New Linear Codes over GF(3), GF(11), and GF(13)

Abstract

Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets F3, F11, and F13 with better parameters. A total of 38 new linear codes over the three alphabets are presented.

A Construction of Binary Punctured Linear Codes and A Supporting Method for Best Code Search

“…There are codes in the database that the minimum distance does not reach the upper bound. Therefore, some studies have been carried out to increase the minimum distance [2]- [4]. The best code construction method using modified codes [7] is one of them.…”

“…However, the minimum distance of many (n, k) codes does not reach a theoretical upper bound. Therefore, various methods are still being tried to construct best codes [2]- [4].…”

Generating generalized necklaces and new quasi-cyclic codes

“…Codes over GF (11) There exist quasi-cyclic codes with parameters: [16,8,8] Codes over GF (13) There exist quasi-cyclic codes with parameters: [14,7,7]…”

“…In [12] and [13] E. Chen and N. Aydin constructed 45 new OC and QT codes over GF (11), 38 such codes over GF (13) and presented databases for small dimensions and n ≤ 150. New QT codes over GF (11) and GF (13) are also presented in [7].…”

“…13 ,[21,7,13] 13 ,[28,7,18] 13 , [7p, 7, 6p − 6] 13 for p = 5, 6, ..., 15 and [7p, 7, 6p − 5] 13 for p= 16, 17, 18, 19. A[14,7,7] 13 code: ccccccb, cccbca7; 0 1 7 2100 8 21336 9 164220 10 983556 114319196 12 12922308 13 23875068 14 20460732 A [21, 7, 13] 13 code: cc484c1, ca650c9, ccb0b70; 0 1 13 6636 14 39396 15 210336 16 957684 17 3356220 18 9025884 19 17011344 20 20458452 21 11682564 A [28, 7, 18] 13 code: cc48499, ca650c7, cc78aca, cccca17; 0 1 18 1848 19 7896 20 49560 21 219168 22 847980 23 2627268 24 6584004 25 12663000 26 17505348 27 15570492 28 6671952 A [35, 7, 24] 13 code: cc4849a, ca650c9, cc78acb, cccca15, cb04650; 0 1 24 2772 25 11172 26 52500 27 209160 28 713412 29 2061696 30 4989516 31 9623040 32 14413560 33 15735888 34 11134032 35 3801768 A [42, 7, 30] 13 code: cc48499, cc78b00, cccca18, cb04647, ccbb381, ca650c9; 0 1 30 2352 31 12768 32 51912 33 194376 34 599592 35 1619184 36 3827880 37 7441812 38 11757564 39 14433048 40 13019076 41 7613844 42 2175108 A [112, 7, 91] 13 code: cccca24, ca650c9, cc484c1, ccc76a8, ccbc4b2, ccbb385, cccbcb4, cc78b01, ccb9656, ccbc163, ccbc962, ccbac74, ccbab1a, ccbab23, ccccbc1, cb38697; 0 1 91 4200 92 14448 93 35784 94 91056 95 196224 96 407736 97 825636 98 1507632 99 2539824 100 3945984 101 5683272 102 7303128 103 8580348 104 8870568 105 8095248 106 6441624 107 4324320 108 2388792 109 1068228 110 336672 111 79716 112 8076 There exist quasi-cyclic codes with parameters: [16, 8, 8] 13 , [24, 8, 14] 13 , [32, 8, 20] 13 , [40, 8, 27] 13 , [48, 8, 33] 13 , [56, 8, 40] 13 , [64, 8, 47] 13 , [72, 8, 53] 13 and [8p, 8, 7p − 10] 13 for p = 10, 11, ..., 19. A [16, 8, 8] 13 code: cccccc77, cc82b0ca; 0 1 8 7116 9 80352 10 583632 11 4004448 12 19892040 13 73427424 14 188896848 15 302188704 16 226650156 A [24, 8, 14] 13 code: ccccccc6, ccb50a09, ccc538ca; 0 1 14 3312 15 32544 16 205248 17 1154400 18 5379504 19 20390304 20 61267056 21 139904640 22 228977184 23 238950528 24 119466000 A [64, 8, 47] 13 code: ccccccca, ccb50a15, ccc7b8b6, ccb75441, cc629b0b, cc847138, ccc538ca, cc6c9aa0; 0 1 47 3168 48 12156 49 50016 50 189360 51 593184 52 1788792 53 4855392 54 11944992 55 25823712 56 50130300 57 84293952 58 122167440 59 148750176 60 149087808 61 117161664 62 68071872 63 25970304 64 4836432 A [80, 8, 60] 13 code: cccccc56, ccb50968, ccc538c8, ccc7c166, ccb75438, cc629a31, cc8471a8, cc6c9a95, cccb9586, ccb57b73; 0 1 60 1584 61 3840 62 19200 63 62112 64 199032 65 583776 66 1584240 67 3965664 68 9137136 69 19061568 70 35892528 71 60720384 72 91043424 73 119819040 74 135862176 75 130555584 76 102976704 77 64252800 78 29645280 79 8981952 80 1362696…”