← 研踪 PaperTrailarXiv:2611.14433 · v2
New linear codes over GF(7), GF(8) and GF(9) from quasi-cyclic codes and Construction X
首次提交 2026-11-11 02:06
版本更新 2026-11-12 22:31
雷达观测 2026-11-13 14:15
摘要
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.
作者备注：12 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
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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.
查看此版本解读 →查看此版本原文 ↗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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