, 571, 573, 583, 591, 605, 607, 733, 739, 743, 749, 761, 763, 799, 811, 813, 827, 839, 847, 851, 855, 861, 871, 889, 907, 911, 923, 935, 939, 951, 953, 957, 971, 979, 981, 985, 997, 1001, 1003, 1007, 1011, 1013, 1015, 1021}

[Formula (300)

T ₁₀ = {15,31,47,99,79,87,95,107,113,125,143,159,75,189,191,207,223,231,239,243,251,253, 375, 379, 383, 407, 411, 415, 423, 429, 431, 435, 445, 375, 375, 378, 343, 347, 351, 359, 447, 461, 467, 471, 475, 479, 482, 493, 495, 499, 501, 505, 509, 535, 539, 573, 575, 603, 607, 619, 621, 627, 637, 639, 655, 743, 747, 749, 751, 755, 791, 807, 815, 827, 831, 712, 713, 713, 717, 723, 733, 738, 739, 983, 989, 991, 997, 999, 1005, 1007, 1009, 991, 992, 993, 873, 871, 875, 879, 885, 893, 895, 907, 935, 939, 941, 943, 959, 1011, 1015, 1023}

[Formula 301]

_{T 11 = {15, 47,} 55, 59, 61, 79, 87, 91, 111, 119, 125, 127, 143, 151, 167, 178, 189, 191, 207, 215, 221, 231, 235, 379, 383, 407, 411, 413, 423, 429, 431, 361, 372, 504, 505, 535, 539, 543, 559, 571, 573, 575, 603, 605, 615, 626, 629, 631, 635, 639, 663, 667, 671, 683, 685, 687, 690, 691, 693, 699, 701, 703, 711, 715, 719, 723, 731, 749, 754, 757, 871, 877, 879, 885, 891, 895, 903, 907, 911, 812, 843, 850, 859, 863, 871, 763, 765, 767, 791, 797, 914, 919, 923, 930, 941, 947, 949, 951, 959, 967, 973, 979, 981, 985, 987, 994, 997, 1001, 1003, 1005, 1009, 1010, 1013, 1015,

[Equation 302]

T ₁₂ = {15, 79, 93, 103, 107, 114, 115, 119, 123, 127, 143, 155, 170, 171, 175, 183, 189, 191, 199, 202, 207, 219, 374, 383, 394, 399, 407, 411, 413, 415, 426, 427, 431, 435, 445, 455, 458, 459, 461, 463, 466, 467, 479, 482, 483, 487, 495, 501, 503, 505, 507, 543, 551, 559, 562, 571, 573, 735, 738, 751, 755, 757, 759, 763, 765, 783, 713, 713, 723, 731, 895, 813, 919, 927, 935, 941, 943, 835, 813, 831, 839, 842, 847, 851, 855, 863, 866, 875, 883, 885, 889, 891, 947, 949, 951, 970, 973, 978, 979, 987, 989, 995, 997, 1001, 1003, 1009, 1013,

[Formula 303]

T ₁₃ = {63, 79, 91, 93, 95, 103, 111, 114, 115, 119, 125, 154, 155, 159, 167, 178, 183, 186, 189, 202, 374, 399, 399, 318, 314, 319, 343, 347, 349, 351, 365, 371, 373, 375, 379, 295, 299, 495, 498, 501, 505, 507, 509, 490, 490, 490, 490, 633, 626, 627, 635, 637, 667, 679, 685, 690, 693, 695, 698, 703, 711, 714, 715, 717, 727, 730, 735, 739, 743, 747, 751, 755, 757, 759, 761, 762, 767, 783, 791, 795, 799, 810, 811, 818, 827, 829, 831, 839, 842, 843, 851, 855, 858, 859, 861, 863, 871, 882, 883, 885, 887, 889, 891, 893, 911, 914, 927, 930, 935, 949, 951, 962, 967, 971, 985, 989, 991, 995, 997, 1002, 1003, 1011, 1013, 1015, 1017, 1019, 1021, 1023}

[Equation 304]

_{T 14 = {15, 31,} 38, 58, 63, 79, 90, 91, 95, 106, 111, 114, 115, 122, 125, 127, 151, 159, 170, 175, 178, 183, 186, 291, 311, 314, 315, 317, 326, 327, 335, 351, 358, 291, 291, 198, 199, 202, 210, 223, 230, 231, 251, 253, 283, 287, 295, 298, 431, 434, 435, 439, 442, 443, 458, 466, 471, 475, 477, 482, 491, 371, 373, 375, 378, 399, 413, 423, 427, 429, 499, 506, 538, 559, 562, 567, 573, 575, 591, 599, 602, 603, 605, 614, 615, 618, 627, 639, 646, 655, 666, 667, 669, 678, 679, 682, 683, 685, 687, 691, 695, 710, 711, 715, 719, 723, 727, 730, 731, 735, 742, 743, 747, 749, 751, 754, 755, 774, 794, 806, 903, 906, 907, 914, 922, 907, 811, 818, 823, 838, 842, 843, 851, 861, 866, 874, 875, 877, 879, 883, 885, 887, 893, 925, 934, 935, 938, 941, 947, 951, 957, 959, 962, 966, 970, 973, 986, 987, 989, 1001, 1003, 1005, 1011, 1013,

Equation (305)

_{T 15 = {38, 47,} 61, 63, 79, 87, 102, 103, 115, 119, 123, 143, 154, 167, 170, 175, 186, 189, 191, 199, 215, 219, 221, 373, 375, 378, 375, 370, 371, 375, 367, 370, 371, 479, 477, 479, 486, 490, 471, 471, 474, 474, 491, 493, 498, 499, 501, 507, 539, 551, 554, 559, 562, 567, 575, 614, 615, 621, 623, 626, 631, 634, 637, 639, 663, 669, 678, 823, 827, 838, 842, 843, 847, 732, 723, 727, 730, 733, 742, 747, 757, 759, 761, 762, 783, 806, 810, 850, 851, 855, 859, 861, 863, 870, 875, 877, 879, 882, 883, 885, 889, 893, 902, 907, 914, 919, 925, 927, 935, 938, 941, 943, 947, 949, 951, 957, 959, 970, 971, 973, 983, 989, 991, 994, 995, 1001, 1005, 1009, 1010, 1021,

[Equation 306]

T ₁₆ = {22, 26, 47, 58, 61, 63, 79, 86, 87, 90, 91, 93, 95, 103, 114, 122, 123, 125, 127, 150, 298, 299, 303, 306, 298, 293, 289, 296, 297, 298, 221, 231, 234, 243, 245, 249, 250, 279, 282, 283, 287, 434, 437, 439, 442, 445, 447, 455, 459, 470, 426, 426, 424, 570, 571, 582, 583, 586, 591, 598, 603, 605, 618, 619, 626, 630, 631, 634, 637, 639, 646, 650, 662, 666, 669, 678, 679, 682, 690, 691, 693, 695, 698, 701, 703, 714, 715, 831, 838, 842, 831, 838, 838, 838, 838, 838, 834, 847, 858, 859, 863, 877, 882, 886, 887, 890, 891, 893, 902, 906, 914, 919, 923, 925, 941, 946, 947, 953, 957, 962, 985, 989, 991, 994, 995, 997, 1002, 1003, 1007, 1009, 1010, 1015, 1017, 1019, 1021}

[Equation 307]

T ₁₇ = {15, 30, 47, 54, 55, 58, 59, 87, 91, 93, 95, 103, 106, 107, 215, 159, 166, 170, 171, 183, 187, 189, 198, 199, 202, 206, 207, 210, 218, 219, 223, 235, 242, 243, 247, 250, 251, 375, 366, 367, 379, 381, 390, 399, 407, 414, 415, 346, 347, 349, 351, 359, 525, 535, 525, 525, 525, 525, 525, 525, 525, 562, 567, 575, 582, 598, 599, 602, 607, 619, 623, 627, 634, 663, 666, 667, 669, 670, 678, 679, 682, 685, 686, 687, 691, 695, 699, 710, 711, 717, 718, 719, 727, 730, 731, 743, 746, 747, 749, 754, 755, 758, 762, 765, 782, 783, 890, 813, 815, 819, 823, 827, 829, 831, 838, 854, 861, 870, 874, 875, 877, 878, 879, 882, 886, 967, 971, 975, 925, 926, 930, 939, 951, 954, 957, 959, 962, 966, 967, 971, 975, 979, 981, 982 , 985, 987, 994, 995, 997, 998, 1001, 1003, 1005, 1006, 1013, 1017, 1018, 1019, 1021, 1023}

[Equation 308]

T ₁₈ = {14, 15, 22, 30, 38, 46, 54, 55, 58, 61, 62, 86, 90, 91, 94, 102, 106, 111, 237, 238, 239, 242, 243, 245, 238, 237, 238, 234, 315, 317, 318, 319, 327, 330, 334, 335, 343, 334, 338, 446, 447, 445, 446, 447, 455, 462, 466, 446, 427, 429, 475, 477, 479, 482, 483, 486, 490, 491, 494, 498, 499, 501, 502, 503, 505, 507, 511, 535, 542, 550, 554, 559, 571, 573, 575, 582, 583, 586, 598, 602, 603, 605, 607, 614, 615, 619, 622, 626, 627, 629, 630, 631, 634, 646, 650, 655, 662, 666, 669, 683, 687, 690, 691, 695, 699, 702, 703, 710, 711, 718, 719, 722, 723, 726, 733, 738, 739, 747, 755, 758, 761, 766, 767, 783, 790, 851, 859, 861, 866, 870, 874, 882, 889, 895, 895, 814, 823, 830, 839, 843, 846, 847, 854, 902, 907, 910, 92 999, 934, 935, 938, 946, 947, 949, 951, 953, 954, 958, 966, 970, 971, 974, 978, 979, 981, 985, 986, 990, 994, 1002, 1003, 1006, 1009, 1011, 1014, 1018, 1021, 1022}

[Equation 309]

T ₁₉ = {14, 22, 30, 46, 54, 55, 58, 62, 86, 90, 91, 93, 95, 102, 103, 107, 110, 215, 219, 221, 223, 231, 235, 239, 245, 253, 364, 377, 375, 378, 381, 388, 346, 347, 350, 358, 390, 394, 398, 406, 407, 410, 411, 413, 415, 423, 427, 429, 431, 437, 438, 439, 445, 446, 447, 455, 458, 461, 471, 477, 479, 566, 567, 575, 583, 586, 598, 602, 603, 607, 622, 623, 626, 629, 631, 634, 635, 650, 654, 669, 670, 671, 679, 682, 686, 687, 691, 693, 695, 698, 702, 703, 710, 711, 715, 749, 751, 755, 759, 761, 767, 774, 783, 795, 797, 798, 799, 810, 712, 723, 726, 727, 730, 733, 734, 738, 742, 743, 814, 818, 823, 827, 836, 843, 847, 850, 851, 859, 862, 863, 870, 871, 883, 886, 889, 890, 894, 895, 906, 910, 914, 934, 935, 939, 941, 942, 943, 946, 947, 951, 959, 966, 967, 974, 975, 978, 979, 981, 982, 987, 989, 990, 998, 999, 1001, 1005, 1007, 1010, 1013, 1015, 1019, 1021, 1022, 1023}

[Equation 310]

T ₂₀ = {22, 26, 38, 46, 47, 54, 59, 63, 78, 86, 87, 90, 93, 94, 95, 100, 103, 107, 111, 118, 254, 255, 270, 271, 278, 272, 274, 274, 278, 342, 343, 346, 349, 351, 358, 362, 373, 375, 338, 334, 335, 338, 349, 474, 475, 452, 454, 458, 459, 461, 463, 466, 470, 474, 475, 507, 510, 511, 534, 535, 542, 543, 550, 558, 559, 562, 566, 567, 547, 478, 482, 483, 484, 490, 498, 499, 501, 503, 505, 623, 627, 630, 634, 637, 639, 644, 646, 650, 654, 662, 663, 666, 740, 740, 732, 740, 740, 742, 746, 740, 740, 740, 749, 751, 754, 757, 765, 766, 767, 772, 782, 783, 790, 795, 797, 798, 806, 807, 810, 813, 814, 818, 822, 826, 827, 831, 836, 843, 84 8, 89, 893, 894, 903, 906, 907, 910, 914, 918, 923, 925, 883, 874, 878, 883, 885, 886, 887, 930, 932, 934, 935, 941, 942, 943, 951, 953, 954, 957, 958, 959, 964, 966, 970, 982, 983, 985, 987, 989, 994, 995, 996, 1002, 1005, 1009, 1011, 1013, 1014, 1015, 1017, 1019, 1022}

[Equation 311]

T ₂₁ = {14,22,30,31,55,58,62,63,79,84,86,95,102,111,114,118,119,123,126,143,151,166,166,164,166,164,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16, 234, 235, 237, 238, 242, 243, 244, 245, 250, 253, 254, 255, 270, 371, 361, 366, 367, 370, 371, 315, 315, 317, 335, 338, 342, 346, 347, 351, 365, 366, 362, 294, 295, 298, 373, 379, 381, 382, 406, 407, 411, 413, 414, 420, 422, 423, 426, 429, 430, 434, 437, 439, 442, 447, 452, 454, 455, 458, 459, 562, 566, 567, 542, 543, 554, 558, 562, 564, 670, 630, 635, 637, 639, 646, 650, 654, 655, 662, 663, 670, 679, 683, 685, 687, 698, 699, 701, 710, 711, 714, 715, 719, 726, 730, 734, 740, 743, 746, 747, 749, 750, 751, 759, 761, 762, 763, 826, 832, 823, 826, 830, 831, 850, 851, 854, 861, 862, 863, 866, 918, 919, 922, 926, 934, 935, 938, 939, 941, 946, 948, 947, 948, 949, 951, 954, 955, 958, 962, 964, 966, 967, 971, 978, 979, 980, 982, 985, 986, 987, 989, 994, 995, 996, 1001, 1002, 1005, 1007, 1009, 1011, 1013, 1015, 1018, 1019, 1021, 1022}

[Equation 312]

T ₂₂ = {12, 14, 22, 30, 31, 36, 38, 44, 47, 54, 55, 58, 59, 61, 62, 63, 84, 86, 87, 91, 107, 108, 116, 126, 127, 132, 142, 143, 148, 150, 151, 154, 166, 167, 170, 171, 174, 175, 178, 182, 187, 189, 199, 212, 214, 219, 234, 235, 237, 238, 243, 245, 250, 251, 253, 255, 270, 276, 278, 283, 287, 292, 295, 298, 302, 306, 311, 315, 317, 318, 319, 382, 334, 335, 338, 340, 349, 350, 351, 365, 366, 367, 371, 372, 374, 378, 381, 383, 390, 394, 404, 407, 411, 412, 413, 414, 415, 423, 426, 427, 428, 429, 430, 431, 438, 439, 443, 445, 446, 454, 455, 459, 463, 466, 467, 470, 471, 475, 477, 483, 554, 564, 567, 574, 575, 588, 590, 596, 599, 603, 685, 686, 690, 691, 682, 683, 682, 630, 631, 634, 639, 644, 660, 667, 668, 669, 676, 679, 692, 693, 694, 698, 699, 701, 710, 714, 715, 716, 718, 719, 722, 723, 724, 726, 734, 739, 747, 749, 750, 754, 757, 759, 761, 762, 765, 772, 774, 78 838, 842, 846, 847, 852, 855, 858, 859, 861, 812, 813, 814, 815, 822, 826, 827, 862, 863, 866, 877, 882, 885, 886, 887, 890, 891, 894, 895, 900, 902, 903, 916, 918, 934, 941, 942, 946, 947, 957, 958, 962, 964, 967, 970, 971, 978, 979, 981, 990, 991, 994, 995, 996, 998, 999, 1002, 1003, 1004, 1005, 1010, 1011, 1012, 1013, 1015, 1018,

[Equation 313]

T ₂₃ = {12, 15, 20, 22, 26, 36, 38, 52, 55, 60, 62, 63, 84, 87, 90, 108, 110, 115, 118, 122, 123, 127, 132, 140, 142, 143, 148, 151, 154, 156, 158, 164, 170, 174, 175, 183, 187, 188, 244, 246, 251, 253, 255, 271, 276, 279, 282, 283, 286, 287, 299, 342, 343, 347, 349, 350, 351, 358, 359, 365, 366, 373, 380, 308, 315, 316, 318, 319, 332, 334, 335, 338, 394, 396, 404, 406, 407, 410, 412, 415, 420, 422, 428, 430, 431, 434, 435, 439, 442, 443, 445, 447, 452, 455, 461, 463, 468, 470, 477, 478, 483, 486, 491, 494, 498, 500, 502, 503, 505, 507, 524, 527, 532, 534, 539, 540, 542, 543, 559, 566, 574, 580, 583, 586, 588, 602, 603, 612, 615, 618, 619, 621, 622, 627, 628, 631, 638, 639, 644, 646, 652, 654, 660, 662, 668, 669, 682, 685, 687, 690, 693, 694, 695, 698, 699, 701, 710, 717, 719, 722, 724, 726, 731, 732, 733, 735, 738, 740, 742, 746, 751, 756, 758, 761, 76 828, 829, 831, 838, 839, 844, 847, 851, 852, 810, 812, 815, 819, 855, 858, 859, 861, 862, 868, 875, 877, 878, 883, 884, 885, 887, 889, 890, 893, 895, 902, 910, 914, 918, 919, 922, 923, 924, 925, 927, 930, 932, 938, 940, 941, 942, 946, 950, 953, 956, 957, 967, 970, 973, 974, 980, 985, 987, 988, 998, 1002, 1003, 1004, 1005, 1014, 1019, 1020, 1021, 1023}

[Equation 314]

T ₂₄ = {28, 72, 236}

[Equation 315]

T ₂₅ = {40, 124}

[Equation 316]

T ₂₆ = {24, 828}

[Equation 317]

T ₂₇ = {776}

(318)

T ₂₈ = {876}

In the embodiments disclosed herein, the arrangement of the components shown may vary depending on the environment or requirements in which the invention is implemented. For example, some components may be omitted or some components may be integrated into one. In addition, the arrangement order and connection of some components may be changed.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments, but, on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention. Accordingly, the technical scope of the present invention should be determined only by the appended claims.

Claims

A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 20 parity check bits and 64 information bits,

Wherein the polarizer is configured to output 256 sign bits in response to the input of the 20 parity check bits, the 64 information bits, and 172 frozen bits,

The 172 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, , 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86 , 87, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 123, 125, 129, 130 , 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, , 163, 164, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181, 183, 185, 187, 189, 193, 194, 195, 196, , 202, 203, 205, 207, 209, 210, 211, 213, 215, 217, 218, 219, 221, 225, 226, 227, 229, 230, 231, 233, 234, 235, 237, 241, 242 , &Lt; / RTI > 243, 245, 249}

The outer encoder outputs the 20 parity check bits to a parity check bit index set P = {159, 191, 122, 182, 110, 30, 62, 52, 76, 44, 28, 124, 72, 40, 24, 184 , 16, 112, 160, 64}. &Lt; / RTI >
The method according to claim 1,

The support set T 1 = {119, 239, 247, 251} for the first parity check bits among the 20 parity check bits and the support set T 2 = {63, 95, 111, 119 , 175}, the support set T 3 = {111, 127, 175, 186, 191, 239, 251, 253, 255} for the third parity check bits, and the support set T 4 = 15, 174, 182, 186, 191, and 182 for the fifth parity check bits, and the support set T 5 = {111, 159, 174, 182, 186, 191, 125, 125, 253, 255} for the sixth parity check bits, and the support set T 6 = {63, 95, 119, 122, 127, 159, 175, 247 , 251, 255}, the support set T 7 = 119, 159, 174, 182, 186, 206, 214, 238, 239, 247, 253, 255 for the seventh parity check bit, not set on bit 8 = T and {111, 118, 119, 122, 127, 159, 214, 246, 253, 255}, a ninth support for the parity check bit house T 9 = a {52, 110, 118, 119, 175, 186, 190, 191, 212, 222, 223, 244, 250, 253, 254, 255}, a support set for the first 10 parity check bits T 10 = And the support set T 11 = {52, 62, 76, 118, 119, 122, 126, 158, The support set T 12 = {28, 52, 62, 95, 108} for the twelfth parity check bits, , 126, 127, 156, 172, 174, 175, 180, 182, 186, 190, 204, 206, 212, 223, 228, 236, 238, 246, 253, 255} The support set T 13 = {136}, the support set T 14 = {72} for the 14th parity check bits, the support set T 15 = {168} for the 15th parity check bits, T 16 = {16, 44, 52, 76, 88, 94, 95, 104, 108, 110, 116, 120, 159, 175, 182, 186, 191, 200, 204, 206, 216, 222 , 223, 232, 244, 246, 252, 254}, and the 17th parity The support set T 17 = {120} for the check bits and the support set T 18 = {122, 127, 174, 175, 186, 191, 214, 238, 239, 247, 255} for the 18th parity check bits , The support set T 19 for the 19th parity check bits is {24, 28, 48, 52, 56, 60, 62, 64, 76, 80, 88, 92, 94, 110, 111, 112, 255, 144, 152, 159, 175, 180, 190, 204, 206, 216, 223, 228, 232, 240, 246, 247, 248, 251, 252, 255} The set T 20 = {24, 28, 56, 60, 88, 124, 144, 152, 208}

Wherein the outer encoder is configured to convert each bit value of the 20 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 20 parity check bits and 85 information bits,

Wherein the polarizer is configured to output 256 code bits in response to the input of the 20 parity check bits, the 85 information bits, and the 151 frozen bits,

The 151 frozen bits are set to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 55, 57 , 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89, 90, , 98, 99, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 143, 145, 146, 147, 149, 150, 151, 153, 154, 155, 157, 161, 162, 163, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181 , 185, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 209, 210, 211, 213, 217, 219, 221, 225, 226, 227, 229, 233 , 235, 237, 241, 242, 243, 245, 249}

The outer encoder outputs the 20 parity check bits to a parity check bit index set P = {247, 31, 191, 186, 86, 142, 46, 30, 164, 84, 140, 44, 28, 136, , 16, 80, 32, 64}. &Lt; / RTI >
The method of claim 3,

The support set T 1 = {123, 187, 253} for the first parity check bits among the 20 parity check bits and the support set T 2 = {111, 183, 215, 231, 247} for the second parity check bits , 251}, the support set T 3 = {31, 63, 95, 111, 123, 175, 187, 231, 247, 253} for the third parity check bits, and the support set T 4 = {63,119,123,127,75,187,191,218,223,251,253,255} and the support set T 5 = {63, 111, 119, 123, 125, 159, 175, 187, 215, 223, 234, 239, 247, 250, 255}, the support set T 6 = {54} for the sixth parity check bits, T 7 = {78}, the support set T 8 = {46} for the eighth parity check bits, the support set T 9 = {30} for the ninth parity check bits, the support for the tenth parity check bits The set T 10 = {100,148}, and for the 11th parity check bit It is not set T 11 = {52, 84}, and the 12 supporting a set of parity check bits T 12 = {52, 76}, and the 13 supporting a set of parity check bits T 13 = {44}, 14 supporting a set of parity check bits T 14 = {92}, and the 15 supporting a set of parity check bits T 15 = {60, 72}, and the support set for the first 16 parity check bits T 16 = {28, 40 }, The support set T 17 = {24} for the 17th parity check bits and the support set T 18 = 30, 31, 44, 52, 62, 63, 76, 86, 94 for the 18th parity check bits , 100, 110, 116, 118, 119, 123, 126, 127, 142, 156, 158, 164, 182, 186, 190, 191, 204, 206, 207, 212, 214, 222, 223, 228, 230 , 231, 234, 238, 239, 253}, the support set T 19 = {16, 24, 40, 48, 56, 72} for the ninth parity check bit, 20 = {30, 40, 46, 54, 62, 63, 76, 78, 80, 88, 92, 94, 95, 96, 108, 110, 111, 112, 124, 127, 142, 144, 152, 156 , 159, 164, 168, 172, 180, 183, 186, 188, 190, 200, 208, 215, 223, 228, 234, 240, 246, 247, 250, 251,

Wherein the outer encoder is configured to convert each bit value of the 20 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 20 parity check bits and 128 information bits,

Wherein the polarizer is configured to output 256 code bits in response to the input of 20 parity check bits, 128 information bits, and 108 frozen bits,

The 108 frozen bits are set to the frozen bit index set F = {1,2,3,4,5,6,7,9,10,11,13,15,17,18,19,21,23,25,26 , 27, 29, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 57, 65, 66, 67, 69, , 75, 77, 81, 82, 83, 85, 89, 93, 97, 98, 99, 101, 105, 109, 113, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135 , 137, 138, 139, 141, 145, 146, 147, 149, 153, 157, 161, 162, 163, 165, 169, 171, 173, 177, 179, 181, 185, 193, 194, 195, 197 , 201, 203, 205, 209, 210, 211, 213, 217, 225, 226, 227, 229, 233, 241}

The outer encoder outputs the 20 parity check bits as a parity check bit index set P = {91, 199, 247, 239, 191, 106, 186, 22, 14, 238, 190, 36, 20, 12, , 24, 16, 80, 64}. &Lt; / RTI >
6. The method of claim 5,

The support set T 1 = {107, 221, 235, 237} for the first parity check bit among the 20 parity check bits and the support set T 2 = {59, 61, 123, 155 , 187, 221, 235, 243, 253} and the support set T 3 = {55, 61, 87, 91, 103, 123, 125, 155, 167, 183, 187, 189, 215, 219, 231, 235, 245, 253}, and the fourth support set for the parity check bit T 4 = {61, 87, 103, 119, 125, 151, 155, 167, 183, 187, 189, 199 , 207, 215, 219, 235, 247}, and the support set T 5 = {31, 55, 79, 87, 91, 95, 107, 125, 151, 159, 167, 183, 187, 189, 207, 223, 235, 237, 239, 243, 245, 247} and the support set T 6 = {47, 55, 91, 95, 111, 119, 125, 143 , 151, 155, 159, 167, 170, 175, 178, 183, 187, 202, 215, 231, 242, 245, 249}, and the support set for the seventh parity check bit T 7 = {55, 58, 59, 63, 79, 90, 95, 103, 106, 111, 123, 125, 1 234, 239, 243, 249, 255}, and the support set T 8 for the eighth parity check bits = {31, , 47, 55, 58, 59, 61, 79, 87, 90, 95, 107, 111, 119, 125, 127, 143, 151, 155, 167, 178, 183, 187, 207, 215, , 223, 230, 231, 235, 239, 243, 245, 249, 253, 255} and the support set T 9 = {22, 31, 54, 55, 58, 59, 86, 183, 189, 199, 219, 221, 231, 242, 234, 234, 245, 250, 251, 253, 255}, and the support set T 10 = {54, 55, 58, 59, 78, 86, 87, 91, 95, 106, 107, 110, 111 , 119, 125, 127, 151, 154, 175, 178, 189, 198, 202, 206, 207, 214, 218, 219, 221, 231, 242, 243, 245, 247, 249, , The support set T 11 for the eleventh parity check bits is {30, 46, 58, 59, 62, 94, 102, 107, 111, 119, 122, 125, 151, 159, 182, 183, 186, 187, 189, 191, 207, 215, 218, 221, 230, 234, 238 }, The support set T 12 = {68} for the twelfth parity check bits, the support set T 13 = {36} for the thirteenth parity check bits, T 14 = {20} for the 14 parity check bits and the support set T 15 = 22, 44, 47, 52, 54, 55, 61, 62, 76, 79, 87 for the 15th parity check bits , 90, 91, 94, 95, 108, 110, 111, 114, 116, 122, 125, 126, 127, 143, 151, 158, 159, 183, 187, 196, 199, 202, 204, 206, 207 , A support set T 16 = {76} for a 16th parity check bit, and a 17th parity check bit for a 16th parity check bit, and 17th, 16th , supporting a set of parity check bits T 17 = {8, 12, 20, 36, 40, 52, 68, 76} , and 18 supporting a set of parity check bits T 18 = {8, 24, 30, 72, 108, 136}, the support set T 19 = {28, 104, 136, 254} for the 19th parity check bits, the support set T 20 = {16, 208} for the 20th parity check bits,

Wherein the outer encoder is configured to convert each bit value of the 20 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 16 parity check bits and 171 information bits,

Wherein the polarizer is configured to output 256 code bits in response to the input of the 16 parity check bits, the 171 information bits, and 69 frozen bits,

The 69 frozen bits correspond to the frozen bit index set F = {1, 2,3, 5,6,7,9,10,11,13,17,18,19,21,25,33,34,35,37 51, 53, 57, 65, 66, 67, 69, 73, 77, 81, 83, 85, 89, 97, 98, 99, 101, 105, 113, 129, 130, 131 , 133, 137, 139, 141, 145, 147, 149, 153, 161, 162, 163, 165, 169, 177, 193, 194, 195, 197, 201, 209, 217, 225, 229, 233, 241 } &Lt; / RTI > in the bit indexes defined by < RTI ID = 0.0 &

The outer encoder codes the 16 parity check bits into a parity check bit index set P = {179, 155, 39, 247, 111, 50, 90, 230, 142, 190, 4, 12, 28, 8, } &Lt; / RTI > in the bit indexes defined by the parity check concatenated polar coder.
8. The method of claim 7,

The second set of parity check bits is a set of the first parity check bits T 1 = {29, 93, 109, 117, 157, 181, 185, 205, 211, 213, 245, 249} And the second set of support bits for the third parity check bit is set to T 2 = {27,61,75,93,121,157,171,173,203,205,111,213,235,237,243} T 3 = {27, 29, 43, 59, 71, 91, 107, 109, 117, 121, 125, 135, 171, 173, 179, 203, 205, 219, 235, 245, 253} supporting a set of parity check bits T 4 = {27, 39, 115, 119, 121, 123, 155, 167, 173, 179, 187, 189, 199, 205, 211, 215, 221, 227, 235, 237 , 243, 249}, and the support set T 5 = {29, 39, 43, 55, 79, 93, 107, 109, 119, 143, 151, 155, 157, 171, 173, 253}, the support set for the sixth parity-check bits T 6 = {29}, {241, 252, , 31, 39, 43, 63, 75, 82, 87, 91, 103, 121, 125, 151, 173, 187, 199, 207, 2 25, 253, 255}, and the support set T 7 = {23,26,27,31} for the seventh parity check bit , 43, 59, 61, 74, 75, 79, 82, 91, 93, 103, 106, 109, 111, 114, 117, 146, 154, 159, 171, 175, 179, 181, 185, 187, 191 23, 239, 243, 247, 249}, and the support set T 8 = {23, 27, 29, 38, 39, 42, 43, 55, 59, 63, 70, 71, 74, 79, 82, 87, 90, 95, 109, 114, 117, 119, 234, 235, 242, 250, 251} for the ninth parity check bits, and the support set T 9 = {15, 26, 29, 39, 42, 47, 50, 54, 55, 58, 59, 61, 63, 70, 71, 79, 82, 86, 87, 91, 93, , 111, 114, 121, 123, 125, 134, 135, 146, 150, 151, 159, 178, 179, 182, 185, 186, 187, 189, 202, 207, 213, 214, 218, 221, 223 , 226, 230, 231, 234, 235, 237, 242, 243, 245, 246, 249, 251, 253, 255} Support set for the check bits T 10 = {14, 15, 22, 23, 27, 30, 47, 50, 54, 55, 58, 61, 62, 71, 75, 78, 82, 86, 90, 94, 176, 179, 181, 183, 185, 187, 199, 199, 117, 111, 114, 115, 117, 119, 121, 122, 142, 146, 150, 155, 158, 159, 167, 171, 22, 242, 245, 247, 250, 255} and the support set T 11 = {14, 22 , 23, 26, 29, 30, 38, 39, 42, 47, 50, 54, 55, 58, 62, 63, 71, 74, 86, 87, 90, 91, 102, , 115, 121, 125, 142, 143, 146, 150, 151, 155, 170, 171, 173, 179, 181, 182, 183, 187, 189, 191, 202, 206, 207, , 218, 221, 222, 223, 226, 227, 238, 239, 247, 250, 253, 254, 255}, the support set T 12 = {20, 30, 36, 68, 132, 182}, the support set T 13 = {14, 36} for the thirteenth parity check bits, the support set T 14 = {12, 118} for the fourteenth parity check bits, Korea A set T 15 = {4, 14, 22, 23, 24, 26, 28, 29, 31, 42, 43, 44, 52, 54, 56, 59, 60, 62, 70, 76, 79, 87, 118, 121, 122, 123, 124, 126, 134, 148, 154, 159, 166, 173, 174, 175, 181, 183, 234, 235, 236, 239, 243, 246, 234, 236, 234, 235, 236, 234, 247, 248, 249, 251, 252, 254}, the support set T 16 = {20} for the 16th parity check bits,

Wherein the outer encoder codes each bit value of the 16 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 22 parity check bits and 128 information bits,

Wherein the polarizer is configured to output 512 sign bits in response to the 22 parity check bits, 128 information bits, and 362 frozen bits,

The 362 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, , 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78 , 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 111 , 113, 114, 115, 117, 118, 119, 121, 122, 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, , 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 159, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 175, 177 , 178, 179, 181, 182, 183, 185, 186, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 206, , 213, 214, 215, 217, 218, 219, 221, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243, 245, 249, 251, 253 , 257, 258, 259, 260, 261, 262, 263, 264, 265, 266 , 267, 268, 269, 270, 271, 273, 274, 275, 276, 277, 278, 279, 281, 282, 283, 285, 287, 289, 290, 291, 292, 293, 294, 295, 297 , 298, 299, 301, 302, 303, 305, 306, 307, 309, 310, 311, 313, 314, 315, 317, 321, 322, 323, 324, 325, 326, 327, , 333, 334, 335, 337, 338, 339, 340, 341, 342, 343, 345, 346, 347, 349, 353, 354, 355, 356, 357, 358, 359, 361, 362, 363, 365 , 369, 370, 371, 373, 377, 379, 381, 385, 386, 387, 388, 389, 390, 391, 393, 394, 395, 397, 398, 399, 401, 402, 403, 404, 405 , 406, 407, 409, 410, 411, 413, 417, 418, 419, 420, 421, 422, 423, 425, 426, 427, 429, 433, 434, 435, 437, 441, 443, 445, 449 , 450, 451, 452, 453, 454, 455, 457, 458, 459, 461, 465, 466, 467, 469, 471, 473, 474, 475, 477, 481, 482, 483, 485, , 490, 491, 493, 497, 498, 499, 501, 505}

The outer encoder outputs the 22 parity check bits to a parity check bit index set P = {255, 438, 430, 286, 62, 254, 180, 396, 428, 236, 92, 380, 296, 488, , 16, 80, 48, 32, 96, 128}.
10. The method of claim 9,

The first set of parity check bits for the first parity check bit is set to T 1 = {63, 239, 247, 319, 351, 431, 463, 495, 503} 2 = {63, 191, 239, 250, 255, 319, 367, 375, 383, 470, 479, 486, 495, 506, 507, 509, 511}, and the third supporting a set of parity check bits T 3 = {127, 174, 247, 319, 375, 378, 383, 439, 442, 447, 462, 463, 479, 502, 503, 506, 507, 511} 4 = {63, 191, 223, 319, 366, 374, 415, 430, 431, 438, 439, 442, 447, 463, 470, 494, 495, 502} T 5 = {63, 94, 127, 158, 191, 239, 246, 247, 250, 255, 319, 350, 351, 366, 374, 375, 378, 383, 414, 430, 431, 439, 442, 462, 463, 470, 478, 479, 486, 494, 495, 502, 503, 511} and the support set T 6 = {63, 110, 127, 158, 191, 239, 318 , 366, 378, 383, 415, 431, 438, 439, 442, 462, 463, 478, 479 , 486, 494, 502, 506, 511}, and the support set T 7 = {62, 63, 94, 110, 127, 158, 191, 212, 222, 238, 239, 246, 447, 462, 463, 478, 506, 507, 510}, and the 8th, 8th, The support set T 8 = {62, 110, 126, 158, 212, 222, 223, 238, 247, 254, 255, 286, 308, 318, 319, 374, 383, 414, 430, 431 , 436, 442, 447, 495, 500, 502, 503, 506} and the support set T 9 = {62, 94, 110, 127, 158, 180, 190, 191, 204, 438, 439, 446, 460, 462, 468, 470, 478, 370, 370, 486, 495, 500, 507, 509, 510, 511}, and the support set for the first 10 parity check bits T 10 = {110, 116, 126, 174, 180, 191, 204, 212, 222, 238, 239 , 246, 250, 254, 300, 318, 332, 350, 364, 366, 375, 383, 430, 438, 439, 446, 447, 460, 468, 470, 484, 486, 500, 506, }, And the 11th Supporting a set of parity check bits T 11 = {62, 110, 116, 126, 127, 190, 191, 212, 222, 236, 308, 318, 332, 364, 367, 378, 382, 383, 396, 415 , 428, 438, 439, 446, 447, 460, 462, 463, 468, 478, 479, 484, 486, 492, 495, 500, 502, 507, 509, 510, 511} not set to T 12 = {62, 92, 94, 116, 124, 188, 212, 236, 238, 239, 300, 308, 318, 332, 348, 350, 367, 372, 374, 375, 383, 503, 507, 511}, and the support set T 13 = {60, 62} for the thirteenth parity check bits , 110, 124, 127, 158, 172, 174, 188, 191, 200, 212, 239, 244, 247, 284, 286, 300, 308, 319, 332, 367, 372, 375, 380, 415, 430 , 444, 447, 456, 462, 468, 470, 478, 479, 492, 502, 503, 508, 509, 510, 511}, the support set T 14 = {62, 63, 286, 296, 300, 318, 328, 360, 364, 367, 375, 382, 414, 415, 424, 4 495, 506, 507, 509, 510, 511}, and the 15th parity check bit is set to " T15 = {104, 116, 127, 168, 172, 180, 190, 191, 200, 212, 220, 223, 232, 238, 239, 250, 252, 254, 284, 286, , 308, 316, 318, 319, 328, 348, 350, 351, 364, 366, 374, 375, 378, 380, 382, 383, 392, 415, 424, 428, 431, 436, 442, 470, 476 , 478, 479, 484, 502, 506, 510}, and the 16 supporting a set of parity check bits T 16 = {24, 56, 60, 104, 110, 116, 124, 156, 168, 172, 190, 364, 367, 372, 374, 362, 364, 362, 364, 368, 362, 344, 348, 351, 507, 508, 509, 510, 511}, and the seventeenth, seventeenth, fourteenth, fourteenth, fourteenth, fourteenth, fourteenth, The support set T 17 = {56, 60, 94, 110, 120, 124, 156, 158, 168, 180, 190, 191, 204, 212, 216, 220, 222, 238, 244, 247 , 248, 255, 280, 284, 286, 296, 318, 319 , 328, 344, 350, 364, 367, 374, 383, 392, 412, 415, 424, 430, 436, 438, 439, 447, 456, 462, 463, 468, 470, 472, 476, 478, 479 , 486, 488, 492, 494, 495, 500, 503, 506, 507, 509}, the support set T 18 = {24, 88, 144, 152, 248, 272} The support set T 20 = {80, 88, 112} for the twentieth parity check bit, the support set T 19 = {80,144,184} for the nineteenth parity check bit, The set T 21 = {16}, the support set T 22 = 22 for the 22nd parity check bits,

Wherein the outer encoder codes each bit value of the 22 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 20 parity check bits and 171 information bits,

Wherein the polarizer is configured to output 512 sign bits in response to the input of 20 parity check bits, 171 information bits, and 321 frozen bits,

The 321 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, , 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85 , 86, 87, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 123, 125, 129 , 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 161 183, 193, 194, 195, 196, 197, 198, 199, 173, 177, 178, 179, 181, 183, 185, 167, 169, 170, 171, , 201, 202, 203, 205, 207, 209, 210, 211, 213, 215, 217, 218, 219, 221, 225, 226, 227, 229, 230, 231, 233, 234, 235, 237, 241 , 242, 243, 245, 249, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274, 275, 276, 277, 278 , 279, 281, 282, 283, 285, 289, 290, 291, 2 314, 315, 317, 321, 322, 323, 324, 325, 326, 327, 329, 361, 362, 363, 365, 369, 370, 343, 345, 346, 347, 349, 353, 354, 355, 357, 358, 359, 391, 395, 397, 399, 401, 402, 403, 405, 406, 407, 409, 410, 411, 413, 393, 391, 393, 454, 455, 457, 458, 457, 458, 453, 454, 458, 419, 421, 422, 423, 425, 426, 427, 429, 433, 434, 435, 437, 441, 445, 459, 461, 465, 466, 467, 469, 473, 477, 481, 482, 483, 485, 489, 491, 493, 497, 499, 501, 505} Lt; / RTI &

The outer encoder outputs the 20 parity check bits to a parity check bit index set P = {175, 223, 223, 506, 118, 430, 30, 382, 340, 396, 44, 28, 60, , 16, 48, 32, 512}. &Lt; / RTI >
12. The method of claim 11,

The support set T 1 = {119, 247, 251, 443, 471, 487, 503, 507} for the first parity check bits among the 20 parity check bits and the support set T 2 = And the support set T 3 = {0, 1, 2, 3, 4, 5, 6, 95, 159, 175, 287, 381, 415, 439, 463, 471, 487, 507, 509}, and the support set T 4 = {63, 95, 122, 127, 159, 191 , 251, 319, 351, 367, 378, 379, 381, 439, 447, 463, 471, 474, 479, 495, 498, 503, 507}, and the support set T 5 = { 507, 507, 507, 507, 507, 507, 507, 507, 507, And the support set T 6 = {63, 95, 119, 122, 159, 174, 175, 186, 250, 251, 253, 287, 302, 310, 319, 342, 367, 378 for the sixth parity check bits , 379, 383, 398, 431, 442, 447, 463, 470, 474, 479, 486, 490, 498, 503} and the support set T 7 = {95, 110, 122, 159, 174, 175, 182, 186, 206, 223, 247, 255, 287, 303, 310, 319 , 366, 367, 374, 381, 383, 430, 431, 438, 442, 447, 463, 471, 486, 487, 490, 498, 503, 506, 507, 509, 511} T8 = {62, 63, 94, 110, 111, 122, 127, 159, 174, 175, 190, 206, 214, 222, 223, 246, 250, 251, 286, 287, 302, 464, 475, 479, 486, 490, 494, 498, 502, 506, 511} And a support set T 9 = {110, 119, 127, 175, 182, 186, 191, 206, 222, 223, 238, 239, 246, 247, 250, 251, 286, 287 for the ninth parity check bits , 302, 318, 319, 356, 366, 375, 378, 379, 381, 383, 398, 414, 415, 431, 442, 443, 446, 447, 462, 471, 479, 486, , 506, 509}, and the support set T 10 = {30, 62, 63, 94, 111, 116, 127, 159, 174, 175, 206, 214, 222, 223, 244, 250, 253, 254, 3 462, 463, 468, 479, 484, 438, 438, 444, 446, 462, 48}, 490, 494, 495, 500, 502, 507, 509}, the support set T 11 = {76} for the 11th parity check bits, the support set T 12 = {44} And the support set T 13 = {63, 94, 108, 110, 111, 116, 119, 126, 158, 159, 174, 175, 182, 186, 191, 204, 214, 239 for the thirteenth parity check bits , 244, 246, 250, 286, 287, 300, 308, 310, 319, 340, 350, 364, 366, 372, 374, 375, 381, 383, 398, 404, 414, 415, 428, 430, 431 , 436, 447, 462, 463, 468, 471, 475, 478, 487, 490, 495, 500, 502, 507, 510, 511}, the support set T 14 = {72, 92, 136}, the support set T 15 = {40, 156} for the 15th parity check bits, the support set T 16 = {316} for the 16th parity check bits, 17 = {72} and the set T, 18 parity check Support set for the bit T 18 = {24, 28, 32, 40, 44, 48, 56, 60, 80, 92, 95, 104, 111, 122, 124, 136, 152, 158, 172, 175, 180 , 182, 184, 186, 188, 191, 200, 204, 214, 222, 228, 236, 239, 244, 247, 248, 251, 253, 272, 286, 287, 300, 316, 318, 332, 342 , 350, 356, 360, 367, 374, 380, 381, 400, 408, 420, 424, 428, 439, 440, 442, 456, 463, 468, 471, 475, 476, 484, 487, 490, 492 , 495, 504, 507, 510, 511}, the support set T 19 = {24, 40, 48, 56, 136, 144} for the ninth parity check bit, 20 = {24,28,30,32,44,60,62,72,80,104,108,111,116,118,119,122,124,126,127,168,180,182,184,188 , 191, 200, 204, 208, 212, 214, 216, 222, 224, 228, 232, 239, 244, 246, 248, 251, 252, 253, 255, 272, 280, 287, 288, 296, 300 , 303, 310, 312, 316, 328, 334, 336, 344, 352, 356, 360, 364, 367, 368, 375, 376, 379, 382, 383, 392, 396, 400, 404, 408, 412 , 414, 416, 420, 424, 428, 430, 432, 436, 438, 439, 442, 462, 468, 475 , 476, 478, 480, 486, 488, 490, 494, 498, 500, 502, 508}

Wherein the outer encoder is configured to convert each bit value of the 20 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 22 parity check bits and 256 information bits,

Wherein the polarizer is configured to output 512 code bits in response to the 22 parity check bits, the 256 information bits, and 234 frozen bits,

The 234 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22 , 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 57, 59, , 67, 68, 69, 70, 71, 73, 74, 75, 77, 81, 82, 83, 85, 89, 91, 93, 97, 98, 99, 101, 103, , 114, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 145, 146, 147, 149, 151, 153, 155, 157, 161, 162 , 163, 165, 167, 169, 170, 171, 173, 177, 178, 179, 181, 185, 193, 194, 195, 197, 198, 199, 201, 202, 203, 205, , 213, 217, 225, 226, 227, 229, 233, 237, 241, 245, 249, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 269, 273, 274, 275 , 277, 279, 281, 282, 283, 285, 289, 290, 291, 293, 295, 297, 298, 299, 301, 305, 306, 307, 309, 313, 321, 322, 323, 325, 326 , 327, 329, 330, 331, 333, 337, 338, 339, 341, 345, 349, 353, 354, 355, 357, 361, 365, 369, 371, 373, 377, 385, 386, , 390, 391, 393, 394, 395, 397, 401, 402, 403, 405, 409, 413, 417, 418, 419, 421, 425, 427, 429, 433, 435, 437, 441, 449, 450, 451, 453, 457, 459, 461, 465, 467, 469, 473, 481, 482, 483, 485, 489, 497}

The outer encoder outputs the 22 parity check bits as parity check bit index sets P = {379, 343, 271, 31, 127, 362, 442, 230, 438, 78, 414, 324, 404, 500, 12, 476 , 8, 40, 24, 16, 32, 64}.
14. The method of claim 13,

Support set for the first parity check bits of the 22 parity check bits T 1 = {219, 221, 235, 347, 363, 411, 443, 445, 493, 499, 501, 505} , and the second parity check And the third set of parity check bits is set to T 2 = {87, 125, 187, 231, 235, 243, 315, 347, 359, 379, 381, 411, 423, 455, 475, 501, 507} not set to T 3 = {87, 123, 189, 219, 231, 235, 247, 251, 253, 311, 343, 347, 375, 381, 407, 475, 477, 487, 501, 505, 507, 509}, and the support set T 4 = {55, 87, 123, 143, 183, 207, 235, 239, 253, 271, 311, 315, 359, 363, 375, 381, 407 , 313, 413, 431, 443, 445, 455, 463, 487, 491, 499, 501, 503}, and the support set T 5 = 359, 367, 399, 407, 411, 439, 445, 455, 463, 471, 477, 479, 491, 501}, and the support set T 6 = {31, 55, 311, 315, 317, 319, 343, 347, 318, 317, 318, 319, 463, 367, 370, 381, 383, 407, 431, 434, 439, 443, 445, 455, 458, 463, 466, 475, 487, 493, 495, 505, 509, 511} support set for the bit T 7 = {47, 63, 127, 154, 183, 186, 221, 243, 247, 251, 255, 271, 303, 311, 315, 347, 351, 362, 363, 367, 375 , 411, 415, 423, 426, 431, 434, 439, 443, 447, 466, 471, 474, 477, 491, 493, 495, 498, 501, 503, 507} The support set T 8 = {31, 63, 79, 87, 90, 95, 102, 111, 119, 127, 143, 166, 186, 187, 189, 207, 218, 219, 221, 223, 231, 242, 431, 434, 439, 442, 445, 447, 455, 458, 431, 432, 423, 47, 47, 479, 490, 493, 499, 506, 507} and the support set T 9 = {47, 54, 58, 79, 87, 95, 111, 119 , 122, 123, 125, 150, 166, 175, 187, 214, 215, 218, 221, 363, 367, 378, 381, 383, 406, 361, 361, 362, 343, 407, 410, 411, 423, 426, 431, 434, 439, 442, 443, 447, 454, 458, 466, 471, 475, 477, 479, 491, 495, 505, 507} The support set T 10 = {47, 58, 63, 79, 90, 102, 106, 111, 118, 142, 143, 166, 182, 183, 186, 187, 207, 214, 219, 230, 231 , 235, 239, 243, 246, 247, 270, 271, 278, 303, 311, 317, 335, 342, 343, 358, 359, 367, 370, 374, 378, 379, 383, 407, 415, 422 , 431, 443, 445, 454, 455, 458, 471, 491, 493, 495, 498, 501, 503, 506, 507}, the support set T 11 = {30, 46, 234, 235, 238, 239, 242, 234, 234, 238, 234, 366, 367, 370, 375, 378, 381, 398, 407, 410, 316, 314, 317, 334, 342, 343, 346, 415, 423, 426, 431, 439, 442, 445, 454, 455, 458, 462, 466, 471, 475, 486, 48, 491, 493, 495, 498, 502, 506, 507} and the support set T 12 = {46, 54, 63, 79, 90, 94, 95, 122, 125, 126 , 127, 142, 143, 174, 175, 183, 189, 190, 206, 207, 215, 218, 219, 221, 230, 234, 235, 238, 239, 242, 243, 246, , 278, 294, 302, 303, 311, 314, 315, 318, 319, 359, 363, 374, 375, 378, 381, 383, 388, 398, 399, 407, 410, 414, 415, 423, 431 , 439, 442, 445, 447, 454, 463, 466, 470, 471, 475, 477, 478, 493, 494, 503, 509}, the support set T 13 = {31, 186, 189, 190, 191, 207, 214, 219, 221, 222, 298, 292, 294, 302, 317, 319, 324, 347, 350, 351, 359, 363, 366, 374, 378, 379, 381, 383, 398, 407, 410, 411, 414, 420, 423, 430, 438, 439, 442, 447, 463, 470, 474, 479, 484, 490, 491, 495, 505, 507, 510, 511}, and support for the 14th parity check bit The sum T 14 = {46, 47, 52, 54, 58, 63, 78, 86, 90, 94, 95, 100, 102, 110, 111, 116, 118, 119, 154, 186, 191, 207, 214 , 215, 218, 221, 222, 223, 234, 242, 246, 251, 254, 255, 276, 278, 294, 302, 310, 314, 315, 318, 319, 324, 346, 350, 358, 362 , 363, 366, 370, 374, 378, 381, 388, 399, 406, 407, 411, 420, 422, 426, 430, 431, 434, 438, 439, 443, 445, 455, 463, 466, 468 31, 47, 52, 55, 58, and 58 for the fifteenth parity check bits, and the support set T 15 = {30, 31, 47, 52, 55, 58, 142, 143, 148, 150, 158, 159, 174, 182, 187, 196, 214, 215, 218, 221, 315, 318, 319, 334, 340, 342, 346, 350, 351, 358, 308, 308, 462, 366, 372, 382, 399, 404, 406, 411, 414, 415, 422, 426, 430, 431, 438, 439, 443, 447, 452, 454, 463, 471, 475, 478, 479, 484, 487, 490, 491, 493, 494, 498, 501, 502, 506, 510} 16 = {12, 30, 31, 44, 46, 52, 62, 78, 84, 86, 92, 94, 95, 102, 108, 110, 119, 140, 148, 150, 158, 159, 164, 172 , 180, 183, 187, 189, 190, 196, 207, 214, 220, 221, 230, 231, 234, 235, 238, 244, 247, 251, 253, 254, 268, 271, 276, 300, 308 , 311, 317, 319, 324, 332, 334, 343, 347, 348, 351, 358, 359, 363, 366, 367, 370, 375, 378, 379, 381, 399, 404, 407, 411, 415 , 422, 423, 431, 438, 439, 442, 443, 445, 446, 447, 452, 460, 462, 466, 468, 470, 471, 474, 475, 478, 479, 486, 487, 490, 492 31, 44, 47, 52, 54, and 54 for the seventeenth parity check bits, and the support set T 17 = {28, 31, 44, 47, 52, 54, 125, 126, 140, 142, 143, 156, 159, 172, 174, 284, 287, 292, 302, 308, 310, and 290, respectively. 314, 316, 318, 319, 332, 334, 335, 340, 347, 348, 350, 351, 356, 359, 362, 363, 374, 375, 379, 380, 381, 382, 383, 396, 39 449, 412, 414, 415, 420, 426, 428, 430, 431, 436, 438, 445, 446, 458, 468, 477, 478, 479, 484, 492, 495, 498, 499, 508, 509, 510, 511}, and 18 supporting a set of parity check bits T 18 = {8, 12, 72, 136, 158, 172, 264, 284, 380, 460} , and , The support set T 20 = {40, 44, 136, 156, 364} for the nineteenth parity check bits and the support set T 20 = {28, 40, 72, 124} The support set T 21 = {24, 184} for the parity check bits 21, the support set T 22 = {16, 220} for the parity check bits 22 ,

Wherein the outer encoder codes each bit value of the 22 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 30 parity check bits and 341 information bits,

Wherein the polarizer is configured to output 512 sign bits in response to the input of the 30 parity check bits, the 341 information bits, and the 141 frozen bits,

The 141 frozen bits are set in the frozen bit index set F = {1,2,3,4,5,6,7,9,10,11,13,17,18,19,21,25,29,33,34 , 35, 37, 41, 43, 45, 49, 51, 53, 57, 65, 66, 67, 69, 71, 73, 75, 77, 81, 82, 83, 85, 89, 97, 98, 99 , 101, 105, 113, 121, 129, 130, 131, 133, 135, 137, 138, 139, 141, 145, 146, 147, 149, 153, 161, 162, 163, 165, 169, 177, 181 , 185, 193, 194, 195, 197, 201, 209, 213, 217, 225, 227, 229, 233, 241, 257, 258, 259, 261, 263, 265, 266, 267, 269, 273, 274 , 275, 277, 281, 289, 290, 291, 293, 297, 305, 309, 313, 321, 322, 323, 325, 329, 333, 337, 341, 345, 353, 355, 357, 361, 369 The bits defined by the bits 385, 386, 387, 389, 393, 397, 401, 403, 405, 409, 417, 419, 421, 425, 433, 449, 450, 451, 453, 457, 465, Input to the polarizer at indexes,

The outer encoder outputs the 30 parity check bits as parity check bit index set P = {493, 125, 435, 235, 27, 39, 23, 15, 367, 95, 210, 330, 26, 58, , 14, 46, 190, 68, 20, 52, 12, 28, 8, 40, 24, 144, 240, 256}.
16. The method of claim 15,

The first set of parity check bits is a support set T 1 = {109, 117, 173, 237, 301, 437, 441, 461, 489, 501} for the first parity check bits, set T 2 = {61, 117, 173, 205, 221, 237, 317, 349, 377, 441, 461, 473, 493, 497} , and the third support set for the parity check bit T 3 = {61, And the fourth set of parity check bits for the fourth parity check bit is T ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 4 = {93, 109, 125, 171, 205, 211, 221, 245, 249, 285, 299, 349, 363, 377, 395, 413, 427, 429, 435, 459, 469, 485, 497, 499 , 505, 509}, and the support set T 5 = {61, 93, 107, 115, 117, 157, 171, 173, 203, 205, 211, 237, 245, 301, 317, 49, 499, 501, 505, 509} and the support set T 6 = {59, 115, 155} for the sixth parity- , 171, 179, 189, 205, 221, 237, 253, 283, 299, 301, 307, 315, 495, 489, 491, 493, 497, 499, 507}, and the seventh, seventh, eighth, T 7 = {27, 91, 109, 115, 117, 123, 125, 155, 157, 171, 189, 211, 219, 231, 235, 243, 249, 283, 285, 295 for parity check bits. , 299, 315, 317, 363, 365, 373, 411, 413, 423, 427, 441, 443, 455, 459, 461, 469, 475, 477, 485, 491, 497, 499, 501, , The support set T 8 = {27, 39, 55, 59, 87, 91, 107, 109, 117, 119, 151, 157, 167, 171, 173, 179, 189, 365, 373, 375, 377, 379, 407, 413, 429, 435, 331, 347, 359, 49, 497, 501, 503}, and the support set T 9 = {15, 27, 39, 47, 59 , 61, 87, 91, 93, 103, 111, 117, 125, 157, 175, 203, 207, 215, 221, 235, 237, 243, 245, 251, 271, 285, 295, 315, 327, 349 , 365, 371, 373, 377, 379, 381, 395, 423, 427, 429, 43 5, 439, 441, 443, 463, 467, 471, 473, 483, 491, 497, 499, 501, 503, 507}, and the support set T 10 = {27, 39, 55 , 61, 79, 93, 103, 107, 111, 117, 119, 125, 155, 157, 173, 175, 187, 199, 203, 205, 211, 237, 243, 245, 247, 249, 251, 253 , 271, 283, 287, 299, 303, 307, 315, 339, 359, 367, 371, 375, 377, 379, 399, 407, 411, 415, 429, 435, 437, 439, 443, 445, 455 , 451, 471, 475, 477, 485, 489, 491, 497, 501, 505, 507}, and the support set T 11 = {15, 23, 27, 31, 47, 61, 237, 245, 251, 255, 295, 299, 239, 239, 239, 371, 373, 375, 381, 383, 395, 402, 411, 413, 415, 418, 441, 447, 459, 469, 473, 343, 347, 349, 359, 509}, the support set T 12 = 50 for the twelfth parity check bit, the thirteenth parity check bit is set to < RTI ID = 0.0 >Lt; RTI ID = 0.0 > T13 = {42}, the support set T 14 = {26, 42} for the 14th parity check bits, the support set T 15 = {26, 70} for the 15th parity check bits, The support set T 16 = {38}, the support set T 17 = {54} for the 17th parity check bits, the support set T 18 = 22, 26, 38, 42} for the 18th parity check bits, T 19 = {15, 22, 23, 26, 27, 30, 31, 39, 42, 55, 59, 62, 70, 74, 78, 79, 86, 87, 91 for the 19th parity check bits. , 93, 94, 95, 102, 106, 110, 111, 114, 115, 127, 143, 150, 151, 166, 167, 171, 174, 175, 179, 182, 186, 198, 202, 206, 207 , 221, 222, 230, 231, 235, 238, 242, 243, 245, 250, 251, 262, 271, 279, 282, 283, 285, 287, 294, 295, 298, , 306, 310, 314, 315, 318, 330, 331, 334, 342, 346, 347, 349, 351, 363, 365, 371, 373, 374, 375, 377, 381, 383, , 413, 418, 422, 423, 426, 427, 429, 430, 431, 435, 437, 438, 439, 447, 454, 458, 459, 461, 462, 466, 462, 471, 474, 479, 486, 489, 491, 494, 495, 501, 506, 511}, the support set T 20 = {132, 278} for the 20th parity check bit, not set T 21 = {36, 68, 86} , and on, and 22 supporting a set of parity check bits T 22 = {206} is the support set for the first 23 parity check bits T 23 = {20}, The support set T 24 = {15, 22, 26, 27, 30, 38, 39, 46, 47, 50, 52, 54, 55, 58, 61, 63, 74, 76, 78 for the 24th parity- , 79, 86, 94, 95, 106, 114, 115, 116, 119, 125, 154, 164, 170, 173, 179, 180, 186, 189, 190, 196, 198, 199, 202, 203, 204 , 205, 210, 211, 219, 222, 228, 231, 235, 236, 238, 242, 244, 245, 246, 247, 249, 250, 251, 254, 262, 268, 270, 278, 283, 292 , 294, 298, 299, 300, 302, 303, 306, 307, 308, 318, 319, 327, 330, 332, 346, 347, 350, 351, 356, 358, 362, 365, 366, 371, 377 , 378, 381, 383, 388, 390, 394, 395, 396, 402, 404, 407, 410, 414, 418, 420, 423, 426, 427, 430, 437, 445, 452, 455, 463, 4 50, 507, 510, 511}, and the support set T 25 = {44, 473, 475, 477, 479, 483, 485, 486, 487, 489, 492, 495, }, The support set T 26 = {14, 22, 23, 27, 28, 31, 38, 39, 42, 46, 50, 54, 55, 60, 61, 62, 63, 68, 70, 74, 78, 79, 84, 86, 87, 92, 94, 100, 102, 109, 115, 117, 118, 119, 124, 127, 132, 134, 136, 186, 187, 188, 190, 196, 198, 202, 203, 204, 210, 214, 218, 262, 264, 268, 270, 278, 279, 282, 283, 292, 348, 358, 367, 371, 375, 378, 380, 382, 334, 334, 335, 339, 340, 343, 346, 347, 348, 463, 429, 431, 435, 437, 438, 444, 445, 452, 459, 461, 468, 469, 470, 474, 475, 476, 477, 482, 483, 484, 485, 486, 490, 491, 494, 495, 500, 503, 505, 508, 510 , 511}, and the support set T 27 = {22, 27, 38, 40, 42, 47, 58, 61, 63, 76, 86, 87, 91, 92, 93, 102, 176, 188, 189, 191, 196, 198, 200, 173, 173, 252, 253, 254, 255, 262, 283, 285, 286, 292, 234, 243, 249, 250, 314, 315, 317, 318, 319, 324, 326, 328, 332, 334, 342, 343, 349, 354, 359, 360, 423, 426, 427, 428, 430, 434, 435, 436, 372, 373, 374, 379, 380, 383, 391, 399, 414, 415, 420, 422, 437, 438, 439, 441, 442, 447, 452, 454, 455, 456, 459, 461, 462, 463, 467, 469, 470, 471, 473, 483, 486, 487, 488, 49, 493, 495, 497, 499, 500, 501, 502, 503, 507, 508, 509, 510, 511}, and the support set T 28 = {12, , 28, 30, 42, 46, 58, 61, 62, 72, 86, 87, 88, 90, 91, 92, 93, 95, 100, 174, 175, 180, 184, 188, 190, 169, 169, 176, 179, 175, 231, 237, 242, 247, 248, 251, 252, 254, 260, 262, 268, 311, 312, 314, 315, 318, 326, 327, 332, 338, 340, 342, 343, 350, 351, 271, 276, 279, 282, 283, 286, 295, 299, 374, 377, 378, 379, 390, 391, 392, 404, 411, 414, 420, 422, 424, 428, 429, 431, 434, 475, 476, 482, 483, 484, 485, 475, 475, 475, 475, 48, 487, 489, 491, 492, 495, 500, 501, 506, 510, 511} and the support set T 29 = {12, 14, 15, 20, 23, 24, 26 , 27, 28, 30, 38, 39, 42, 46, 47, 52, 54, 58, 59, 60, 62, 70, 79, 84, 86, 87, 88, 90, 91, 92, 93, 95 , 102, 103, 118, 119, 120, 123, 124, 126, 127, 143, 151, 154, 155, 156, 157, 167, 170, 171, 173, 174, 180, 183, 186, 189, 191 , One 268, 270, 271, 276, 279, 272, 274, 272, 274, 328, 335, 336, 338, 340, 347, 350, 351, 354, 359, 316, 314, 315, 316, 318, 372, 374, 375, 377, 378, 379, 381, 383, 392, 396, 398, 400, 402, 404, 408, 411, 412, 415, 420, 442, 443, 444, 446, 447, 455, 458, 459, 460, 461, 466, 467, 469, 470, 473, 423, 426, 427, 429, 430, 434, 435, 437, 475, 477, 478, 482, 483, 485, 486, 489, 490, 491, 492, 493, 494, 495, 498, 499, 500, 501, 502, 503, 505} T30 = {8, 12, 14, 23, 26, 27, 31, 32, 38, 39, 40, 44, 46, 47, 52, 56, 59, 63, 64, 70, 78, 80 , 84, 88, 91, 93, 94, 95, 96, 102, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 123, 124, 125, 134, 143, 148 150, 151, 152, 154, 155, 156, 157, 159, 160, 164, 168, 170, 178, 183, 184, 189, 191, 196, 199, 200, 204, 206, , 216, 218, 222, 223, 224, 231, 237, 238, 239, 240, 244, 246, 248, 249, 254, 262, 272, 276, 278, 283, 285, 286, 294, 295, 296 , 306, 307, 310, 311, 315, 316, 318, 320, 324, 327, 330, 332, 334, 335, 338, 340, 346, 348, 349, 351, 354, 356, 358, 359, 362 , 363, 365, 367, 368, 370, 371, 373, 377, 378, 379, 383, 384, 390, 392, 394, 396, 398, 399, 400, 402, 404, 408, 411, 412, 413 , 414, 415, 416, 420, 423, 426, 427, 429, 432, 434, 435, 438, 440, 444, 445, 446, 447, 448, 454, 466, 468, 469, 471, 475, 477 , 478, 479, 482, 484, 486, 487, 490, 492, 493, 495, 497, 500, 501, 503, 504, 505, 506,

Wherein the outer encoder is configured to convert each bit value of the 30 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

Wherein the outer encoder is configured to output 32 parity check bits and 256 information bits,

Wherein the polarizer is configured to output 1,024 sign bits in response to the input of the 32 parity check bits, the 256 information bits, and 736 frozen bits,

The 736 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 , 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, , 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75 , 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106 , 107, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138 , 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 153, 154, 155, 157, 158, 159, 161, 162, 163, 164, 165, 166, 167 , 169, 170, 171, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 185, 186, 187, 189, 193, 194, 195, 196, , 202, 203, 204, 205, 206, 207, 209, 210, 211, 212, 213, 214, 215, 217, 218, 219, 221, 225, 226, 227, 228, 229, 230, 231, 233 , 234, 235, 237, 241, 242, 243, 245, 247, 249, 251, 2 272, 272, 277, 278, 279, 281, 282, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 273, 274, 275, 283, 285, 286, 287, 289, 290, 291, 292, 293, 294, 295, 297, 298, 299, 300, 301, 302, 303, 305, 306, 307, 308, 309, 310, 311, 333, 334, 335, 337, 338, 339, 340, 341, 342, 343, 323, 324, 325, 326, 327, 329, 365, 369, 370, 371, 373, 375, 377, 378, 379, 381, 385, 355, 356, 357, 358, 359, 361, 362, 363, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 401, 402, 403, 404, 405, 406, 407, 409, 441, 442, 443, 445, 449, 450, 451, 452, 423, 425, 426, 427, 429, 431, 433, 434, 435, 437, 439, 441, 483, 485, 486, 487, 489, 485, 485, 485, 485, 490, 491, 493, 497, 498, 499, 501, 505, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 529, 530, 5 545, 546, 547, 548, 549, 550, 551, 553, 554, 555, 556, 557, 558, 559, 532, 533, 534, 535, 537, 538, 539, 541, 542, 543, 562, 563, 564, 565, 566, 567, 569, 570, 571, 573, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 593, 594, 595, 596, 597, 598, 599, 601, 602, 603, 605, 609, 610, 611, 612, 613, 614, 615, 617, 618, 619, 621, 623, 625, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 657, 658, 646, 627, 629, 631, 633, 634, 635, 637, 641, 642, 689, 660, 661, 662, 663, 665, 666, 667, 669, 673, 674, 675, 676, 677, 678, 679, 681, 682, 683, 685, 687, 689, 690, 691, 693, 709, 710, 711, 713, 714, 715, 717, 719, 721, 722, 723, 725, 726, 727, 729, 730, 731, 733, 737, 738, 739, 741, 742, 743, 745, 746, 747, 749, 753, 754, 755, 757, 761, 765, 769, 770, 771, 772, 776, 777, 778, 779, 780, 781, 782, 783, 785, 786, 787, 788, 789, 790, 791, 793, 794, 795, 797, 799, 801, 802, 803, 8 818, 813, 813, 815, 817, 818, 819, 821, 822, 823, 825, 826, 827, 829, 833, 834, 835, 836, 837, 838, 839, 841, 842, 843, 845, 846, 847, 849, 850, 851, 853, 854, 855, 857, 858, 859, 861, 865, 866, 867, 868, 869, 870, 871, 873, 874, 907, 901, 906, 907, 909, 910, 911, 913, 914, 915, 916, 917, 918, 919, 921, 922, 923, 925, 929, 930, 931, 932, 933, 934, 935, 937, 938, 939, 941, 945, 946, 947, 949, 953, 955, 957, 961, 962, 963, 964, 965, 966, 967, 969, 970, 971, 973, 977, 978, 979, 981, 983, 985, 987, 989, 993, 994, 995, 997, 999, 1001, 1002, 1003, 1005, 1009, 1010, 1011, 1013, 1017}

The outer encoder outputs the 32 parity check bits as parity check bit index set P = {887, 495, 735, 959, 1023, 762, 1014, 366, 926, 62, 446, 484, 948, 908, 108, 796 , The parity check concatenated poles are configured to output at the bit indices defined by the parity check bits, Encoder.
18. The method of claim 17,

The support set T 1 = {507, 891, 1019, 1021} for the first parity check bits among the 32 parity check bits and the support set T 2 = {239, 367, 507, 751 , 759, 887, 943, 975, 1019, 1021} and the support set T 3 = {223, 367, 495, 507, 509, 759, 879, 887, 891, 943, 975, 1007, 1015, 1019}, and the support set T 4 = {191, 223, 239, 319, 367, 479, 503, 509, 575, 671, 703, 831, 863, 879, 887 , 951, 1007, 1015, 1021} and the support set T 5 = {127, 191, 255, 351, 367, 383, 447, 479, 503, 507, 509, 511, 639, The support set T 6 = {223, 250, 351, 367, 383, 415, 503, 509, 575 for the sixth parity check bit) , 639, 759, 763, 863, 890, 1021, 1023} and the support set T 7 = 127, 191, 223, 239, 246, 374, 383, 479, 495, 511, 607, 694, 735, And the support set T 8 = {127, 191, 223} for the eighth parity check bit is set to " 0, " , 250, 319, 351, 367, 383, 415, 430, 438, 447, 479, 502, 506, 509, 511, 575, 607, 630, 671, 718, 735, 762, 763, 767, 814, 831 , 886, 890, 891, 895, 927, 943, 982, 986, 998, 1007, 1018, 1019, 1021}, the support set T 9 = {127, 191, 223, 238, 708, 750, 758, 767, 798, 831, 879, 712, 612, 630, 639, 671, 694, 887, 891, 927, 942, 951, 954, 959, 982, 991, 998, 1007, 1014, 1019}, and the support set T 10 = 127, 191, 222, 223, 239 , 250, 255, 350, 351, 415, 430, 438, 462, 478, 479, 494, 495, 506, 509, 606, 607, 630, 686, 694, 703, 750, 751, 758, 762, 763 , 798, 831, 879, 890, 927, 950, 954, 959, 975, 982, 991, 1006, 1018}, the support set T 11 = {127, 630, 639, 670, 694, 702, 703, 718, 734, 735, 758, 762, 912, 951, 975, 990, 991, 998, 1006, 1007, 1021, 1023} for the 12th parity check bits, 12 = {62, 126, 127, 223, 239, 254, 319, 350, 374, 382, 430, 438, 462, 479, 495, 502, 506, 507, 575, 606, 638, 639, 670, 686 , 702, 703, 735, 740, 759, 762, 798, 814, 862, 878, 886, 890, 891, 926, 927, 942, 958, 959, 975, 986, 991, 998, 1015, 1019, 1022 }, The support set T 13 = {62, 94, 126, 222, 246, 250, 255, 318, 319, 366, 383, 430, 436, 446, 468, 479, 503, 508, 575, 622, 638, 639, 694, 702, 734, 740, 750, 751, 758, 759, 767, 798, 831, 863, 878, 879, 887, 890, 891, 927, 950, 954, And the support set T 14 = {62, 126, 127, 222, 246} for the 14th parity check bits , 318, 350, 351, 382, 38 3, 414, 415, 430, 462, 478, 494, 502, 506, 575, 628, 638, 670, 671, 686, 692, 694, 703, 718, 724, 735, 750, 751, 756, 758, 942, 943, 958, 959, 974, 982, 990, 991, 996, 998, 1006, 1014, 1021, 1022} The support set T 15 = {126, 222, 223, 239, 246, 319, 350, 366, 372, 382, 415, 447, 460, 494, 500, 502, 506, 507, 509 , 511, 574, 575, 606, 607, 622, 628, 630, 639, 670, 671, 686, 756, 758, 762, 763, 812, 820, 830, 831, 844, 862, 863, 878, 879 , 884, 890, 891, 908, 926, 943, 950, 951, 958, 972, 975, 982, 986, 990, 991, 1006, 1014, 1015, 1021, 1022} The support set T 16 = {62, 94, 172, 222, 238, 246, 250, 255, 284, 319, 364, 374, 382, 414, 415, 446, 460, 462, 479, 484, 495, 503, 506, 507, 511, 540, 574, 606, 607, 638, 639, 670, 684, 686, 694, 703, 716, 718, 724, 748, 756, 758, 759, 763, 766, 767, 820, 830, 831, 844, 878, 879, 886, 894, 908, 926, 927, 942, 948, 925, 959, 974, 980, 990, 991, 998, 1004, 1015, 1019, 1021, 1022, 1023} and the support set T 17 = {94, 108, 126, 172, 190 , 191, 222, 236, 238, 246, 254, 318, 319, 350, 351, 372, 382, 412, 430, 436, 446, 460, 462, 468, 478, 479, 484, 492, 506, 509, 510, 606, 628, 668, 670, 684, 692, 694, 703, 718, 735, 740, 750, 751, 756, 758, 763, 766, 796, 798, 812 , 820, 830, 831, 844, 863, 876, 879, 884, 891, 895, 908, 926, 927, 942, 943, 948, 951, 954, 958, 972, 980, 986, 991, 1004, 1006 , 1012, 1015, 1019, 1021, 1022, 1023} and the support set T 18 = {88, 94, 126, 127, 152, 168, 172, 190, 191, 200, 222, 238, 239, 244, 246, 252, 254, 255, 318, 319, 328, 344, 360, 366, 367, 374, 380, 436, 438, 446, 447, 460, 488, 494, 507, 536, 540, 552, 607, 616, 628, 636, 668, 670, 680, 684, 686, 692, 703, 712, 728, 734, 740, 750, 751, 759, 766, 767, 792, 796, 798, 808, 820, 828, 830, 831, 840, 844, 852, 863, 872, 87 984, 986, 991, 1000, 1012, 1014, 1015, 810, 910, 950, 951, 956, 958, 959, 972, 974, 975, 982, 984, 1018, 1022}, and the support set T 19 = {222, 246, 250, 622, 670, 694, 750, 798, 814, 890, 926, 954, 982, 1006, 1018} for the 19th parity check bits , A support set T 20 = {62, 94, 222, 246, 250, 414, 430, 438, 478, 502, 506, 622, 670, 686, 694, 734, 750, 758, And a support set T 21 = {255, 351, 415, 479, 507, 509, 511} for the 21st parity check bits , 625, 607, 671, 703, 763, 767, 887, 895, 951, 975, 991, 1007, 1015, 1023}, the support set T 22 = 314, 350, 366, 372, 374, 380, 382, 412, 414, 506, 507, 509, 511, 536, 540, 572, 574, 607, 616, 628, 634, 424, 428, 436, 444, 446, 447, 462, 492, 494, 832, 830, 860, 863, 732, 732, 732, 732, 764, 735, 744, 748, 758, 762, 766, 638, 639, 670, 692, 700, 703, 712, 716, 718, 922, 895, 904, 924, 940, 942, 948, 950, 958, 959, 968, 972, 990, 1012, 1014, 1015, 1022} The support set T 23 = 127, 191, 223, 239, 319, 367, 495, 503, 507, 511, 639, 671, 735, 767, 863, 879, 887, 951, 975, 991, 1007 , 1015} and the support set T 24 = {62, 88, 92, 108, 126, 127, 152, 168, 172, 191, 200, 220, 238, 244, 252, 328, 508, 507, 510, 511, 536, 540, 552, 532, 532, 532, 724, 732, 734, 735, 740, 758, 792, 808, 814, 820, 828, 830, 840, 844, 860, 872, 876, 878, 887, 890, 891, 894, 895, 904, 920, 940, 943, 950, 956, 958, 959, 975, 980, 984, 988, 991, 998, 1000, 1004, 1006, 1021, 1022}, the 25th parity check Support set for the bit T 25 = {60, 62, 94, 104, 108, 120, 124, 126, 127, 152, 156, 168, 188, 190, 191, 200, 216, 244, 246, 254, 284 , 296, 316, 319, 360, 364, 366, 374, 382, 414, 424, 428, 436, 438, 440, 444, 446, 456, 460, 462, 472, 476, 478, 484, 488, 492 , 495, 500, 503, 568, 572, 575, 604, 607, 616, 632, 638, 664, 668, 680, 692, 694, 703, 718, 724, 732, 740, 744, 748, 750, 758 , 759, 760, 763, 764, 766, 767, 808, 812, 820, 828, 831, 862, 863, 872, 876, 878, 884, 886, 887, 891, 892, 894, 895, 904, 920 , 927, 943, 950, 951, 952, 954, 956, 958, 968, 982, 986, 991, 998, 1012, 1022, 1023}, the support set T 26 = {144, 272}, the support set T 27 = {80} for the 27th parity check bits and the support set T 28 = {60, 62, 88, 104, 120, 168, 200, 222, 412, 415, 424, 428, 446, 447, 456, 460, 412, 412, 464, 472, 476, 478, 612, 628, 638, 656, 670, 671, 680, 694, 696, 700, 703, 712, 716, 720, 724, 734, 744, 748, 750, 759, 763, 764, 766, 767, 784, 792, 796, 798, 808, 812, 814, 820, 828, 831, 840, 844, 924, 924, 926, 927, 940, 951, 954, 956, 959, 968, 980, 982, 988, 991, 1000, 1004, 1006, 908, 890, 894, 100, 1018, 1019, 1021, 1022, 1023}, the support set T 29 = 210 for the 29th parity check bits, the support set T 30 = 60, 62, 92, 244, 246, 248, 250, 252, 254, 284, 296, 312, 336, 500, 502, 503, 504, 506, 507, 508, 507, 508, 508, 509, 511, 536, 540, 560, 568, 574, 604, 607, 622, 628, 636, 638, 639, 656, 668, 670, 671, 686, 688, 694, 696, 700, 702, 703, 712, 716, 720, 724, 734, 740, 750, 751, 758, 759, 796, 798, 880, 814, 816, 820, 830, 831, 840, 852, 860, 876, 878, 879, 880, 891, 895, 908, 920, 926, 927, 936, 940, 942, 948, 950, 952, 954, 958, 968, 972, 974, 980, 990, 1000, 1007, 1012, 1014, 1015, 1018, 1019, 1021, 1023}, and the support set for the first 31 parity check bits T 31 = 230, 248, 250, 272, 280, 228, 224, 232, 236, 239, 240, 248, , 284, 318, 328, 336, 348, 350, 351, 366, 383, 408, 412, 415, 428, 444, 447, 460, 472, 476, 479, 480, 502, 503, 504, 507, 509 , 560, 568, 575, 607, 616, 620, 630, 632, 636, 664, 670, 671, 680, 684, 686, 692, 703, 716, 724, 728, 732, 734, 735, 736, 744 , 748, 750, 752, 756, 762, 763, 764, 784, 792, 796, 800, 812, 816, 820, 830, 840, 844, 848, 852, 878, 884, 887, 895, 904, 912 920, 924, 926, 927, 936, 942, 944, 950, 952, 958, 974, 975, 984, 986, 988, 991, 1004, 1006, 1007, 1008, 1012, 1016, 1020, 1021, 1022 }, The support set T 32 = 32, 60 for the 32nd parity check bits, 320, 328, 344, 338, 334, 336, 338, 334, 336, 508, 503, 504, 508, 507, 508, 508, 508, 508, 508, 508, 509, 510, 511, 536, 540, 544, 568, 572, 574, 576, 592, 604, 620, 630, 636, 638, 672, 688, 692, 694, 696, 700, 702, 703, 712, 716, 718, 720, 724, 728, 736, 740, 744, 751, 756, 759, 762, 763, 764, 766, 767, 792, 798, 800, 812, 820, 831, 840, 860, 972, 975, 992, 924, 926, 942, 948, 950, 952, 954, 956, 958, 960, 972, 975, 992, 996, 1000, 1007, 1023}

Wherein the outer encoder codes each bit value of the 32 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

The outer encoder is configured to output 26 parity check bits and 341 information bits,

Wherein the polarizer is configured to output 1,024 sign bits in response to the 26 parity check bits, the 341 information bits, and the 657 frozen bits,

The 657 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, , 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79 , 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114 , 115, 117, 119, 121, 122, 123, 125, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, , 148, 149, 150, 151, 153, 154, 155, 157, 159, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 173, 175, 177, 178, 179, 181 , 183, 185, 186, 187, 189, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 207, 209, 210, 211, 213, 214, 215, 217, 218 , 219, 221, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 241, 242, 243, 245, 249, 253, 257, 258, 259, 260, , 263, 264, 265, 266, 267, 268, 269, 270, 271, 2 292, 293, 294, 295, 297, 298, 299, 301, 303, 305, 287, 282, 283, 285, 287, 289, 290, 291, 292, 293, 334, 335, 337, 338, 339, 332, 333, 334, 335, 336, 361, 362, 363, 365, 369, 370, 371, 373, 377, 381, 385, 352, 343, 345, 346, 347, 349, 353, 354, 355, 356, 357, 386, 387, 388, 389, 390, 391, 393, 394, 395, 397, 398, 399, 401, 402, 403, 404, 405, 406, 407, 409, 410, 411, 413, 417, 418, 449, 450, 451, 452, 453, 454, 455, 457, 458, 431, 422, 423, 425, 426, 427, 429, 433, 434, 435, 437, 441, 443, 491, 493, 497, 498, 499, 501, 505, 513, 471, 471, 473, 475, 477, 481, 482, 483, 485, 487, 489, 490, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 529, 530, 531, 532, 533, 534, 535, 537, 538, 543, 545, 546, 547, 548, 549, 550, 551, 553, 554, 555, 557, 559, 561, 562, 563, 565, 566, 567, 569, 570, 571, 573, 577, 578, 5 595, 597, 598, 599, 601, 602, 603, 605, 609, 610, 611, 612, 591, 592, 583, 585, 586, 587, 589, 613, 614, 615, 617, 618, 619, 621, 625, 626, 627, 629, 633, 635, 637, 641, 642, 643, 644, 645, 646, 647, 649, 650, 651, 653, 683, 685, 689, 660, 661, 662, 663, 665, 666, 667, 669, 673, 674, 675, 676, 677, 678, 679, 681, 682, 683, 690, 691, 693, 697, 699, 701, 705, 706, 707, 708, 709, 710, 711, 713, 714, 715, 717, 721, 722, 723, 725, 727, 729, 731, 733, 737, 738, 739, 741, 743, 745, 746, 747, 749, 753, 754, 755, 757, 761, 769, 770, 771, 772, 773, 774, 775, 777, 778, 781, 782, 783, 785, 786, 787, 788, 789, 790, 791, 793, 794, 795, 797, 801, 802, 803, 804, 805, 806, 807, 809, 810, 811, 813, 837, 838, 839, 841, 842, 843, 845, 849, 850, 851, 853, 855, 857, 821, 823, 825, 827, 829, 833, 834, 835, 836, 858, 859, 861, 865, 866, 867, 869, 871, 873, 874, 875, 877, 881, 882, 883, 885, 889, 897, 898, 899, 900, 901, 902, 903, 905, 9 939, 913, 913, 914, 921, 922, 923, 925, 929, 930, 931, 933, 934, 935, 937, 938, 939, 941, 945, 946, 977, 977, 978, 979, 981, 985, 989, 993, 994, 995, 997, 1001, 1005, 967, 967, 1009, 1011, 1013, 1017} to the polar encoders,

The outer encoder outputs the 26 parity check bits as parity check bit index sets P = {695, 847, 927, 63, 474, 742, 630, 302, 366, 990, 254, 852, 884, 556, 1004, 476 , 636, 200, 40, 24, 120, 16, 48, 32, 96, 192}.
20. The method of claim 19,

The second set of parity check bits is a set of the first parity check bits T 1 = {251, 379, 507, 509, 765, 893, 955, 957, 999, 1003, for supporting a set T 2 = a {379, 631, 695, 763, 765, 891, 951, 955, 983, 987, 999, 1019, 1021}, a third support set for the parity check bit T 3 = {247, 913, 1015, 1019, 1021}, and the fourth parity check is performed on the basis of the first parity check value < RTI ID = 0.0 > support set for the bit T 4 = {223, 247, 375, 379, 415, 495, 503, 507, 687, 695, 759, 763, 799, 847, 879, 893, 943, 951, 955, 983, 991 , 999, 1007, 1015, 1019, 1021}, and the support set T 5 = {63, 247, 255, 319, 351, 383, 415, 431, 439, 447, 503, 507, 509, 575, 631, 639, 671, 687, 719, 735, 759, 763, 765, 767, 887, 893, 895, 927, 943, 951, 955, 983, 987, 991, 1003, 1007, 1015, 1019, 1021, 1023}, and the support for the sixth parity check bit Set T 6 = {63, 95, 191, 247, 250, 251, 375, 379, 431, 439, 447, 463, 495, 506, 511, 575, 631, 634, 639, 698, 703, 719, 730 , 735, and 751, 759, 762, 815, 863, 887, 891, 893, 943, 954, 957, 975, 987, 999, 1007, 1010, 1018}, a seventh supporting a set of parity check bits T 7 = 63, 182, 191, 319, 351, 375, 378, 438, 439, 447, 470, 479, 486, 507, 623, 634, 639, 703, 735, 751, 765, 767, 799, 847, 830, 879, 887, 891, 893, 943, 959, 983, 986, 987, 991, 999, 1002, 1007} and the support set T 8 = 118, 239, 247 , 250, 251, 255, 319, 379, 439, 442, 447, 462, 463, 486, 495, 502, 503, 506, 507, 558, 623, 630, 631, 671, 694, 703, 718, 719 , 730, 735, 742, 751, 758, 763, 765, 767, 799, 815, 847, 870, 886, 918, 927, 943, 955, 959, 975, 982, 987, 998, 999, }, And the support set T 9 = {127, 206, 223, 319, 351, 367, 375, 378, 383, 430, 431, 439, 447, 470, 495, 502, 507, 511, 622, 623, 634, 639, 671, 686, 877, 890, 891, 895, 918, 950, 954, 957, 959, 810, 822, 831, 846, 925, 982, 983, 991, 999, 1002, 1010, 1014, 1019, 1023} and the support set T 10 = {63, 118, 127, 191, 222, 238, 239, , 251, 351, 374, 383, 415, 430, 439, 442, 447, 462, 463, 470, 474, 478, 486, 494, 495, 502, 503, 506, 507, 509, 558, 575, 606 , 607, 623, 686, 695, 698, 703, 718, 719, 726, 734, 735, 742, 762, 765, 767, 799, 814, 831, 846, 847, 854, 863, 879, 886, 895 , 910, 918, 950, 954, 955, 957, 974, 986, 987, 1002, 1003, 1018, 1021} and the support set T 11 = {63, 94, 95, 110, 462, 463, 470, 486, 495, 503, and 440, respectively, 758, 762, 763, 765, 767, 799, 814, 822, 826, 830, 854, 862, 863, 879, 886, 890, 891, 894, 895, 927, 942 , 943, 950, 951, 957, 974, 982, 983, 991, 998, 999, 1002, 1003, 1007, 1010, 1014, 1018}, the support set T 12 = {63, 434, 414, 418, 438, 439, 442, 463, 424, 428, 474, 486, 494, 502, 503, 507, 509, 511, 542, 574, 575, 596, 606, 607, 623, 630, 634, 694, 698, 702, 703, 718, 730, 734, 742, 878, 879, 886, 887, 890, 895, 918, 943, 951, 954, 957, 974, 987, And the support set T 13 = {94, 110, 116, 158, 174, 190, 222, 223, 239, 246 for the thirteenth parity check bits , 255, 286, 308, 351, 367, 372, 374, 379, 383, 414, 415, 431, 436, 468, 470, 479, 486, 494, 502, 506, 558, 564, 596, 606, 623 , 634, 638, 686, 687, 692, 698, 702, 703, 718, 724, 734, 740, 742, 759, 763, 766, 767, 815, 822, 826, 830, 831, 846, 852, 854 , 863, 868, 878, 887, 890, 891, 894, 895, 910, 918, 942, 950, 951, 954, 955, 982, 983, 986, 987, 990, 996, 998, 1002, 1003, 1006, 1010, 1014, 1021}, the support set T 14 = {62 , 63, 116, 127, 180, 182, 190, 191, 206, 222, 223, 239, 244, 250, 251, 254, 255, 286, 308, 318, 332, 340, 350, 366, 375, 378 , 379, 382, 383, 415, 431, 438, 446, 462, 479, 486, 494, 495, 500, 503, 506, 509, 510, 542, 558, 575, 588, 606, 607, 622, 628 , 638, 671, 686, 687, 702, 703, 716, 719, 724, 726, 734, 735, 740, 742, 751, 756, 759, 763, 765, 766, 767, 798, 799, 814, 815 , 822, 826, 830, 847, 854, 862, 863, 870, 884, 890, 893, 908, 916, 918, 932, 942, 943, 950, 951, 955, 957, 964, 974, 983, 991 , 996, 1006, 1012, 1014, 1015, 1022} and the support set T 15 = {62, 95, 110, 116, 126, 172, 180, 182, 206, 212, 222, 443, 442, 446, 447, 460, 462, 374, 372, 371, 362, 372, 375, 378, 379, 382, 396, 414, 430, 431, 468, 470, 474, 478, 479, 484, 486, 495, 502, 503, 506, 5 730, 730, 730, 730, 730, 730, 730, 730, 730, 732, 845, 822, 826, 831, 846, 862, 868, 879, 884, 886, 893, 895, 908, 910, 918, 932, 942, 943, 958, 959, 964, 972, 983, 986, 987, 996, 999, 1003, 1006, 1010, 1014, 1015} The set T 16 = {158,180,182,901,191,206,202,223,236,239,246,250,251,255,286,302,31,332,348,503,351,374,382 , 383, 396, 412, 415, 430, 431, 460, 462, 468, 470, 474, 478, 492, 503, 507, 509, 540, 556, 558, 564, 574, 588, 607, 622, 623 , 628, 630, 639, 652, 668, 670, 671, 684, 686, 687, 694, 695, 698, 716, 719, 726, 730, 735, 748, 751, 756, 758, 762, 763, 765 , 766, 796, 798, 799, 814, 815, 826, 830, 844, 846, 847, 852, 854, 862, 863, 868, 870, 879, 887, 891, 893, 910, 916, 918, 927 , 932, 940, 943, 950, 957, 975, 980, 982, 983, 986, 987, 990, 991, 996, 998, 1002, 1006, 1 The support set T 17 = 94, 95, 110, 124, 156, 172, 188, 204, 220 for the 17th parity check bits , 236, 238, 239, 244, 246, 250, 286, 302, 318, 332, 340, 350, 351, 364, 366, 367, 374, 378, 382, 396, 414, 415, 428, 438, 439 , 442, 444, 446, 447, 463, 468, 470, 474, 478, 495, 507, 510, 511, 542, 558, 574, 588, 604, 606, 607, 620, 622, 623, 628, 630 , 634, 638, 639, 668, 671, 684, 686, 687, 694, 695, 700, 702, 716, 718, 719, 724, 726, 734, 735, 740, 751, 756, 759, 762, 765 , 767, 796, 799, 820, 822, 826, 828, 844, 846, 847, 854, 870, 879, 886, 890, 891, 893, 894, 895, 908, 910, 916, 918, 924, 926 , 940, 943, 950, 951, 955, 956, 957, 964, 975, 980, 988, 991, 996, 998, 999, 1003, 1004, 1007, 1015, 1023} The support set T 18 = 108, 118, 126, 127, 156, 158, 174, 180, 191, 238, 239, 244, 246, 247, 250, 286, 300, 316, 319, 340, 348, 350, 351, 367, 372, 375, 378, 382, 392, 396, 412, 430, 438 , 444, 447, 460, 463, 468, 470, 474, 476, 478, 486, 492, 494, 502, 508, 509, 510, 542, 572, 574, 575, 584, 588, 604, 622, 636 , 638, 639, 648, 670, 671, 684, 687, 692, 698, 700, 702, 719, 726, 730, 732, 734, 735, 740, 742, 750, 751, 756, 762, 763, 764 , 766, 767, 776, 799, 812, 822, 828, 831, 840, 862, 863, 876, 878, 890, 892, 893, 894, 895, 904, 910, 924, 927, 940, 942, 948 , 955, 957, 958, 964, 972, 974, 980, 982, 983, 990, 999, 1003, 1004, 1007, 1010, 1015, 1019, 1022}, and the support set for the first 19 parity check bits T 19 = 94, 110, 118, 127, 206, 238, 239, 250, 251, 319, 351, 374, 379, 431, 438, 439, 442, 446, 463, 470, 478, 494, 506, 542, 708, 708, 703, 718, 719, 734, 751, 758, 759, 762, 763, 767, 826, 831, 854, 863, 870, 878, And the support set T 20 for the twentieth parity check bit is T 40 = {40, 48, 60, 601, 601, 601, 601, , 62, 80, 92, 104, 108, 116, 124, 126, 127, 144, 172, 180, 282, 296, 300, 308, 312, 318, 364, 366, 367, 375, 382, 506, 507, 508, 509, 510, 542, 552, 556, 564, 486, 492, 494, 495, 640, 612, 623, 628, 634, 636, 638, 639, 648, 652, 668, 670, 671, 686, 687, 695, 698, 700, 712, 716, 719, 724, 766, 767, 776, 798, 799, 812, 822, 828, 831, 840, 844, 863, 732, 732, 735, 740, 742, 744, 748, 751, 756, 759, 762, 763, 870, 876, 879, 884, 886, 890, 892, 904, 924, 932, 936, 940, 942, 950, 951, 982, 986, 988, 991, 996, 998, 1000, 1002, 1004, 1007, 1012, 1015, 1023}, and the support set T 21 = {24, 56, 62, 94, 95, 110, 118, 152, 156, 158, 168, 172, 174, 180, 188 , 191, 206, 222, 232, 247, 254, 255, 280, 296, 312, 328, 340, 348, 350, 360, 366, 367, 374, 375, 378, 379, 380, 392, 412, 415 , 424, 430, 431, 438, 439, 442, 444, 460, 462, 463, 470, 472, 476, 479, 486, 494, 500, 502, 506, 507, 510, 540, 542, 552, 572, 584, 588, 600, 607, 620, 622, 623, 628, 652, 670, 671, 694, 696, 698, 703, 712, 732, 735, 740, 742, 744, 748, 756, 758, 763, 764, 765, 766, 776, 792, 796, 798, 812, 814, 820, 824, 826, 828, 840, 846, 854, 972, 974, 975, 980, 927, 940, 942, 948, 951, 954, 957, 958, 964, 972, 974, 887, 890, 893, 894, 908, 920, 98, 990, 996, 998, 999, 1003, 1012, 1015, 1020, 1022, 1023} and the support set T 22 = {40, 56, 80, 88, 92, 94, 96 , 118, 120, 126, 127, 152, 158, 168, 172, 180, 182, 188, 190, 191, 204, 206, 216, 220, 222, 232, 239, 247, 248, 250, 251, 254 , 280, 300, 302, 312, 316, 318, 328, 350, 351, 360, 364, 366, 380, 396, 408, 424, 428, 431, 440, 446, 463, 468, 470, 479, 486 508, 509, 510, 511, 536, 540, 552, 556, 564, 574, 588, 596, 604, 607, 616, 623, 628, 630, 631 , 632, 648, 671, 680, 684, 687, 692, 696, 712, 718, 732, 751, 756, 759, 760, 776, 798, 799, 812 820, 822, 824, 828, 830, 831, 840, 854, 860, 862, 863, 868, 870, 879, 894, 904, 916, 918, 924, 932, 940, 943, 951, 954, 956 , 958, 964, 974, 980, 990, 999, 1000, 1004, 1006, 1010, 1012, 1014, 1016, 1020, 1022, 1023}, the support set T 23 = {60, 80, 144, 216, 312, 872}, the support set T 24 = {16, 48, 120, 664} for the 24th parity check bits, the support set T 25 = 80, 600}, the support set T 26 = {32} for the 26th parity check bits,

Wherein the outer encoder codes each bit value of the 26 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

The outer encoder is configured to output 26 parity check bits and 512 information bits,

Wherein the polarizer is configured to output 1,024 sign bits in response to the input of the 26 parity check bits, the 512 information bits, and the 486 frozen bits,

The 486 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20 , 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 49, 50, 51, 53, 55, 57, 58 , 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89, 90, , 99, 101, 102, 103, 105, 106, 107, 109, 113, 114, 115, 117, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, , 145, 146, 147, 149, 151, 153, 154, 155, 157, 161, 162, 163, 165, 166, 167, 169, 170, 171, 173, 177, 178, 179, 181, 185, 189 , 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 205, 209, 210, 211, 213, 217, 221, 225, 226, 227, 229, 233, 235, 237, 241 , 243, 245, 249, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 269, 271, 273, 274, 275, 277, 278, 279, 281, , 289, 290, 291, 293, 294, 295, 297, 298, 299, 301, 305, 306, 307, 309, 313, 317, 321, 322, 323, 324, 325, 326, , 331, 333, 337, 338, 339, 341, 345 , 349, 353, 354, 355, 357, 361, 363, 365, 369, 371, 373, 377, 385, 386, 387, 388, 389, 390, 391, 393, 394, 395, , 403, 405, 409, 411, 413, 417, 418, 419, 421, 423, 425, 427, 429, 433, 434, 435, 437, 441, 449, 450, 451, 453, 455, 457, 458 519, 517, 518, 519, 521, 522, 523, 461, 465, 466, 467, 469, 473, 481, 482, 483, 485, 489, 497, 501, 505, 513, , 525, 526, 527, 529, 530, 531, 533, 534, 535, 537, 538, 539, 541, 545, 546, 547, 548, 549, 550, 551, 553, 554, 555, 557, 561 , 562, 563, 565, 569, 573, 577, 578, 579, 580, 581, 582, 583, 585, 586, 587, 589, 593, 594, 595, 597, 601, 603, 605, 609, 610 , 611, 613, 617, 619, 621, 625, 626, 627, 629, 633, 641, 642, 643, 644, 645, 646, 647, 649, 650, 651, 653, 657, 658, , 665, 667, 669, 673, 674, 675, 677, 679, 681, 683, 685, 689, 690, 691, 693, 697, 705, 706, 707, 709, 711, 713, 714, 715, 717 , 721, 722, 723, 725, 729, 737, 738, 739, 741, 745, 753, 757, 761, 769, 770, 771, 772, 773, 774, 775, 777, 778, 779, 781, 785 , 786, 787, 789, 791, 793, 795, 797, 801, 802, 803, 805, 807, 809, 810, 811, 813, 817, 818, 819, 821, 825, 833, 834, 835, 837 , 839, 841, 842, 843, 845, 849, 850, 851, 853, 857, 865, 866, 867, 869, 873, 877, 881, 885, 889, 897, 898, 899, 901, 902, 903 , 905, 906, 907, 909, 913, 914, 915, 917, 921, 929, 930, 931, 933, 937, 941, 945, 947, 949, 953, 961, 962, 963, 965, , 973, 977, 979, 981, 985, 993, 995, 997, 1001, 1009}

The outer encoder sets the 26 parity check bits to a parity check bit index set P = {987, 871, 471, 207, 543, 191, 426, 858, 678, 214, 782, 46, 30, 638, , 76, 44, 28, 904, 40, 24, 16, 80, 32, 256}.
22. The method of claim 21,

And the second set of parity check bits for the first parity check bit is a set of T 1 = {219, 253, 445, 475, 491, 493, 509, 859, 861, 923, 957, 1011, 1021} The support set T 2 = {123, 125, 219, 253, 381, 443, 507, 571, 615, 635, 699, 701, 731, 733, 747, 763, 861, 883, 893, 925 , 935, 939, 967, 989, 1011, 1013}, and the support set for the three parity check bits T 3 = {123, 125, 187, 219, 231, 251, 253, 315, 347, 359, 477, 499, 571, 615, 635, 663, 699, 743, 749, 755, 763, 829, 855, 859, 891, 919, 923, 939, 987, 989, 1003, 1005, 1011, 1017, 1021} The support set T 4 = {119, 123, 125, 187, 215, 231, 247, 399, 407, 443, 445, 475, 477, 507, 509, 591, 615, 635, 637 for the fourth parity- , 695, 727, 731, 749, 763, 783, 827, 859, 871, 875, 883, 893, 923, 925, 935, 939, 951, 955, 957, 967, 987, 1005, 1015, , The support set T 5 = {111, 125, 215, 231, 247, 251, 253, 3 for the fifth parity check bits 509, 567, 571, 599, 615, 623, 655, 663, 687, 731, 749, 751, 759, 913, 935, 939, 943, 951, 955, 957, 975, 987, 1007, 1013, 1015, 1017, 1021} The support set T 6 = {111, 159, 175, 207, 219, 223, 231, 247, 253, 287, 311, 319, 343, 347, 367, 381, 407, 415, 431, 439 , 445, 471, 475, 477, 479, 487, 491, 495, 503, 559, 571, 591, 599, 615, 631, 655, 663, 671, 695, 719, 727, 733, 735, 743, 763 , 827, 829, 831, 847, 871, 875, 879, 911, 923, 927, 935, 939, 943, 951, 957, 967, 975, 983, 987, 991, 1015} not set to T 7 = {31, 95, 125, 183, 191, 207, 223, 247, 251, 253, 255, 287, 303, 319, 335, 347, 351, 375, 381, 383, 407, 415, 431, 439, 443, 445, 463, 471, 479, 493, 499, 503, 507, 509, 559, 571, 599, 615, 631, 635, 639, 671, 695, 701, 703, 727, 733, 735, 747, 749, 751, 763, 823, 829, 831, 847, 855, 861, 863, 979, 882, 887, 911, 923, 925, 927, 939, 946, 955, 957, 959, 967, 975, 983, 987, 994, 999, 1010, 1013, 1015, 1019, 1021, 1023} The support set T 8 = {123, 125, 183, 187, 191, 207, 242, 251, 253, 287, 303, 319, 335, 343, 346, 347, 359, 362, 370 for the eighth parity- , 375, 381, 399, 410, 415, 426, 431, 439, 443, 447, 463, 475, 490, 493, 499, 511, 543, 559, 607, 615, 635, 639, 655, 666, 671 , 682, 703, 719, 731, 733, 743, 746, 749, 754, 759, 767, 794, 823, 829, 831, 847, 855, 859, 861, 875, 879, 882, 887, 893, 919 , 923, 927, 935, 946, 955, 957, 959, 967, 975, 989, 991, 999, 1003, 1005, 1010, 1015, 1017, 1019, 1023} T 9 = {63, 111, 122, 123, 159, 183, 186, 187, 191, 215, 218, 223, 231, 234, 239, 242, 253, 255, 287, 303, 319, 335, 343, 347, 475, 493, 507, 509, 511, 602, 615, 631, 663, 671, 698, 703, 727, 730, 735, 743, 749, 751, 759, 762, 794, 806, 8 922, 923, 925, 938, 939, 946, 955, 966, 970, 991, 994, 999, 100, 1003, 1005, 1007, 1011, 1023}, and the support set T 10 = {31, 63, 86, 119, 122, 123, 125, 127, 150, 159, 183, 186 , 207, 215, 223, 231, 234, 247, 250, 251, 253, 311, 315, 319, 346, 351, 359, 367, 375, 378, 379, 381, 383, 406, 415, 422, 442 , 443, 445, 447, 471, 477, 479, 487, 490, 491, 499, 503, 506, 543, 559, 570, 575, 614, 615, 618, 623, 635, 662, 671, 687, 695 , 699, 703, 710, 719, 727, 735, 742, 746, 747, 749, 754, 755, 759, 763, 767, 783, 790, 794, 799, 806, 855, 858, 871, , 879, 882, 893, 919, 922, 923, 925, 934, 946, 955, 957, 959, 967, 986, 987, 989, 998, 999, 1007, 1011, 1013, 1015, 1017, 1019, 1021 , 1023} and the support set T 11 = {54, 63, 86, 95, 111, 118, 123, 125, 159, 182, 183, 186, 187, 214, 242, 246, 247, 251, 253, 287, 310, 314, 315, 335, 342, 343, 346, 351 , 362, 370, 375, 379, 399, 407, 410, 415, 422, 431, 439, 445, 447, 477, 479, 486, 491, 493, 495, 498, 499, 503, 509, 511, 543 , 566, 567, 598, 602, 615, 618, 635, 637, 639, 655, 662, 663, 666, 671, 678, 694, 695, 699, 701, 703, 710, 727, 730, 733, 735 , 742, 743, 746, 747, 755, 758, 759, 783, 794, 806, 822, 823, 826, 838, 854, 870, 879, 882, 886, 891, 911, 919, 923, 925, 927 934, 935, 938, 943, 946, 950, 951, 954, 959, 966, 967, 982, 983, 986, 987, 991, 994, 998, 999, 1002, 1005, 1007, 1010, 1013, 1014 , 1015, 1017, 1018, 1021, 1023} and the support set T 12 = {54, 63, 78, 86, 95, 111, 118, 119, 123, 125, 127, 142, 371, 375, 378, 381, 383, 398, 399, 370, 370, 370, 507, 511, 566, 570, 571, 591, 598, 599, 490, 495, 499, 602, 614, 615, 623, 630, 631, 634, 635, 637, 639, 666, 678, 687, 695, 703, 710, 718, 719, 735, 747, 749, 762, 765, 767, 783, 799, 806, 815, 822, 847, 854, 855, 863, 879, 886, 887, 890, 895, 918, 923, 925, 939, 951, 954, 955, 957, 966, 967, 975, 991, 999, 1003, 1005, 1011, 1013, 1014, 1019, 1021, 1023}, and the support set for the first 13 parity check bits T 13 = {302} and the support set T 14 = {31, 54, 94, 95, 119, 127, 174, 182, 183, 186, 187, 190, 206, 207, 218, 219, 318, 314, 319, 334, 335, 347, 350, 362, 367, 381, 415, 430, 431, 498, 498, 499, 502, 506, 507, 511, 542, 543, 558, 559, 566, 575, 590, 591, 606, 438, 439, 443, 446, 470, 478, 479, 490, 754, 758, 759, 763, 765, 742, 623, 631, 634, 654, 655, 663, 670, 671, 682, 694, 710, 727, 731, 734, 735, 743, 746, 862, 863, 874, 883, 886, 887, 893, 918, 925, 934, 832, 824, 955, 957, 959, 966, 967, 970, 975, 982, 983, 987, 994, 999, 1002, 1005, 1006, 1011, 1015, 1017, 1019, 1021}, and claim 15, the support set of the parity check bits T 15 = {47, 63, 94, 95, 100, 118, 123, 142, 150, 158 , 164, 174, 182, 183, 187, 190, 207, 214, 238, 242, 246, 250, 251, 254, 255, 286, 287, 292, 303, 311, 314, 318, 334, 342, 343 , 359, 362, 382, 399, 406, 407, 410, 420, 422, 431, 438, 439, 443, 447, 452, 462, 463, 474, 477, 478, 486, 490, , 503, 506, 509, 543, 567, 570, 575, 590, 599, 602, 612, 614, 615, 618, 623, 630, 639, 662, 670, 676, 678, 682, 686, 687, 698 , 702, 703, 718, 719, 740, 743, 746, 747, 751, 754, 755, 762, 763, 765, 766, 767, 790, 794, 799, 822, 827, 836, 846, 854, 855 , 858, 859, 861, 862, 874, 875, 878, 879, 882, 886, 890, 895, 911, 919, 922, 934, 938, 939, 942, 950, 955, 958, 970, 975, 978 , 982, 986, 987, 989, 990, 991, 994, 999, 1006, 1010, 1011, 1014, 1015} and the support set T 16 = {31, 47, 62, 94, 111, 118, 119, 127, 142, 148, 159, 186, 187, 191, 206, 215 , 218, 219, 230, 239, 247, 250, 251, 286, 292, 303, 310, 311, 314, 318, 340, 342, 343, 347, 350, 351, 358, 362, 366, 367, 374 , 378, 379, 381, 399, 404, 406, 407, 410, 414, 415, 420, 430, 431, 438, 442, 446, 447, 462, 463, 474, 486, 491, 494, 495, 498 , 499, 510, 511, 558, 559, 566, 571, 574, 575, 591, 602, 606, 612, 614, 615, 618, 623, 630, 631, 634, 635, 638, 639, 655, 660 , 662, 663, 666, 670, 678, 682, 686, 687, 694, 702, 726, 734, 735, 740, 742, 746, 749, 750, 754, 762, 766, 767, 783, 845, 823, 830, 831, 838, 846, 852, 855, 859, 861, 868, 870, 874, 878, 879, 883, 887, 890, 891, 893, 910, 916, 918, 922, 923, 925, 926, 927, 932, 934, 938, 939, 950, 951, 957, 958, 959, 964, 967, 974, 978, 980, 983, 987, 989 , 990, 996, 998, 1002, 1003, 1006, 1015, 1019, 1022} and the support set T 17 = {30, 31, 52, 63, 84, 100, 110, 111, 116, 119, 123, 125, 126, 140, 142, 148, 159, 175, 183, 191, 207, 214, 218, 222, 230, 234, 238, 366, 372, 374, 379, 383, 399, 346, 347, 351, 362, 366, 372, 440, 440, 440, 440, 440, 440, 440, 440, 440, 440, 440, 440, 440, 426, 438, 439, 442, 446, 462, 474, 475, 478, 484, 486, 491, 493, 499, 502, 507, 509, 524, 542, 558, 564, 566, 567, 571, 590, 591, 607, 614, 618, 623, 634, 635, 637, 638, 639, 655, 662, 670, 678, 682, 687, 692, 699, 702, 708, 710, 719, 836, 838, 852, 854, 826, 824, 830, 831, 836, 838, 852, 854, 935, 932, 934, 935, 938, 939, 943, 933, 935, 935, 950, 951, 954, 957, 958, 970, 986, 987, 989, 990, 991, 998, 999, 1005, 1010, 1011, 1012, 1014, 1018, 1023} T 18 = {30, 47, 52, 62, 76, 84, 86, 100, 110, 111, 116, 118, 119, 123, 125, 127, 158, 175, 182, 183, 190, 191, 204 , 206, 207, 215, 218, 230, 234, 239, 246, 250, 251, 255, 268, 286, 310, 311, 332, 343, 350, 3 472, 474, 475, 479, 486, 491, 493, 434, 432, 424, 640, 652, 654, 655, 671, 676, 682, 687, 612, 624, 752, 754, 763, 767, 782, 804, 806, 815, 829, 836, 844, 847, 852, 732, 799, 726, 730, 733, 734, 742, 747, 749, 751, 755, 756, 924, 923, 925, 926, 934, 942, 946, 924, 925, 926, 948, 951, 954, 966, 970, 972, 974, 975, 978, 980, 982, 986, 989, 991, 994, 999, 1003, 1006, 1010, 1011, 1012, 1014, 1015, And a support set T 19 = {31, 46, 52, 54, 95, 108, 110, 111, 122, 123, 125, 127, 158, 164, 174, 175, 180, 182 , 183, 186, 187, 190, 204, 206, 207, 212, 214, 215, 218, 219, 223, 231, 234, 236, 238, 239, 242, 244, 246, 247, , 292, 300, 302, 303, 310, 311, 314, 318, 319, 332, 334, 335, 340, 347, 351, 358, 362, 364, 370, 372, 375, 378, 379, 382, 383, 404, 406, 407, 410, 414, 415, 420, 426, 431, 439, 442, 445, 452, 468, 470, 475, 477, 479, 494, 495, 498, 499, 500, 502, 503, 507, 511, 524, 564, 571, 574, 575, 590, 591, 596, 598, 602, 607, 612, 620, 623, 631, 635, 655, 660, 662, 663, 670, 671, 676, 687, 692, 695, 699, 701, 708, 716, 719, 724, 726, 727, 733, 734, 746, 747, 748, 749, 751, 758, 759, 762, 765, 766, 767, 780, 790, 794, 798, 804, 814, 829, 831, 836, 854, 858, 861, 862, 868, 871, 922, 923, 926, 927, 934, 935, 939, 940, 942, 943, 946, 948, 951, 954, 958, 964, 966, The support set T 20 for the twentieth parity check bit is set to T 20 = 10, 10, 10, 175, 180, 186, 188, 190 < RTI ID = 0.0 > , 191, 204, 206, 212, 214, 215, 218, 220, 223, 230, 231, 234, 236, 238, 244, 247, 250, 254 , 255, 270, 292, 302, 308, 311, 314, 315, 334, 335, 340, 342, 348, 351, 356, 358, 362, 367, 370, 372, 374, 380, 381, , 398, 406, 407, 410, 412, 414, 415, 420, 426, 431, 436, 438, 439, 443, 445, 447, 462, 463, 470, 475, 476, 477, 479, 484, 486 , 491, 493, 494, 495, 498, 499, 502, 503, 511, 520, 540, 543, 558, 566, 567, 574, 590, 591, 598, 599, 607, 615, 620, 622, 628 630, 631, 635, 636, 637, 638, 639, 652, 660, 662, 670, 676, 682, 686, 687, 694, 695, 699, 701, 716, 719, 724, 726, 727, 731 , 732, 734, 743, 747, 748, 749, 750, 751, 754, 755, 759, 766, 776, 782, 783, 788, 794, 799, 804, 806, 814, 815, 820, , 844, 846, 847, 852, 854, 860, 861, 862, 870, 874, 884, 890, 894, 895, 919, 922, 923, 924, 927, 932, 934, 935, 938, 939, 940 , 942, 943, 948, 950, 956, 958, 959, 964, 966, 967, 970, 972, 978, 987, 988, 994, 1004, 1005, 1006, 1010, 1015, 1019, 1020, 1021, 1023 }, and the claim 21 and supporting a set T 21 = {72} for the parity check bits, 22 parity check bits A support set T 22 = {40}, and 23 supporting a set of parity check bits T 23 = {24}, and the support set for the first 24 parity check bits T 24 = {30, 31, 46, 60, 76 , 94, 95, 100, 111, 116, 126, 127, 136, 144, 148, 150, 152, 159, 164, 174, 175, 180, 186, 188, 190, 200, 207, 216, , 222, 228, 231, 234, 236, 238, 239, 244, 248, 250, 252, 253, 264, 268, 276, 280, 284, 286, 292, 296, 300, 302, 310, 311, 312 , 316, 318, 319, 332, 343, 347, 348, 358, 359, 362, 364, 367, 378, 381, 396, 398, 404, 406, 407, 408, 412, 414, 420, 422, 428 , 438, 452, 454, 456, 463, 468, 470, 472, 474, 475, 476, 477, 479, 488, 491, 492, 493, 495, 500, 503, 508, 509, 528, 532, 536 , 543, 552, 556, 558, 559, 567, 572, 574, 584, 588, 590, 596, 598, 602, 606, 607, 616, 622, 623, 628, 631, 637, 639, 654, 655 , 656, 660, 662, 670, 676, 678, 682, 684, 686, 692, 695, 700, 701, 702, 703, 708, 712, 718, 719, 731, 734, 743, , 755, 758, 760, 762, 763, 766, 780, 782, 783, 788, 790, 794, 79 880, 806, 814, 823, 824, 827, 828, 829, 830, 836, 838, 844, 847, 852, 854, 855, 856, 858, 859, 860, 861, 862, 863, 868, 875, 882, 884, 886, 887, 890, 891, 894, 895, 900, 904, 912, 916, 918, 919, 922, 924, 926, 932, 935, 936, 938, 939, 940, 942, 943, 946, 948, 950, 951, 952, 955, 959, 968, 970, 974, 975, 978, 980, 989, 991, 994, 996, 998, 999, 1003, 1004, 1005, 1006, 1011, 1012, 1013, 1015, 1017, 1021, 1022, 1023}, and the support set for the first 25 parity check bits T 25 = {28, 30, 31, 47, 62, 76, 88, 94, 108, 110, 116 , 118, 119, 120, 124, 126, 148, 152, 156, 158, 172, 174, 176, 180, 184, 187, 191, 200, 204, 207, 208, 212, 214, 219, 223, 228 , 230, 232, 239, 240, 242, 247, 251, 252, 255, 264, 270, 280, 284, 300, 302, 303, 310, 311, 312, 316, 347, 348, 356, 358, 359 , 360, 364, 368, 372, 378, 383, 392, 396, 398, 407, 410, 414, 415, 420, 424, 428, 430, 431, 438, 439, 440, 442, 443, 445, 452 , 462, 470, 477, 487, 490, 494, 499, 500, 502, 504, 506, 509, 510, 524, 528, 532, 542, 552, 656, 660, 663, 664, 660, 632, 624, 630, 635, 638, 639, 654, 655, 656, 666, 671, 676, 678, 680, 686, 687, 694, 699, 700, 701, 702, 716, 719, 724, 726, 732, 733, 734, 735, 740, 742, 746, 747, 748, 824, 826, 827, 828, 830, 836, 838, 838, 828, 813, 840, 844, 847, 848, 854, 856, 859, 861, 862, 874, 876, 880, 882, 883, 892, 895, 904, 910, 912, 920, 923, 926, 927, 932, 934, 935, 936, 939, 940, 942, 943, 946, 948, 951, 952, 956, 957, 958, 966, 970, 972, 974, 978, 986, 987, 990, 991, 994, 996, 998, 1000, 1006, 1008, 1012, 1013, 1019, 1020}, and claim 26, the support set of the parity check bits T 26 = {52, 63, 76, 78, 92, 94, 100, 108, 110, 123, 124 , 125, 126, 127, 148, 150, 156, 159, 164, 172, 174, 175, 182, 183, 188, 190, 191, 204, 207, 212, 214, 215, 219, 220, 222, 223 , 230, 238, 250, 251, 254, 255, 264, 270, 276, 284, 287, 288, 296, 302, 303, 310, 311, 312, 3 381, 383, 396, 399, 404, 406, 408, 415, 416, 416, 416, 440, 443, 446, 456, 460, 472, 475, 476, 477, 478, 479, 480, 484, 486, 487, 492, 496, 498, 500, 502, 508, 580, 590, 511, 528, 536, 543, 544, 556, 558, 560, 564, 572, 574, 575, 584, 588, 590, 591, 592, 596, 600, 606, 607, 608, 615, 616, 632, 634, 638, 639, 640, 655, 660, 670, 672, 676, 678, 692, 694, 695, 699, 700, 701, 708, 710, 724, 727, 732, 735, 736, 740, 742, 743, 744, 747, 752, 755, 756, 758, 759, 763, 765, 766, 790, 792, 794, 796, 798, 799, 804, 806, 808, 814, 816, 820, 823, 829, 831, 832, 836, 838, 840, 846, 847, 852, 855, 856, 860, 861, 862, 863, 864, 868, 871, 875, 876, 878, 880, 886, 887, 891, 892, 894, 895, 904, 908, 911, 912, 920, 922, 923, 925, 926, 934, 935, 936, 938, 939, 942, 943, 946, 950, 955, 957, 959, 960, 964, 970, 974, 976, 983, 989, 990, 991, 992, 994, 998, 999, 1000, 1002, 1003, 1005, 1011, 1014, 1016, 1017 , 1018, 1020, 1022, 1023}

Wherein the outer encoder codes each bit value of the 26 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.
A parity check concatenated code encoder comprising:

An outer encoder, and

And a polarizer connected to said outer encoder,

The outer encoder is configured to output 28 parity check bits and 683 information bits,

Wherein the polarizer is configured to output 1,024 sign bits in response to the 28 parity check bits, the 683 information bits, and 313 frozen bits,

The 313 frozen bits correspond to the frozen bit index set F = {1,2,3,4,5,6,7,9,10,11,13,17,18,19,21,23,25,27,29 , 33, 34, 35, 37, 39, 41, 42, 43, 45, 49, 50, 51, 53, 57, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 81 , 82, 83, 85, 89, 97, 98, 99, 101, 105, 109, 113, 117, 121, 129, 130, 131, 133, 134, 135, 137, 138, , 147, 149, 153, 157, 161, 162, 163, 165, 169, 173, 177, 179, 181, 185, 193, 194, 195, 197, 201, 203, 205, 209, 211, 213, 217 , 225, 226, 227, 229, 233, 241, 257, 258, 259, 260, 261, 262, 263, 265, 266, 267, 269, 273, 274, 275, 277, 281, , 291, 293, 297, 301, 305, 307, 309, 313, 321, 322, 323, 325, 329, 331, 333, 337, 339, 341, 345, 353, 354, 355, 357, 361, 369 , 377, 385, 386, 387, 389, 391, 393, 395, 397, 401, 402, 403, 405, 409, 417, 418, 419, 421, 425, 433, 441, 449, 450, 451, 453 , 457, 465, 469, 473, 481, 485, 489, 497, 513, 514, 515, 516, 517, 518, 519, 521, 522, 523, 525, 529, 530, 531, 533, 537, 541 , 545, 546, 547 , 549, 553, 555, 557, 561, 563, 565, 569, 577, 578, 579, 581, 585, 587, 589, 593, 594, 595, 597, 601, 609, 610, 611, 613, 617 , 625, 633, 641, 642, 643, 645, 647, 649, 651, 653, 657, 658, 659, 661, 665, 673, 674, 675, 677, 681, 689, 697, 705, , 709, 713, 721, 725, 729, 737, 741, 745, 753, 769, 770, 771, 773, 775, 777, 778, 779, 781, 785, 786, 787, 789, 793, 801, 802 , 803, 805, 809, 817, 821, 825, 833, 834, 835, 837, 841, 845, 849, 853, 857, 865, 867, 869, 873, 881, 897, 898, 899, 901, 905 , 909, 913, 915, 917, 921, 929, 931, 933, 937, 945, 961, 963, 965, 969, 977, 993}

The outer encoder encodes the 28 parity check bits into a parity check bit index set P = {797, 893, 459, 603, 327, 407, 1015, 15, 575, 210, 202, 26, 1018, 486, , 94, 132, 36, 20, 12, 92, 8, 40, 24, 16, 48, 96}.
24. The method of claim 23,

A set of support bits T 1 = 249, 365, 373, 493, 501, 505, 685, 717, 757, 761, 813, 877, 885, 889, 973, 1001, 1005, 1013}, and the support set T 2 = {125, 189, 245, 249, 349, 373, 413, 437, 621, 669, 693, 701, 757, 797, 813 , 889, 953, 957, 997, 1001, 1009, 1017} and the support set T 3 = {61, 115, 125, 189, 221, 243, 245, 249, 253, 317, 349, 365, 371, 429, 435, 445, 467, 499, 501, 505, 509, 573, 605, 629, 637, 717, 733, 739, 749, 755, 757, 765, 813, 829, 843, The support set T 4 = {61, 91, 93, 107, 115} for the fourth parity check bits , 155, 221, 235, 237, 245, 253, 349, 365, 371, 381, 411, 427, 429, 461, 477, 483, 491, 493, 501, 505, 509, 539, 605, 619, 627 , 667, 669, 685, 691, 693, 717, 739, 747, 755, 761, 795, 797, 811, 819, 829, 851, 861, 885, 889, 907, 941 , 949, 953, 957, 971, 985, 989, 1001, 1003, 1011, 1013, 1017, 1021} and the support set T 5 = {59, 61, 107, 115, 125, 373, 379, 381, 427, 445, 475, 477, 483, 493, 571, 583, 603, 619, 627, 637, 667, 683, 699, 907, 939, 949, 953, 957, 971, 973, 987, 989, 1003, 1017, 1019}, and the support set T 6 = {61, 91, 103, 107, 125, 151, 167, 171, 187, 199, 219, 231, 235, 237, 243, 245, 251 , 295, 299, 315, 317, 327, 347, 349, 379, 381, 423, 427, 429, 443, 475, 477, 483, 499, 501, 505, 509, 551, 583, 603, 605, 615 , 619, 621, 627, 635, 637, 669, 679, 685, 691, 693, 701, 715, 717, 723, 747, 749, 757, 761, 765, 795, 819, 829, 851, 861, 875 , 877, 883, 885, 889, 891, 893, 903, 907, 923, 925, 941, 957, 967, 971, 973, 979, 985, 987, 989, 999, 1001, 1003, 1005, }, The support for the seventh parity check bit T 7 = {55, 61, 107, 115, 123, 151, 155, 167, 189, 219, 231, 237, 279, 295, 299, 327, 343, 349, 373, 407, 413, 427, 429, 437, 439, 467, 477, 483, 503, 505, 535, 551, 573, 583, 599, 605, 627, 629, 631, 663, 669, 679, 691, 695, 701, 711, 715, 717, 893, 903, 907, 923, 939, 951, 957, 813, 813, 819, 823, 827, 851, 871, 885, 889, 891, 893, And the support set T 8 = {55, 59, 93, 107, 123, 151, 155 for the eighth parity check bits} , 167, 199, 219, 221, 237, 249, 311, 327, 347, 349, 359, 365, 373, 375, 381, 407, 413, 423, 435, 443, 445, 467, 475, 477, 483 , 487, 491, 503, 535, 539, 567, 571, 599, 615, 619, 621, 627, 631, 663, 667, 669, 679, 683, 691, 695, 699, 717, 731, , 757, 761, 763, 765, 791, 795, 797, 811, 813, 819, 823, 827, 839, 843, 851, 855, 859, 861, 885, 891, 903, 919, 925, 941, 949 , 953, 955, 957, 967, 971, 973, 981, 983, 997, 1001, 1003, 1019} Support set for the bit T 9 = {15, 31, 55, 59, 63, 87, 103, 111, 115, 119, 155, 159, 167, 171, 183, 189, 199, 221, 231, 235, 237 , 239, 245, 247, 249, 283, 287, 299, 303, 315, 359, 367, 381, 399, 411, 415, 429, 435, 439, 443, 445, 463, 479, 483, 491, 495 , 499, 535, 539, 543, 559, 571, 573, 583, 591, 605, 607, 615, 627, 629, 635, 637, 671, 679, 685, 687, 691, 695, 699, , 723, 727, 733, 739, 743, 749, 761, 763, 799, 811, 813, 827, 839, 847, 851, 855, 861, 871, 889, 907, 911, 923, 935, 939, 951 , 953, 957, 971, 979, 981, 985, and 997, 1001, 1003, 1007, 1011, 1013, 1015, 1021}, a set support to the 10th parity check bit T 10 = {15, 31, 47, 239, 243, 251, 253, 255, 279, 287, 311, 315, 327, 338, 343, 347, 351, 359, 363, 367, 373, 375, 379, 383, 407, 411, 415, 423, 429, 431, 435, 445, 447, 461, 467, 471, 475, 479, 482, 493, 495, 499, 501, 505, 509, 535, 539, 573, 575, 603, 607, 619, 621, 627, 63 7, 639, 655, 663, 667, 669, 679, 691, 693, 695, 701, 703, 711, 717, 723, 733, 738, 739, 743, 747, 749, 751, 755, 791, 807, 815, 827, 831, 843, 871, 875, 879, 885, 893, 895, 907, 935, 939, 941, 943, 959, 967, 973, 979, 985, 987, 989, 991, 100, 100, 1009, 1011, 1015, 1023}, and the support set T 11 = {15, 47, 55, 59, 61, 79, 87, 91, 111, 119, 125, 127 , 143, 151, 167, 178, 189, 191, 207, 215, 221, 231, 235, 239, 243, 245, 249, 253, 279, 303, 306, 319, 335, 343, 349, 365, 367 , 370, 371, 375, 379, 383, 407, 411, 413, 423, 429, 431, 434, 459, 461, 463, 466, 471, 475, 479, 483, 491, 493, 499, 503, 505 , 509, 535, 539, 543, 559, 571, 573, 575, 603, 605, 615, 626, 629, 631, 635, 639, 663, 667, 671, 683, 685, 687, 690, 691, 693 , 699, 701, 703, 711, 715, 719, 723, 731, 749, 754, 757, 761, 763, 765, 767, 791, 797, 807, 811, 815, 827, 839, 842, 843, 850 , 859, 863, 871, 877, 879, 885, 891, 895, 903, 907, 911, 914, 919, 923, 930, 941, 947, 949, 951, 9 100, 100, 100, 100, 100, 1010, 1013, 1015, 1017} and the support set T 12 = {15 , 79, 93, 103, 107, 114, 115, 119, 123, 127, 143, 155, 170, 171, 175, 183, 189, 191, 199, 202, 207, 219, 223, 234, 235, 237 , 243, 251, 271, 287, 299, 306, 317, 327, 330, 349, 363, 365, 379, 383, 394, 399, 407, 411, 413, 415, 426, 427, 431, 435, 445 , 455, 458, 459, 461, 463, 466, 467, 479, 482, 483, 487, 495, 501, 503, 505, 507, 543, 551, 559, 562, 571, 573, 575, 599, 603 , 623, 626, 631, 635, 639, 669, 695, 701, 714, 717, 723, 731, 733, 735, 738, 751, 755, 757, 759, 763, 765, 783, 795, 811, 813 , 819, 827, 829, 831, 839, 842, 847, 851, 855, 863, 866, 875, 883, 885, 889, 891, 914, 919, 927, 935, 941, 943, 947, 949, 951 , 970, 973, 978, 979, 987, 989, 995, 997, 1001, 1003, 1009, 1013, 1017}, the support set T 13 = {63, 79, 95, 103, 111, 114, 115, 119, 125, 154, 155, 159, 167, 178, 282, 283, 287, 295, 299, 314, 317, 319, 343, 347, 349, 351, 365, 371, 373, 375, 379, 394, 399, 407, 413, 427, 429, 435, 443, 445, 447, 455, 463, 467, 471, 474, 475, 477, 479, 487, 490, 491, 495, 498, 501, 505, 507, 509, 535, 543, 551, 567, 570, 583, 602, 615, 618, 621, 623, 626, 627, 635, 637, 667, 679, 685, 690, 693, 695, 698, 703, 711, 714, 715, 717, 727, 730, 735, 739, 743, 747, 751, 755, 757, 759, 761, 762, 767, 783, 871, 882, 883, 885, 887, 889, 891, 893, 911, 811, 818, 827, 829, 831, 839, 842, 843, 851, 855, 858, 859, 861, 863, 914, 927, 930, 935, 949, 951, 962, 967, 971, 985, 989, 991, 995, 997, 1002, 1003, 1011, 1013, 1015, 1017, 1019, 1021, 1023} The support set for the parity check bits T 14 = {15, 31, 38, 58, 63, 79, 90, 91, 95, 106, 111, 114, 115, 122, 125, 127, 151, 159, 170, 175 , 178, 183, 186, 189, 191, 198, 199, 202, 210, 223, 230, 231, 251, 253, 283, 287, 295, 298, 30 371, 373, 375, 378, 399, 413, 423, 427, 429, 431, 434, 435, 608, 603, 605, 614, 615, 615, 618, 618, 618, 618, 627, 639, 646, 655, 666, 667, 669, 678, 679, 682, 683, 685, 687, 691, 695, 710, 711, 715, 719, 723, 727, 730, 731, 735, 742, 743, 747, 749, 751, 754, 755, 774, 794, 806, 807, 811, 818, 823, 838, 842, 843, 851, 861, 866, 874, 875, 877, 879, 883, 985, 887, 893, 895, 902, 903, 906, 907, 914, 922, 925, 934, 935, 938, 941, 947, 951, 957, 959, 962, 966, 970, 973, 98, 1001, 1003, 1005, 1011, 1013, 1017} and the support set T 15 = {38, 47, 61, 63, 79, 87, 102, 103, 115, 119, 123 , 143, 154, 167, 170, 175, 186, 189, 191, 199, 215, 219, 221, 230, 234, 242, 243, 245, 247, 249, 251, 253, 255, 271, 287, 315 , 317, 343, 346, 347, 359, 365, 367, 370, 371, 373, 375, 378, 379, 381, 394, 399, 413, 426, 427, 429, 434, 435, 439, 442, 445, 447, 454, 458, 459, 461, 467, 471, 474, 477, 479, 486, 490, 491, 493, 498, 499, 501, 507, 539, 551, 554, 559, 562, 567, 575, 614, 615, 621, 623, 626, 631, 634, 637, 639, 663, 669, 678, 693, 695, 710, 714, 722, 830, 823, 827, 838, 842, 843, 847, 850, 851, 855, 859, 861, 863, 870, 730, 733, 742, 747, 757, 759, 761, 762, 783, 875, 877, 879, 882, 883, 885, 889, 893, 902, 907, 914, 919, 925, 927, 935, 938, 941, 943, 947, 949, 951, 957, 959, The support set T 16 = {22, 26, 47, 58, 61, 63} for the 16th parity check bits , 79, 86, 87, 90, 91, 93, 95, 103, 114, 122, 123, 125, 127, 150, 151, 154, 155, 166, 167, 170, 178, 183, 189, 198, 199 , 202, 218, 221, 231, 234, 243, 245, 249, 250, 279, 282, 283, 287, 298, 299, 303, 306, 310, 319, 326, 342, 346, 349, 358, 362 , 363, 365, 383, 390, 399, 406, 423, 429, 434, 437, 439, 442, 445, 447, 455, 459, 470, 475, 477, 482 , 486, 487, 495, 498, 501, 502, 506, 527, 534, 535, 551, 559, 567, 570, 571, 582, 583, 586, 591, 598, 603, 605, 618, 619, 626 630, 631, 634, 637, 639, 646, 650, 662, 666, 669, 678, 679, 682, 690, 691, 693, 695, 698, 701, 703, 714, 715, 719, 726, 733 , 735, 738, 739, 747, 749, 751, 754, 767, 774, 783, 790, 791, 794, 797, 799, 806, 815, 819, 829, 831, 838, 842, 847, 858, 859 , 863, 877, 882, 886, 887, 890, 891, 893, 902, 906, 914, 919, 923, 925, 941, 946, 947, 953, 957, 962, 966, 975, 985, 989, 991 , 994, 995, 997, 1002, 1003, 1007, 1009, 1010, 1015, 1017, 1019, 1021}, and the support set T 17 = {15, 30, 47, 54, 55, 58, 59, 87, 91, 93, 95, 103, 106, 107, 111, 115, 118, 123, 127, 142, 143, 154, 155, 158, 159, 166, 170, 171, 183, 187, 258, 282, 294, 295, 302, 303, 310, and 298, 315, 326, 330, 346, 347, 349, 351, 359, 365, 366, 367, 379, 381, 390, 399, 407, 414, 415, 422, 42 495, 506, 526, 527, 535, 542, 490, 494, 497, 490, 490, 663, 667, 669, 670, 678, 679, 682, 685, 685, 686, 687, 691, 695, 699, 710, 711, 717, 718, 719, 727, 730, 731, 743, 746, 747, 749, 754, 755, 758, 762, 765, 782, 783, 790, 831, 838, 854, 861, 870, 874, 875, 877, 878, 879, 882, 886, 887, 813, 815, 819, 969, 967, 971, 975, 979, 925, 926, 930, 939, 951, 954, 957, 959, 962, 1018, 1019, 1021, 1023}, and the support set T 18 for the 18th parity check bit is set to 18 = 981, 982, 985, 987, 994, 995, 997, 998, 1001, 1003, 1005, 1006, , {14,15,22,30,38,46,54,55,58,61,62,86,90,91,94,102,106,111,111,119,122,125,126,127,142 , 150, 155, 170, 174, 178, 182, 183, 186, 187, 189, 191, 198, 206, 221, 230, 231, 235, 237, 238, 239, 315, 317, 318, 319, 327, 330, and 318, respectively, 424, 427, 429, 437, 443, 445, 446, 447, 442, 364, 366, 367, 371, 379, 390, 398, 410, 411, 414, 490, 491, 494, 498, 499, 501, 502, 503, 505, 507, 511, 535, 542, 550, 554, 559, 625, 622, 626, 627, 629, 630, 631, 634, 646, 650, 655, 662, 608, 602, 603, 605, 607, 614, 615, 619, 666, 669, 683, 687, 690, 691, 695, 699, 702, 703, 710, 711, 718, 719, 722, 723, 726, 733, 738, 739, 747, 755, 758, 761, 766, 867, 870, 843, 830, 839, 843, 846, 847, 854, 855, 858, 859, 861, 866, 870, 874, 982, 889, 895, 902, 907, 910, 922, 930, 934, 935, 938, 946, 947, 949, 951, 953, 954, 958, 966, 970, 971, 974, 978, 1004, 1011, 1014, 1018, 1021, 1022}, and the 19th parity check bit is set to " The support set T 19 = {14,22,30,46,45,55,58,62,86,90,91,93,95,102,103,107,111,111,111,119,123 , 127, 150, 154, 167, 170, 171, 175, 178, 182, 187, 189, 198, 199, 202, 206, 207, 210, 215, 219, 221, 223, 231, 235, 239, 245 , 253, 254, 255, 270, 282, 283, 287, 295, 302, 303, 314, 315, 317, 324, 327, 338, 346, 347, 350, 359, 366, 367, 375, 378, 381 , 388, 390, 394, 398, 406, 407, 410, 411, 413, 415, 423, 427, 429, 431, 437, 438, 439, 445, 446, 447, 455, 458, 461, 471, 477 , 479, 486, 490, 494, 498, 502, 505, 507, 509, 511, 527, 535, 539, 542, 543, 550, 562, 566, 567, 575, 583, 586, 598, 602, 603 , 607, 622, 623, 626, 629, 631, 634, 635, 650, 654, 669, 670, 671, 679, 682, 686, 687, 691, 693, 695, 698, 702, 703, 710, 711 , 715, 719, 722, 723, 726, 727, 730, 733, 734, 738, 742, 743, 746, 749, 751, 755, 759, 761, 767, 774, 783, 795, 797, 810, 814, 818, 823, 827, 836, 843, 847, 850, 851, 859, 862, 863, 870, 871, 883, 886, 889, 890, 894, 895, 906, 910, 914, 934 , 935, 939, 941, 942, 943, 946, 947, 951, 959, 966, 967, 974, 975, 978, 979, 981, 982, 987, 989, 990, 998, 999, 1001, 1005, 1007, 10, 1013, 1015, 1019, 1021, 1022, 1023} and the support set T 20 = {22, 26, 38, 46, 47, 54, 59, 63, 78, 86, 87 , 90, 93, 94, 95, 100, 103, 107, 111, 118, 123, 126, 151, 154, 155, 158, 164, 166, 170, 174, 175, 183, 198, 214, 219, 228 , 230, 231, 235, 238, 242, 249, 253, 254, 255, 270, 271, 278, 279, 282, 283, 294, 295, 299, 303, 306, 311, 314, 318, 319, 330 , 334, 335, 338, 342, 343, 346, 349, 351, 358, 362, 373, 375, 379, 383, 390, 398, 411, 414, 422, 427, 429, 430, 434, 437, 442 , 446, 447, 452, 454, 458, 459, 461, 463, 466, 470, 474, 475, 477, 478, 482, 483, 484, 490, 498, 499, 501, 503, 505, 506, 507 , 510, 511, 534, 535, 542, 543, 550, 558, 559, 562, 566, 567, 573, 580, 582, 583, 590, 591, 599, 602, 603, 618, 619, 621, 623 , 627, 630, 634, 637, 639, 644, 646, 650, 654, 662, 663, 666, 667, 669, 670, 671, 685, 686, 687, 69 0, 691, 693, 694, 699, 702, 703, 708, 714, 718, 726, 727, 730, 731, 735, 740, 742, 746, 749, 751, 754, 757, 765, 766, 767, 821, 821, 814, 818, 822, 826, 827, 829, 830, 831, 836, 843, 846, 847, 855, 859, 891, 893, 894, 903, 906, 907, 910, 914, 918, 923, 925, 930, 932, 934, 935, 941, 942, 943, 951, 953, 954, 957, 958, 959, 964, 966, 970, 982, 983, 985, 987, 989, 994, 995, 996, 1002, 1005, 1009, 1011, 1013, 101, 101, 1015, 1017, 1019, 1022}, and the support set T 21 = {14, 22, 30, 31, 55, 58, 62, 63, 79, 84, 86, 95, 102 , 111, 114, 118, 119, 123, 126, 143, 150, 164, 166, 174, 178, 180, 183, 186, 202, 214, 215, 218, 222, 223, 230, 234, 235, 237 , 238, 242, 243, 244, 245, 250, 253, 254, 255, 270, 271, 276, 282, 287, 292, 294, 295, 298, 302, 303, 311, 314, 315, 317, 335 , 338, 342, 346, 347, 351, 365, 366, 367, 370, 371, 373, 379, 381, 382, 406, 407, 411, 413, 414, 4 454, 452, 458, 459, 462, 466, 474, 477, 478, 482, 484, 486, 491, 422, 423, 426, 429, 430, 434, 437, 439, 442, 447, 452, 565, 575, 573, 591, 596, 602, 605, 607, 614, 619, 623, 626, 627, 629, 630, 635, 637, 639, 646, 650, 654, 655, 662, 663, 670, 679, 683, 685, 687, 698, 699, 701, 710, 711, 714, 715, 719, 726, 730, 734, 740, 743, 746, 747, 749, 750, 751, 759, 761, 762, 763, 766, 767, 772, 774, 782, 794, 795, 804, 806, 807, 811, 814, 819, 820, 822, 823, 826, 830, 831, 850, 851, 854, 861, 862, 863, 866, 868, 875, 878, 879, 935, 939, 941, 946, 947, 948, 949, 951, 954, 955, 958, 962, 964, 966, 967, 971, 978, 979, 980, 982, 985, 986, 987, 989, 994, 995, 996, 997, 998, 1001, 1002, 1005, 1007, 1009, 1011, 10, 1015, 1018, 1019, 1021, 1022}, and the support set T 22 = {12, 14, 22, 30, 31, 36, 38, 44, 47, 54, 55, 58, 59, 61, 62, 63, 84, 86, 87, 91, 107, 108, 116, 126, 127, 132, 142, 234, 235, 237, 238, 243, 245, 238, 234, 235, 235, 315, 317, 318, 319, 326, 332, 334, 335, 338, 340, 349, 350, 351, 365, 366, 367, 371, 372, 374, 378, 381, 383, 390, 394, 404, 407, 411, 412, 413, 414, 415, 423, 426, 498, 498, 498, 490, 490, 438, 439, 443, 445, 446, 454, 455, 459, 463, 466, 467, 470, 471, 475, 477, 483, 499, 500, 503, 507, 538, 548, 550, 551, 554, 559, 564, 567, 574, 575, 588, 590, 596, 599, 603, 605, 607, 614, 618, 620, 621, 693, 694, 698, 699, 701, 612, 683, 688, 630, 631, 634, 639, 644, 660, 667, 668, 669, 676, 679, 682, 683, 710, 714, 715, 716, 718, 719, 722, 723, 724, 726, 734, 739, 747, 749, 750, 754, 757, 759, 761, 762, 765, 772, 774, 783, 788, 794, 798, 804, 8 858, 859, 861, 862, 863, 866, 877, 882, 864, 812, 813, 814, 815, 822, 826, 827, 831, 838, 842, 846, 978, 978, 978, 968, 964, 967, 970, 971, 978, 934, 941, 942, 946, 947, 957, 958, 1002, 1003, 1004, 1005, 1010, 1011, 1012, 1013, 1015, 1018, 1023}, and support for the 23rd parity check bit The set T 23 = {12,15,20,22,26,36,38,52,55,60,62,63,84,87,90,91,92,93,94,95,100,103,107 , 108, 110, 115, 118, 122, 123, 127, 132, 140, 142, 143, 148, 151, 154, 156, 158, 164, 170, 174, 175, 183, 187, 188, 190, 199 , 204, 214, 219, 221, 223, 228, 230, 231, 235, 236, 239, 243, 244, 246, 251, 253, 255, 271, 276, 279, 282, 283, 286, 287, 299 , 302, 303, 306, 308, 315, 316, 318, 319, 332, 334, 335, 338, 340, 342, 343, 347, 349, 350, 351, 358, 359, 365, 366, 373, 380 , 394, 396, 404, 406, 407, 410, 412, 415, 420, 422, 428, 430, 431, 434, 435, 439, 442, 443, 445, 447, 45 505, 507, 524, 527, 532, 534, 539, 540, 542, 543, 642, 652, 654, 642, 624, 628, 631, 638, 639, 644, 646, 652, 654, 730, 732, 732, 733, 735, 738, 732, 732, 732, 734, 740, 742, 746, 747, 750, 751, 756, 758, 761, 762, 767, 772, 774, 780, 782, 788, 791, 794, 795, 799, 810, 812, 815, 819, 826, 828, 829, 831, 838, 839, 844, 847, 851, 852, 855, 858, 859, 861, 862, 868, 875, 877, 878, 883, 884, 885, 887, 889, 890, 893, 895, 902, 910, 914, 918, 919, 922, 923, 924, 925, 927, 930, 932, 938, 940, 941, 942, 946, 950, 953, 956, 957, The support set T 24 = {28, 72, 236} for the 24th parity-check bits, }, The support set T 25 = {40, 124} for the 25th parity check bit, 26 support for the parity check bits T 26 = {24, 828}, the support set T 27 = {776} for the 27th parity check bits, the support set T 28 = {876} for the 28th parity check bits, ego,

Wherein the outer encoder codes each bit value of the 28 parity check bits into a value of an information bit in at least one bit index included in the corresponding support set or a value of information bits in the at least one bit index, 2 < / RTI > adder.