On the largest minimum distances of [n,6] LCD codes
Yang Liu, Ruihu Li
TL;DR
The paper determines the largest minimum distance $d_l(n,6)$ of binary LCD codes for all lengths $n\ge 51$ by combining nonexistence proofs (via defining vectors, generalized anti-codes, and reduced codes) with constructive protocols that piece together simplex-blocks and smaller LCD codes. It delivers exact values of $d_l(n,6)$ for most lengths $n=63s+t$, except a short set of congruence classes, and ties these to the corresponding optimal linear distances $d_a(n,6)$ through precise inequalities. The authors provide explicit constructions yielding optimal or near-optimal LCD codes for $51\le n\le 68$ and a general juxtaposition strategy that extends to all $n\ge 68$, illustrated by concrete examples. Together, these results advance the understanding of LCD-code parameters in dimension $6$, with clear implications for designing codes robust against side-channel and fault attacks. All mathematical expressions in the paper are maintained within $ $ delimiters to ensure precise interpretation.
Abstract
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$t\in{25,33,45\}$.