arXiv 2406.020652024-06-04cs.ITmath.IT

关于[n,6] LCD码的最大最小距离

On the largest minimum distances of [n,6] LCD codes

Yang Liu, Ruihu Li

备注：: optimal linear code, LCD code,generalized anti-code, defining vector, reduced code
发表信息：: Heliyon 2024
DOI：: 10.1016/j.heliyon.2024.e38238

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AI总结：

本文研究抵御攻击的线性互补对偶（LCD）码，利用定义向量和嵌套码等理论，精确确定了 $n\geq 51$ 时维度为 6 的 LCD 码最大最小距离 $d_{l}(n,6)$ 的值或取值范围。

摘要：

线性互补对偶（LCD）码可用于抵御侧信道攻击和故障非侵入式攻击。设 $d_{a}(n,6)$ 和 $d_{l}(n,6)$ 分别为所有长度为 $n$、维度为 6 的二元最优线性码和 LCD 码的最小权重。本文旨在通过研究给定参数下 LCD 码的不存在性与构造，求得 $n\geq 51$ 时 $d_{l}(n,6)$ 的值。设 $s \ge 0$ 和 $0\leq t\leq 62$ 为两个整数且 $n=63s+t$。利用定义向量、广义反码、缩减码和嵌套码理论，我们精确确定了 $t \notin\{21,22,25,26,33,34,37,38,45,46\}$ 时的 $d_{l}(n,6)$，同时证明了当 $t\in\{21,22,26,34,37,38,46\}$ 时 $d_{l}(n,6)\in\{d_{a}(n,6)-1,d_{a}(n,6)\}$，当 $t\in\{25,33,45\}$ 时 $d_{l}(n,6)\in\{d_{a}(n,6)-2,d_{a}(n,6)-1\}$。

英文摘要：

Linear complementary dual (LCD) codes can be used to against side-channel attacks and fault noninvasive attacks. Let $d_{a}(n,6)$ and $d_{l}(n,6)$ be the minimum weights of all binary optimal linear codes and LCD codes with length $n$ and dimension 6, respectively.In this article, we aim to obtain the values of $d_{l}(n,6)$ for $n\geq 51$ by investigating the nonexistence and constructions of LCD codes with given parameters. Suppose that $s \ge 0$ and $0\leq t\leq 62$ are two integers and $n=63s+t$. Using the theories of defining vectors, generalized anti-codes, reduced codes and nested codes, we exactly determine $d_{l}(n,6)$ for $t \notin\{21,22,25,26,33,34,37,38,45,46\}$, while we show that $d_{l}(n,6)\in$$\{d_{a}(n,6)$ $-1,d_{a}(n,6)\}$ for $t\in\{21,22,26,34,37,38,46\}$ and $ d_{l}(n,6)\in$$ \{d_{a}(n,6)-2,$ $d_{a}(n,6)-1\}$ for$t\in{25,33,45\}$.

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