On the minimum weights of binary LCD codes and ternary LCD codes

TL;DR

This paper investigates the maximum possible minimum weights of binary and ternary LCD codes, providing partial and complete results for specific parameters to advance understanding of their optimal properties.

Contribution

It determines several new bounds and exact values for the largest minimum weights of binary and ternary LCD codes for various lengths and dimensions.

Findings

Partially determines $d_2(n,5)$ and $d_3(n,4)$.

Establishes $d_2(n,n-5)$, $d_3(n,n-i)$ for $i eq 1$.

Finds $d_3(n,k)$ for $n=11$ to $19$.

Abstract

Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.