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arXiv:2611.14433 · v1

New linear codes over GF(7), GF(8) and GF(9) from quasi-cyclic codes and Construction X

由准循环码与Construction X构造的GF(7)、GF(8)和GF(9)上的新线性码

Sebastian Cochinescu
cs.IT
首次提交 2026-11-11 02:06
版本更新 2026-11-11 02:06
雷达观测 2026-11-11 13:07
arXiv ↗PDF ↗

摘要

We present linear codes with parameters [88,8,64] and [90,8,65] over GF(7), [98,7,75] and [97,8,72] over GF(8), and [44,7,31] over GF(9). Each has a minimum distance one larger than the lower bound listed in Grassl's tables of best known linear codes; codes with parameters [89,8,64] over GF(7) and [96,8,71], [95,8,70], [94,8,69] over GF(8) follow. Four of the five codes are obtained from a quasi-cyclic code and a vector that is constant on each block, by Construction X with a repetition code. The remaining one, the [90,8,65] code, is obtained from the first by appending two coordinates. The parameters and all weight counts were obtained by exhaustive enumeration, and a short program that repeats these computations is included.

作者备注：9 pages, 1 table. Ancillary files: verification program and generator matrices. Codes and programs archived at doi:10.5281/zenodo.24098667 doi:10.5281/zenodo.24098671 doi:10.5281/zenodo.24098673 doi:10.5281/zenodo.24112842 doi:10.5281/zenodo.24112844

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AI 论文解读

基于摘要 · 非全文审阅

作者报告通过准循环码结合Construction X（以及坐标扩展）在GF(7)、GF(8)、GF(9)上构造出若干最小距离比Grassl已知最优线性码表下界大1的线性码。

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版本差异

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摘要变化

We present linear codes with parameters [88,8,64] and_7, [90,8,65] over GF(7)_7, [98,7,75] and_8, [97,8,72] over GF(8)_8, and [44,7,31]_9, over[97,8,71]_7, GF(9)[85,6,67]_8 and [78,8,56]_7. Each has a minimum distance one larger than the lower bound listed in Grassl's tables of best known linear codes; codes with parameters [89,8,64] over GF(7) and_7, [96,8,71]_8, [95,8,70]_8, [94,8,69] over_8, GF([77,8),55]_7 and [86,6,68]_8 follow. FourSeven of the fiveeight codes are obtained from a quasi-cyclic code and a vector that is constant on each block, by Construction X with a repetition code. The remaining one, the [90,8,65]_7 code, is obtained from the first by appending two coordinates. The parameters and all weight counts were obtained by exhaustive enumeration, and a short program that repeats these computations is included.

作者备注变化

912 pages, 1 table. Ancillary files: verification program and generator matrices. v2: three more codes and two more derived codes added. Codes and programs archived at doi:10.5281/zenodo.24098667 doi:10.5281/zenodo.24098671 doi:10.5281/zenodo.24098673 doi:10.5281/zenodo.24112842 doi:10.5281/zenodo.24112844 doi:10.5281/zenodo.24132260 doi:10.5281/zenodo.24132262 doi:10.5281/zenodo.24136306

观测到的版本

相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。

v2 · 2026-11-12 22:31 · 最新观测版本

New linear codes over GF(7), GF(8) and GF(9) from quasi-cyclic codes and Construction X

We present linear codes with parameters [88,8,64]_7, [90,8,65]_7, [98,7,75]_8, [97,8,72]_8, [44,7,31]_9, [97,8,71]_7, [85,6,67]_8 and [78,8,56]_7. Each has a minimum distance one larger than the lower bound listed in Grassl's tables of best known linear codes; codes with parameters [89,8,64]_7, [96,8,71]_8, [95,8,70]_8, [94,8,69]_8, [77,8,55]_7 and [86,6,68]_8 follow. Seven of the eight codes are obtained from a quasi-cyclic code and a vector that is constant on each block, by Construction X with a repetition code. The remaining one, the [90,8,65]_7 code, is obtained from the first by appending two coordinates. The parameters and all weight counts were obtained by exhaustive enumeration, and a short program that repeats these computations is included.

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v1 · 2026-11-11 02:06

New linear codes over GF(7), GF(8) and GF(9) from quasi-cyclic codes and Construction X

We present linear codes with parameters [88,8,64] and [90,8,65] over GF(7), [98,7,75] and [97,8,72] over GF(8), and [44,7,31] over GF(9). Each has a minimum distance one larger than the lower bound listed in Grassl's tables of best known linear codes; codes with parameters [89,8,64] over GF(7) and [96,8,71], [95,8,70], [94,8,69] over GF(8) follow. Four of the five codes are obtained from a quasi-cyclic code and a vector that is constant on each block, by Construction X with a repetition code. The remaining one, the [90,8,65] code, is obtained from the first by appending two coordinates. The parameters and all weight counts were obtained by exhaustive enumeration, and a short program that repeats these computations is included.

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