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Optimization of Non Binary Parity Check Coefficients

The tables of optimized coefficients are computed from the binary image of a parity checks of degree dc. This coefficients ensure a binary hamming weight greater on equal to three. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction over high order Galois Field, and thus helps the construction of new generations of NB-LDPC codes with better decoding performance. More details on the coefficients construction and properties are given in: Optimized coefficients .
List of optimal coefficient's exponents for GF(64)
dc GF(64)
3 {0, 15, 41} ; {0, 22, 37}; {0, 26, 48}
4 {0, 9, 22, 37}
5 {0, 7, 18, 44, 53}; {0, 6, 17, 43, 52}
6 {0, 6, 13, 20, 46, 55}; {0, 6, 12, 19, 45, 54}
7 {0, 6, 13, 21, 28, 44, 54}; {0, 6, 12, 20, 27, 43, 53}
8 {0, 6, 13, 21, 28, 36, 44, 54}; {0, 6, 12, 20, 27, 35, 43, 53}
9 {0, 6, 14, 21, 27, 35, 42, 48, 56}
10 {0, 6, 12, 18, 24, 30, 37, 44, 50, 56}
List of optimal coefficient's exponents for GF(128)
dc GF(128)
3 {0, 15, 53} ; {0, 15, 54}; {0, 73, 88} ; {0, 74, 89} ; {0, 38, 112} ; {0, 39, 112}
4 {0, 12, 84, 101};
5 {0, 11, 55, 84, 101}
6 {0, 9, 21, 60, 94, 112}
7 {0, 7, 24, 39, 48, 60, 99}
8 {0, 7, 30, 37, 54, 69, 78, 90}
9 {0, 7, 19, 30, 37, 54, 69, 78, 90}
10 {0, 7, 38, 45, 59, 68, 75, 92, 107, 116}
11 {0, 7, 22, 30, 37, 48, 55, 69, 78, 89, 96}
12 {0, 7, 18, 25, 39, 48, 59, 66, 88, 97, 104, 119}
13 {0, 7, 17, 24, 38, 48, 58, 65, 72, 87, 96, 103, 118}
14 {0, 7, 14, 29, 38, 45, 55, 62, 69, 76, 86, 93, 107, 116}
15 {0, 7, 18, 25, 32, 39, 47, 54, 61, 69, 78, 89, 96, 103, 118}
16 {0, 7, 17, 24, 31, 38, 47, 54, 61, 68, 75, 82, 89, 96, 103, 118}
17 {0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 119}
18 {0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119}
List of optimal coefficient's exponents for GF(256)
dc GF(256)
3 {0, 72, 80} ; {0, 8, 183} ; {0, 175, 247}
4 {0, 8, 172, 183} ; {0, 8, 172, 182} ; {0, 8, 171, 182}
5 {0, 8, 66, 172, 180}
6 {0, 8, 75, 83, 91, 150}
7 {0, 8, 76, 84, 92, 131, 150}
8 {0, 8, 36, 75, 83, 91, 129, 149}
9 {0, 8, 37, 76, 84, 92, 130, 150, 234}
10 {0, 8, 16, 54, 74, 139, 158, 179, 188, 215}
11 {0, 8, 27, 92, 109, 131, 139, 169, 208, 216, 224}
12 {0, 8, 27, 39, 92, 109, 132, 140, 169, 208, 216, 224}
13 {0, 8, 18, 38, 46, 65, 77, 130, 147, 170, 178, 207, 246}
14 {0, 8, 16, 42, 82, 90, 98, 107, 128, 136, 155, 167, 220, 237}
15 {0, 8, 29, 37, 76, 84, 92, 103, 123, 131, 150, 162, 192, 215, 233}
16 {0, 8, 16, 34, 42, 79, 87, 95, 106, 126, 134, 153, 165, 196, 218, 236}
17 {0, 8, 45, 53, 61, 69, 77, 94, 102, 121, 133, 164, 186, 203, 221, 229, 237}
18 {0, 8, 19, 27, 35, 52, 60, 92, 100, 108, 116, 126, 147, 155, 173, 185, 216, 237}
19 {0, 8, 26, 39, 70, 91, 109, 117, 126, 134, 142, 161, 169, 183, 202, 210, 218, 226, 236}
20 {0, 8, 22, 30, 38, 52, 61, 75, 93, 101, 109, 117, 127, 147, 155, 174, 186, 206, 216, 238}
List of optimal coefficient's exponents for GF(512)
dc GF(512)
3 {0, 27, 109}
4 {0, 41, 122, 442}
5 {0, 39, 155, 320, 436}
6 {0, 22, 122, 162, 393, 478}
7 {0, 19, 45, 64, 210, 243, 409}
8 {0, 19, 75, 119, 159, 228, 312, 367}
9 {0, 19, 75, 119, 159, 228, 264, 312, 367}
10 {0, 14, 64, 213, 232, 288, 332, 372, 441, 479}
11 {0, 14, 64, 213, 232, 288, 332, 355, 372, 441, 479}
12 {0, 13, 27, 77, 226, 245, 301, 345, 366, 386, 454, 492}
13 {0, 13, 27, 77, 226, 245, 301, 345, 366, 385, 424, 454, 492}
14 {0, 12, 27, 76, 185, 225, 244, 302, 344, 365, 385, 423, 454, 491}
15 {0, 12, 27, 63, 76, 185, 225, 244, 302, 344, 365, 385, 423, 454, 491}
16 {0, 12, 27, 63, 76, 185, 225, 244, 302, 332, 344, 365, 385, 423, 454, 491}
17 {0, 12, 27, 63, 76, 185, 225, 244, 302, 332, 344, 365, 385, 423, 438, 454, 491}
18 {0, 12, 27, 63, 76, 116, 185, 225, 244, 302, 332, 344, 367, 385, 423, 438, 454, 491}
19 {0, 10, 25, 41, 78, 98, 110, 125, 161, 174, 214, 283, 323, 342, 400, 430, 442, 463, 483}
20 {0, 9, 40, 59, 117, 147, 159, 180, 200, 228, 238, 253, 269, 306, 326, 338, 353, 389, 402, 443}
List of optimal coefficient's exponents for GF(1024)
dc GF(1024)
3 {0, 171, 449} ; {0, 575, 745} ; {0, 278, 852}
4 {0, 105, 433, 918}
5 {0, 30, 328, 358, 448}
6 {0, 32, 125, 291, 672, 729}
7 {0, 67, 208, 592, 685, 829, 956}
8 {0, 38, 86, 585, 640, 728, 776, 828}
9 {0, 35, 121, 288, 327, 489, 554, 744, 830, 888}
10 {0, 22, 249, 410, 666, 730, 845, 901, 939, 986}
11 {0, 35, 288, 327, 391, 489, 554, 687, 744, 831, 888}
12 {0, 26, 161, 196, 230, 449, 488, 552, 650, 848, 905, 992}
13 {0, 24, 219, 258, 322, 420, 618, 675, 762, 793, 819, 954, 989}
14 {0, 24, 219, 258, 322, 420, 576, 618, 676, 762, 793, 819, 955, 990}
15 {0, 24, 133, 219, 258, 322, 420, 576, 618, 676, 762, 793, 819, 955, 990}
16 {0, 24, 133, 219, 258, 321, 420, 546, 575, 618, 675, 761, 793, 819, 954, 991}
17 {0, 24, 52, 133, 219, 258, 321, 420, 546, 575, 618, 675, 761, 793, 819, 954, 991}
18 {0, 24, 52, 133, 219, 258, 321, 420, 546, 575, 618, 675, 761, 793, 819, 888, 954, 991}
19 {0, 24, 52, 133, 219, 258, 321, 420, 518, 546, 575, 618, 675, 761, 793, 819, 888, 954, 991}
20 {0, 16, 126, 155, 198, 255, 341, 373, 398, 469, 534, 571, 603, 627, 655, 736, 799, 822, 861, 925}

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