Method and apparatus for low density parity check channel coding in wireless communication system
A low density parity check (LDPC) channel encoding method for use in a wireless communications system includes a communication device encoding an input bit sequence by using a LDPC matrix to obtain an encoded bit sequence for transmission. The LDPC matrix is obtained based on a lifting factor Z and a base matrix. The encoding method can be used in various communications systems including the fifth generation (5G) telecommunication systems, and can support various encoding requirements for information bit sequences with different code lengths.
1. A method for channel coding in wireless communication, comprising:
obtaining, by a communication apparatus, an input sequence;
decoding, by the communication apparatus, the input sequence using a matrix H to obtain a decoded sequence, wherein the decoded sequence comprises K bits, and wherein K>1; and
outputting, by the communication apparatus, the decoded sequence;
wherein the matrix H is determined according to a base matrix and a lifting factor Z, and wherein Z is a positive integer;
wherein the base matrix comprises a plurality of rows and columns of a first matrix;
wherein each element in the first matrix corresponds to a respective row index i and a respective column index j, where i≥0 and j≥0;
wherein each element in the first matrix is either a zero-element or a non-zero-element;
wherein each zero-element in the first matrix corresponds to an all-zero matrix of size Z×Z in the matrix H, and each non-zero-element at row i and column j corresponds to a circular permutation matrix of size Z×Z, wherein the circular permutation matrix is equal to a matrix obtained by circularly shifting an identity matrix of size Z×Z to the right for P i,j times, wherein P i,j =mod (V i,j ,Z), wherein V i,j is a value corresponding to a non-zero-element at row i and column j, and wherein:
with respect to row index i=0, for column indexes j=0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, V i,j is respectively 211, 198, 188, 186, 219, 4, 29, 144, 116, 216, 115, 233, 144, 95, 216, 73, 261, 1, 0;
with respect to row index i=1, for column indexes j=0, 2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, V i,j is respectively 179, 162, 223, 256, 160, 76, 202, 117, 109, 15, 72, 152, 158, 147, 156, 119, 0, 0, 0;
with respect to row index i=2, for column indexes j=0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, V i,j is respectively 258, 167, 220, 133, 243, 202, 218, 63, 0, 3, 74, 229, 0, 216, 269, 200, 234, 0, 0;
with respect to row index i=3, for column indexes j=0, 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 25, V i,j is respectively 187, 145, 166, 108, 82, 132, 197, 41, 162, 57, 36, 115, 242, 165, 0, 113, 108, 1, 0;
with respect to row index i=4, for column indexes j=0, 1, 26, V i,j is respectively 246, 235, 0;
with respect to row index i=5, for column indexes j=0, 1, 3, 12, 16, 21, 22, 27, V i,j is respectively 261, 181, 72, 283, 254, 79, 144, 0;
with respect to row index i=6, for column indexes j=0, 6, 10, 11, 13, 17, 18, 20, 28, V i,j is respectively 80, 144, 169, 90, 59, 177, 151, 108, 0;
with respect to row index i=7, for column indexes j=0, 1, 4, 7, 8, 14, 29, V i,j is respectively 169, 189, 154, 184, 104, 164, 0;
with respect to row index i=8, for column indexes j=0, 1, 3, 12, 16, 19, 21, 22, 24, 30, V i,j is respectively 54, 0, 252, 41, 98, 46, 15, 230, 54, 0;
with respect to row index i=9, for column indexes j=0, 1, 10, 11, 13, 17, 18, 20, 31, V i,j is respectively 162, 159, 93, 134, 45, 132, 76, 209, 0;
with respect to row index i=10, for column indexes j=1, 2, 4, 7, 8, 14, 32, V i,j is respectively 178, 1, 28, 267, 234, 201, 0;
with respect to row index i=11, for column indexes j=0, 1, 12, 16, 21, 22, 23, 33, V i,j is respectively 55, 23, 274, 181, 273, 39, 26, 0;
with respect to row index i=12, for column indexes j=0, 1, 10, 11, 13, 18, 34, V i,j is respectively 225, 162, 244, 151, 238, 243, 0;
with respect to row index i=13, for column indexes j=0, 3, 7, 20, 23, 35, V i,j is respectively 231, 0, 216, 47, 36, 0;
with respect to row index i=14, for column indexes j=0, 12, 15, 16, 17, 21, 36, V i,j is respectively 0, 186, 253, 16, 0, 79, 0;
with respect to row index i=15, for column indexes j=0, 1, 10, 13, 18, 25, 37, V i,j is respectively 170, 0, 183, 108, 68, 64, 0;
with respect to row index i=16, for column indexes j=1, 3, 11, 20, 22, 38, V i,j is respectively 270, 13, 99, 54, 0, 0;
with respect to row index i=17, for column indexes j=0, 14, 16, 17, 21, 39, V i,j is respectively 153, 137, 0, 0, 162, 0;
with respect to row index i=18, for column indexes j=1, 12, 13, 18, 19, 40, V i,j is respectively 161, 151, 0, 241, 144, 0;
with respect to row index i=19, for column indexes j=0, 1, 7, 8, 10, 41, V i,j is respectively 0, 0, 118, 144, 0, 0;
with respect to row index i=20, for column indexes j=0, 3, 9, 11, 22, 42, V i,j is respectively 265, 81, 90, 144, 228, 0;
with respect to row index i=21, for column indexes j=1, 5, 16, 20, 21, 43, V i,j is respectively 64, 46, 266, 9, 18, 0;
with respect to row index i=22, for column indexes j=0, 12, 13, 17, 44, V i,j is respectively 72, 189, 72, 257, 0;
with respect to row index i=23, for column indexes j=1, 2, 10, 18, 45, V i,j is respectively 180, 0, 0, 165, 0;
with respect to row index i=24, for column indexes j=0, 3, 4, 11, 22, 46, V i,j is respectively 236, 199, 0, 266, 0, 0;
with respect to row index i=25, for column indexes j=1, 6, 7, 14, 47, V i,j is respectively 205, 0, 0, 183, 0;
with respect to row index i=26, for column indexes j=0, 2, 4, 15, 48, V i,j is respectively 0, 0, 0, 277, 0;
with respect to row index i=27, for column indexes j=1, 6, 8, 49, V i,j is respectively 45, 36, 72, 0;
with respect to row index i=28, for column indexes j=0, 4, 19, 21, 50, V i,j is respectively 275, 0, 155, 62, 0;
with respect to row index i=29, for column indexes j=1, 14, 18, 25, 51, V i,j is respectively 0, 180, 0, 42, 0;
with respect to row index i=30, for column indexes j=0, 10, 13, 24, 52, V i,j is respectively 0, 90, 252, 173, 0;
with respect to row index i=31, for column indexes j=1, 7, 22, 25, 53, V i,j is respectively 144, 144, 166, 19, 0;
with respect to row index i=32, for column indexes j=0, 12, 14, 24, 54, V i,j is respectively 0, 211, 36, 162, 0;
with respect to row index i=33, for column indexes j=1, 2, 11, 21, 55, V i,j is respectively 0, 0, 76, 18, 0;
with respect to row index i=34, for column indexes j=0, 7, 15, 17, 56, V i,j is respectively 197, 0, 108, 0, 0;
with respect to row index i=35, for column indexes j=1, 6, 12, 22, 57, V i,j is respectively 199, 278, 0, 205, 0;
with respect to row index i=36, for column indexes j=0, 14, 15, 18, 58, V i,j is respectively 216, 16, 0, 0, 0;
with respect to row index i=37, for column indexes j=1, 13, 23, 59, V i,j is respectively 72, 144, 0, 0;
with respect to row index i=38, for column indexes j=0, 9, 10, 12, 60, V i,j is respectively 190, 0, 0, 0, 0;
with respect to row index i=39, for column indexes j=1, 3, 7, 19, 61, V i,j is respectively 153, 0, 165, 117, 0;
with respect to row index i=40, for column indexes j=0, 8, 17, 62, V i,j is respectively 216, 144, 2, 0;
with respect to row index i=41, for column indexes j=1, 3, 9, 18, 63, V i,j is respectively 0, 0, 0, 183, 0;
with respect to row index i=42, for column indexes j=0, 4, 24, 64, V i,j is respectively 27, 0, 35, 0;
with respect to row index i=43, for column indexes j=1, 16, 18, 25, 65, V i,j is respectively 52, 243, 0, 270, 0;
with respect to row index i=44, for column indexes j=0, 7, 9, 22, 66, V i,j is respectively 18, 0, 0, 57, 0; and
with respect to row index i=45, for column indexes j=1, 6, 10, 67, V i,j is respectively 168, 0, 144, 0.
2. The method according to claim 1 , wherein K is equal to 22×Z.
3. The method according to claim 1 , wherein Z is 9, 18, 36, 72, 144, or 288.
4. The method according to claim 1 , wherein the matrix H is determined according to a transformed matrix of the base matrix, and wherein the transformed matrix is obtained by performing row transformation and/or column transformation on the base matrix.
5. The method according to claim 1 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, and n=m+22.
6. The method according to claim 1 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, 5≤m≤46, and 27≤n≤68.
7. A communication apparatus comprising at least one processor and at least one non-transitory memory having processor-executable instructions stored thereon, wherein the at least one processor is configured to execute the processor-executable instructions to perform the following:
obtaining an input sequence;
decoding the input sequence using a matrix H to obtain a decoded sequence, wherein the decoded sequence comprises K bits, and wherein K>1; and
outputting the decoded sequence;
wherein the matrix H is determined according to a base matrix and a lifting factor Z, and wherein Z is a positive integer;
wherein the base matrix comprises a plurality of rows and columns of a first matrix;
wherein each element in the first matrix corresponds to a respective row index i and a respective column index j, where i≥0 and j≥0;
wherein each element in the first matrix is either a zero-element or a non-zero-element;
wherein each zero-element in the first matrix corresponds to an all-zero matrix of size Z×Z in the matrix H, and each non-zero-element at row i and column j corresponds to a circular permutation matrix of size Z×Z, wherein the circular permutation matrix is equal to a matrix obtained by circularly shifting an identity matrix of size Z×Z to the right for P i,j times, wherein P i,j =mod (V i,j ,Z), wherein V i,j is a value corresponding to a non-zero-element at row i and column j, and wherein:
with respect to row index i=0, for column indexes j=0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, V i,j is respectively 211, 198, 188, 186, 219, 4, 29, 144, 116, 216, 115, 233, 144, 95, 216, 73, 261, 1, 0;
with respect to row index i=1, for column indexes j=0, 2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, V i,j is respectively 179, 162, 223, 256, 160, 76, 202, 117, 109, 15, 72, 152, 158, 147, 156, 119, 0, 0, 0;
with respect to row index i=2, for column indexes j=0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, V i,j is respectively 258, 167, 220, 133, 243, 202, 218, 63, 0, 3, 74, 229, 0, 216, 269, 200, 234, 0, 0;
with respect to row index i=3, for column indexes j=0, 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 25, V i,j is respectively 187, 145, 166, 108, 82, 132, 197, 41, 162, 57, 36, 115, 242, 165, 0, 113, 108, 1, 0;
with respect to row index i=4, for column indexes j=0, 1, 26, V i,j is respectively 246, 235, 0;
with respect to row index i=5, for column indexes j=0, 1, 3, 12, 16, 21, 22, 27, V i,j is respectively 261, 181, 72, 283, 254, 79, 144, 0;
with respect to row index i=6, for column indexes j=0, 6, 10, 11, 13, 17, 18, 20, 28, V i,j is respectively 80, 144, 169, 90, 59, 177, 151, 108, 0;
with respect to row index i=7, for column indexes j=0, 1, 4, 7, 8, 14, 29, V i,j is respectively 169, 189, 154, 184, 104, 164, 0;
with respect to row index i=8, for column indexes j=0, 1, 3, 12, 16, 19, 21, 22, 24, 30, V i,j is respectively 54, 0, 252, 41, 98, 46, 15, 230, 54, 0;
with respect to row index i=9, for column indexes j=0, 1, 10, 11, 13, 17, 18, 20, 31, V i,j is respectively 162, 159, 93, 134, 45, 132, 76, 209, 0;
with respect to row index i=10, for column indexes j=1, 2, 4, 7, 8, 14, 32, V i,j is respectively 178, 1, 28, 267, 234, 201, 0;
with respect to row index i=11, for column indexes j=0, 1, 12, 16, 21, 22, 23, 33, V i,j is respectively 55, 23, 274, 181, 273, 39, 26, 0;
with respect to row index i=12, for column indexes j=0, 1, 10, 11, 13, 18, 34, V i,j is respectively 225, 162, 244, 151, 238, 243, 0;
with respect to row index i=13, for column indexes j=0, 3, 7, 20, 23, 35, V i,j is respectively 231, 0, 216, 47, 36, 0;
with respect to row index i=14, for column indexes j=0, 12, 15, 16, 17, 21, 36, V i,j is respectively 0, 186, 253, 16, 0, 79, 0;
with respect to row index i=15, for column indexes j=0, 1, 10, 13, 18, 25, 37, V i,j is respectively 170, 0, 183, 108, 68, 64, 0;
with respect to row index i=16, for column indexes j=1, 3, 11, 20, 22, 38, V i,j is respectively 270, 13, 99, 54, 0, 0;
with respect to row index i=17, for column indexes j=0, 14, 16, 17, 21, 39, V i,j is respectively 153, 137, 0, 0, 162, 0;
with respect to row index i=18, for column indexes j=1, 12, 13, 18, 19, 40, V i,j is respectively 161, 151, 0, 241, 144, 0;
with respect to row index i=19, for column indexes j=0, 1, 7, 8, 10, 41, V i,j is respectively 0, 0, 118, 144, 0, 0;
with respect to row index i=20, for column indexes j=0, 3, 9, 11, 22, 42, V i,j is respectively 265, 81, 90, 144, 228, 0;
with respect to row index i=21, for column indexes j=1, 5, 16, 20, 21, 43, V i,j is respectively 64, 46, 266, 9, 18, 0;
with respect to row index i=22, for column indexes j=0, 12, 13, 17, 44, V i,j is respectively 72, 189, 72, 257, 0;
with respect to row index i=23, for column indexes j=1, 2, 10, 18, 45, V i,j is respectively 180, 0, 0, 165, 0;
with respect to row index i=24, for column indexes j=0, 3, 4, 11, 22, 46, V i,j is respectively 236, 199, 0, 266, 0, 0;
with respect to row index i=25, for column indexes j=1, 6, 7, 14, 47, V i,j is respectively 205, 0, 0, 183, 0;
with respect to row index i=26, for column indexes j=0, 2, 4, 15, 48, V i,j is respectively 0, 0, 0, 277, 0;
with respect to row index i=27, for column indexes j=1, 6, 8, 49, V i,j is respectively 45, 36, 72, 0;
with respect to row index i=28, for column indexes j=0, 4, 19, 21, 50, V i,j is respectively 275, 0, 155, 62, 0;
with respect to row index i=29, for column indexes j=1, 14, 18, 25, 51, V i,j is respectively 0, 180, 0, 42, 0;
with respect to row index i=30, for column indexes j=0, 10, 13, 24, 52, V i,j is respectively 0, 90, 252, 173, 0;
with respect to row index i=31, for column indexes j=1, 7, 22, 25, 53, V i,j is respectively 144, 144, 166, 19, 0;
with respect to row index i=32, for column indexes j=0, 12, 14, 24, 54, V i,j is respectively 0, 211, 36, 162, 0;
with respect to row index i=33, for column indexes j=1, 2, 11, 21, 55, V i,j is respectively 0, 0, 76, 18, 0;
with respect to row index i=34, for column indexes j=0, 7, 15, 17, 56, V i,j is respectively 197, 0, 108, 0, 0;
with respect to row index i=35, for column indexes j=1, 6, 12, 22, 57, V i,j is respectively 199, 278, 0, 205, 0;
with respect to row index i=36, for column indexes j=0, 14, 15, 18, 58, V i,j is respectively 216, 16, 0, 0, 0;
with respect to row index i=37, for column indexes j=1, 13, 23, 59, V i,j is respectively 72, 144, 0, 0;
with respect to row index i=38, for column indexes j=0, 9, 10, 12, 60, V i,j is respectively 190, 0, 0, 0, 0;
with respect to row index i=39, for column indexes j=1, 3, 7, 19, 61, V i,j is respectively 153, 0, 165, 117, 0;
with respect to row index i=40, for column indexes j=0, 8, 17, 62, V i,j is respectively 216, 144, 2, 0;
with respect to row index i=41, for column indexes j=1, 3, 9, 18, 63, V i,j is respectively 0, 0, 0, 183, 0;
with respect to row index i=42, for column indexes j=0, 4, 24, 64, V i,j is respectively 27, 0, 35, 0;
with respect to row index i=43, for column indexes j=1, 16, 18, 25, 65, V i,j is respectively 52, 243, 0, 270, 0;
with respect to row index i=44, for column indexes j=0, 7, 9, 22, 66, V i,j is respectively 18, 0, 0, 57, 0; and
with respect to row index i=45, for column indexes j=1, 6, 10, 67, V i,j is respectively 168, 0, 144, 0.
8. The apparatus according to claim 7 , wherein K is equal to 22×Z.
9. The apparatus according to claim 7 , wherein Z is 9, 18, 36, 72, 144, or 288.
10. The apparatus according to claim 7 , wherein the matrix H is determined according to a transformed matrix of the base matrix, and wherein the transformed matrix is obtained by performing row transformation and/or column transformation on the base matrix.
11. The apparatus according to claim 7 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, and n=m+22.
12. The apparatus according to claim 7 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, and 5≤m≤46, and 27≤n≤68.
13. The apparatus according to claim 7 , wherein the at least one processor comprises one or more of the following:
a central processing unit;
a digital signal processor;
a digital signal processing device;
an application-specific integrated circuit;
a programmable logic device;
a field programmable gate array; or
a programmable logical apparatus.
14. The apparatus according to claim 7 , wherein the at least one non-transitory memory is configured to store the base matrix, the first matrix, the lifting factor Z or one or more circular permutation matrices.
15. The apparatus according to claim 7 , wherein the at least one non-transitory memory is configured to store parameters associated with the matrix H.
16. The apparatus according to claim 7 , further comprising:
a transceiver configured to receive a signal comprising the input sequence.
17. The apparatus according to claim 7 , wherein the apparatus is a network device or a terminal.
18. Anon-transitory computer-readable storage medium having processor-executable instructions stored thereon, wherein the processor-executable instructions, when executed by a processor, facilitate performance of the following:
obtaining an input sequence;
decoding the input sequence using a matrix H to obtain a decoded sequence, wherein the decoded sequence comprises K bits, and wherein K>1; and
outputting the decoded sequence;
wherein the matrix H is determined according to a base matrix and a lifting factor Z, wherein Z is a positive integer;
wherein the base matrix comprises a plurality of rows and columns of a first matrix;
wherein each element in the first matrix corresponds to a respective row index i and a respective column index j, where i≥0 and j≥0;
wherein each element in the first matrix is either a zero-element or a non-zero-element;
wherein each zero-element in the first matrix corresponds to an all-zero matrix of size Z×Z in the matrix H, and each non-zero-element at row i and column j corresponds to a circular permutation matrix of size Z×Z, wherein the circular permutation matrix is equal to a matrix obtained by circularly shifting an identity matrix of size Z×Z to the right for P i,j times, wherein P i,j =mod (V i,j ,Z), wherein V i,j is a value corresponding to a non-zero-element at row i and column j, and wherein:
with respect to row index i=0, for column indexes j=0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, V i,j is respectively 211, 198, 188, 186, 219, 4, 29, 144, 116, 216, 115, 233, 144, 95, 216, 73, 261, 1, 0;
with respect to row index i=1, for column indexes j=0, 2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, V i,j is respectively 179, 162, 223, 256, 160, 76, 202, 117, 109, 15, 72, 152, 158, 147, 156, 119, 0, 0, 0;
with respect to row index i=2, for column indexes j=0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, V i,j is respectively 258, 167, 220, 133, 243, 202, 218, 63, 0, 3, 74, 229, 0, 216, 269, 200, 234, 0, 0;
with respect to row index i=3, for column indexes j=0, 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 25, V i,j is respectively 187, 145, 166, 108, 82, 132, 197, 41, 162, 57, 36, 115, 242, 165, 0, 113, 108, 1, 0;
with respect to row index i=4, for column indexes j=0, 1, 26, V i,j is respectively 246, 235, 0;
with respect to row index i=5, for column indexes j=0, 1, 3, 12, 16, 21, 22, 27, V i,j is respectively 261, 181, 72, 283, 254, 79, 144, 0;
with respect to row index i=6, for column indexes j=0, 6, 10, 11, 13, 17, 18, 20, 28, V i,j is respectively 80, 144, 169, 90, 59, 177, 151, 108, 0;
with respect to row index i=7, for column indexes j=0, 1, 4, 7, 8, 14, 29, V i,j is respectively 169, 189, 154, 184, 104, 164, 0;
with respect to row index i=8, for column indexes j=0, 1, 3, 12, 16, 19, 21, 22, 24, 30, V i,j is respectively 54, 0, 252, 41, 98, 46, 15, 230, 54, 0;
with respect to row index i=9, for column indexes j=0, 1, 10, 11, 13, 17, 18, 20, 31, V i,j is respectively 162, 159, 93, 134, 45, 132, 76, 209, 0;
with respect to row index i=10, for column indexes j=1, 2, 4, 7, 8, 14, 32, V i,j is respectively 178, 1, 28, 267, 234, 201, 0;
with respect to row index i=11, for column indexes j=0, 1, 12, 16, 21, 22, 23, 33, V i,j is respectively 55, 23, 274, 181, 273, 39, 26, 0;
with respect to row index i=12, for column indexes j=0, 1, 10, 11, 13, 18, 34, V i,j is respectively 225, 162, 244, 151, 238, 243, 0;
with respect to row index i=13, for column indexes j=0, 3, 7, 20, 23, 35, V i,j is respectively 231, 0, 216, 47, 36, 0;
with respect to row index i=14, for column indexes j=0, 12, 15, 16, 17, 21, 36, V i,j is respectively 0, 186, 253, 16, 0, 79, 0;
with respect to row index i=15, for column indexes j=0, 1, 10, 13, 18, 25, 37, V i,j is respectively 170, 0, 183, 108, 68, 64, 0;
with respect to row index i=16, for column indexes j=1, 3, 11, 20, 22, 38, V i,j is respectively 270, 13, 99, 54, 0, 0;
with respect to row index i=17, for column indexes j=0, 14, 16, 17, 21, 39, V i,j is respectively 153, 137, 0, 0, 162, 0;
with respect to row index i=18, for column indexes j=1, 12, 13, 18, 19, 40, V i,j is respectively 161, 151, 0, 241, 144, 0;
with respect to row index i=19, for column indexes j=0, 1, 7, 8, 10, 41, V i,j is respectively 0, 0, 118, 144, 0, 0;
with respect to row index i=20, for column indexes j=0, 3, 9, 11, 22, 42, V i,j is respectively 265, 81, 90, 144, 228, 0;
with respect to row index i=21, for column indexes j=1, 5, 16, 20, 21, 43, V i,j is respectively 64, 46, 266, 9, 18, 0;
with respect to row index i=22, for column indexes j=0, 12, 13, 17, 44, V i,j is respectively 72, 189, 72, 257, 0;
with respect to row index i=23, for column indexes j=1, 2, 10, 18, 45, V i,j is respectively 180, 0, 0, 165, 0;
with respect to row index i=24, for column indexes j=0, 3, 4, 11, 22, 46, V i,j is respectively 236, 199, 0, 266, 0, 0;
with respect to row index i=25, for column indexes j=1, 6, 7, 14, 47, V i,j is respectively 205, 0, 0, 183, 0;
with respect to row index i=26, for column indexes j=0, 2, 4, 15, 48, V i,j is respectively 0, 0, 0, 277, 0;
with respect to row index i=27, for column indexes j=1, 6, 8, 49, V i,j is respectively 45, 36, 72, 0;
with respect to row index i=28, for column indexes j=0, 4, 19, 21, 50, V i,j is respectively 275, 0, 155, 62, 0;
with respect to row index i=29, for column indexes j=1, 14, 18, 25, 51, V i,j is respectively 0, 180, 0, 42, 0;
with respect to row index i=30, for column indexes j=0, 10, 13, 24, 52, V i,j is respectively 0, 90, 252, 173, 0;
with respect to row index i=31, for column indexes j=1, 7, 22, 25, 53, V i,j is respectively 144, 144, 166, 19, 0;
with respect to row index i=32, for column indexes j=0, 12, 14, 24, 54, V i,j is respectively 0, 211, 36, 162, 0;
with respect to row index i=33, for column indexes j=1, 2, 11, 21, 55, V i,j is respectively 0, 0, 76, 18, 0;
with respect to row index i=34, for column indexes j=0, 7, 15, 17, 56, V i,j is respectively 197, 0, 108, 0, 0;
with respect to row index i=35, for column indexes j=1, 6, 12, 22, 57, V i,j is respectively 199, 278, 0, 205, 0;
with respect to row index i=36, for column indexes j=0, 14, 15, 18, 58, V i,j is respectively 216, 16, 0, 0, 0;
with respect to row index i=37, for column indexes j=1, 13, 23, 59, V i,j is respectively 72, 144, 0, 0;
with respect to row index i=38, for column indexes j=0, 9, 10, 12, 60, V i,j is respectively 190, 0, 0, 0, 0;
with respect to row index i=39, for column indexes j=1, 3, 7, 19, 61, V i,j is respectively 153, 0, 165, 117, 0;
with respect to row index i=40, for column indexes j=0, 8, 17, 62, V i,j is respectively 216, 144, 2, 0;
with respect to row index i=41, for column indexes j=1, 3, 9, 18, 63, V i,j is respectively 0, 0, 0, 183, 0;
with respect to row index i=42, for column indexes j=0, 4, 24, 64, V i,j is respectively 27, 0, 35, 0;
with respect to row index i=43, for column indexes j=1, 16, 18, 25, 65, V i,j is respectively 52, 243, 0, 270, 0;
with respect to row index i=44, for column indexes j=0, 7, 9, 22, 66, V i,j is respectively 18, 0, 0, 57, 0; and
with respect to row index i=45, for column indexes j=1, 6, 10, 67, V i,j is respectively 168, 0, 144, 0.
19. The non-transitory computer-readable storage medium according to claim 18 , wherein K is equal to 22×Z.
20. The non-transitory computer-readable storage medium according to claim 18 , wherein Z is 9, 18, 36, 72, 144, or 288.
21. The non-transitory computer-readable storage medium according to claim 18 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, and n=m+22.
22. The non-transitory computer-readable storage medium according to claim 18 , wherein the base matrix comprises m rows and n columns, wherein m and n are integers, and 5≤m≤46, and 27≤n≤68.