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

TL;DR

This paper determines the exact minimum weights of binary LCD codes with length n and dimension 6 for large n, using advanced coding theory techniques, and identifies cases where the minimum weight differs from optimal linear codes.

Contribution

The paper precisely computes the minimum weights of [n,6] LCD codes for n ≥ 51, extending known results and analyzing specific parameter cases with new theoretical methods.

Findings

Exact values of d_l(n,6) for most n ≥ 51.

Identification of parameter cases with minimum weight close to optimal.

Development of new theoretical tools for code nonexistence and construction.

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$t\in{25,33,45\}$.