F2[x]/〈x3-x〉上的CD编码的子字段编码

IF 0.7 3区数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-08-27 DOI:10.1016/j.disc.2024.114223

Anuj Kumar Bhagat, Ritumoni Sarma, Vidya Sagar

引用次数: 0

摘要

从有限维交换 F 代数到 F 域的非零 F 线性映射，如果其内核不包含任何非零理想，则称为 F 值迹。在本文中，我们利用 F2-代数 R2:=F2[x]/〈x3-x〉的 F2 值踪迹来研究 CD:={(x⋅d)d∈D:x∈R2m} 的二进制子域码 CD(2)，对于每个定义集 D 都是从某个单纯复数导出的。对于 m∈N 和 X⊆{1,2,...,m}，定义 ΔX:={v∈F2m:Supp(v)⊆X}和 D:=(1+u2)D1+u2D2+(u+u2)D3，R2m 的一个子集，其中 u=x+〈x3-x〉,D1∈{ΔL,ΔLc},D2∈{ΔM,ΔMc}和 D3∈{ΔN,ΔNc}，对于 L,M,N⊆{1,2,...,m}。这些二进制子字段码在 L、M 和 N 的卡片数的某些温和条件下是最小的。因此，我们得到了一些最小、自正交和距离最优的二进制线性编码无穷族，它们要么是 2 权码，要么是 4 权码。值得一提的是，我们还得到了几种新的距离最优二元线性编码。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Subfield codes of CD-codes over F2[x]/〈x3−x〉

A non-zero $F$ -linear map from a finite-dimensional commutative $F$ -algebra to the field $F$ is called an $F$ -valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an $F_{2}$ -valued trace of the $F_{2}$ -algebra $R_{2} : = F_{2} [x] / 〈 x^{3} - x 〉$ to study binary subfield code $C_{D}^{(2)}$ of $C_{D} : = {{(x \cdot d)}_{d \in D} : x \in R_{2}^{m}}$ for each defining set D derived from a certain simplicial complex. For $m \in N$ and $X \subseteq {1, 2, \dots, m}$ , define $Δ_{X} : = {v \in F_{2}^{m} : Supp (v) \subseteq X}$ and $D : = (1 + u^{2}) D_{1} + u^{2} D_{2} + (u + u^{2}) D_{3}$ , a subset of $R_{2}^{m}$ , where $u = x + 〈 x^{3} - x 〉, D_{1} \in {Δ_{L}, Δ_{L}^{c}}, D_{2} \in {Δ_{M}, Δ_{M}^{c}}$ and $D_{3} \in {Δ_{N}, Δ_{N}^{c}}$ , for $L, M, N \subseteq {1, 2, \dots, m}$ . The parameters and the Hamming weight distribution of the binary subfield code $C_{D}^{(2)}$ of $C_{D}$ are determined for each D. These binary subfield codes are minimal under certain mild conditions on the cardinalities of $L, M$ and N. Moreover, most of these codes are distance-optimal. Consequently, we obtain a few infinite families of minimal, self-orthogonal and distance-optimal binary linear codes that are either 2-weight or 4-weight. It is worth mentioning that we have obtained several new distance-optimal binary linear codes.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.

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