On the minimum weights of binary linear complementary dual codes

TL;DR

This paper investigates the maximum minimum weight of binary linear complementary dual codes, providing exact values for specific parameters and expanding known results for codes up to length 24.

Contribution

It determines the largest minimum weight $d(n,k)$ for certain lengths and dimensions of binary linear complementary dual codes, extending existing knowledge.

Findings

Exact values of $d(n,4)$ for specified congruences.

Exact values of $d(n,5)$ for specified congruences.

Determined $d(n,k)$ for all $n \

Abstract

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We determine $d(n,4)$ for $n \equiv 2,3,4,5,6,9,10,13 \pmod{15}$, and $d(n,5)$ for $n \equiv 3,4,5,7,11,19,20,22,26 \pmod{31}$. Combined with known results, the values $d(n,k)$ are also determined for $n \le 24$.