Minimum distances of binary optimal LCD codes of dimension five are completely determined

TL;DR

This paper completely determines the minimum distances of binary optimal LCD codes of dimension five for a specific class of lengths, revealing their relationship with optimal linear codes and identifying exact distances.

Contribution

It provides a complete characterization of the minimum distances of binary optimal LCD codes of dimension five for lengths of the form 31s+t, filling a gap in the understanding of these codes.

Findings

Optimal LCD codes have minimum distances closely related to those of optimal linear codes.

For t ≠ 16, optimal LCD codes have minimum distance one less than optimal linear codes.

When t=16, the minimum distance of optimal LCD codes is exactly 16s+6, two less than the optimal linear codes.

Abstract

Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ and $d_{l}(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_{a}(n,5)]$ optimal linear code is not an LCD code, there is an $[n,5,d_{l}(n,5)]=[n,5,d_{a}(n,5)-1]$ optimal LCD code if $t\neq 16$, and an optimal $[n,5,d_{l}(n,5)]$ optimal LCD code has $d_{l}(n,5)=16s+6=d_{a}(n,5)-2$ for $t=16$. Combined with known results on optimal LCD code, $d_{l}(n,5)$ of all $[n,5]$ LCD codes are completely determined.