• • 上一篇    

有限链环上的广义拟循环码

崔梦瑶, 高健, 马芳卉, 孟祥蕊   

  1. 山东理工大学数学与统计学院, 淄博 255000
  • 收稿日期:2022-03-02 修回日期:2022-05-29 发布日期:2022-12-13
  • 通讯作者: 高健, Email: dezhougaojian@163.com
  • 作者简介:崔梦瑶, 女, 硕士研究生, Email:cmy17864309561@163.com
  • 基金资助:
    山东省自然科学基金 (ZR2022MA024),国家自然科学基金(12071264, 11701336, 11626144, 11671235), 山东省自然科学基金(ZR2021QA047), 山东省高等学校“青创人才引育计划”资助课题.

崔梦瑶, 高健, 马芳卉, 孟祥蕊. 有限链环上的广义拟循环码[J]. 系统科学与数学, 2022, 42(11): 3134-3148.

CUI Mengyao, GAO Jian, MA Fanghui, MENG Xiangrui. On Generalized Quasi-Cyclic Codes over Finite Chain Rings[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(11): 3134-3148.

On Generalized Quasi-Cyclic Codes over Finite Chain Rings

CUI Mengyao, GAO Jian, MA Fanghui, MENG Xiangrui   

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000
  • Received:2022-03-02 Revised:2022-05-29 Published:2022-12-13
有限链环上一般指标广义拟循环码的显式生成元对于确定广义拟循环码的生成矩阵、对偶码、计数、最小距离、对偶包含的条件、重量分布具有重要意义. 设有限链环$R=\mathbb{F}_{q}+u\mathbb{F}_{q}+\cdots+u^{s-1}\mathbb{F}_{q}$, 其中$q$是某个素数的幂次, $s$是正整数, $s\geq2$和$u^s=0$. 环$R$ 上分块码长为$(r_{1},r_{2},\cdots,r_{l})$, 指标为$l$的广义拟循环码是$R[x]/\langle x^{r_{1}}-1\rangle\times R[x]/\langle x^{r_{2}}-1\rangle\times\cdots\times R[x]/\langle x^{r_{l}}-1\rangle$的$R[x]$-子模. 文章确定了有限链环$R$上一般指标广义拟循环码的生成元、极小生成元集以及广义拟循环码与对偶码之间生成元的关系.
Explicit generators of generalized quasi-cyclic (GQC) codes of general index over the finite chain ring are of great significance for determining the generator matrix, the duality, the enumeration, the minimum distance bound, the hull and the weight distribution of GQC codes. Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}+\cdots+u^{s-1}\mathbb{F}_{q}$, where $q$ is a prime power, $s$ is a positive integer, $s\geq2$ and $u^s=0$. A GQC code of block lengths $(r_{1},r_{2},\cdots,r_{l})$ with index $l$ over $R$ can be viewed as an $R[x]$-submodule of $R[x]/\langle x^{r_{1}}-1\rangle\times R[x]/\langle x^{r_{2}}-1\rangle\times\cdots\times R[x]/\langle x^{r_{l}}-1\rangle$. In this paper, we determine the generators and the minimum generating sets of GQC codes with general index over the finite chain ring $R$. We also determine the relationship of generators between GQC codes and their dual codes.

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