On Some Ternary LCD Codes

TL;DR

This paper explores ternary LCD codes, providing an alternative proof of Massey's theorem, and investigates bounds on their minimum distance, especially when these bounds are achieved for q=3.

Contribution

It offers a new proof of Massey's theorem for LCD codes and analyzes conditions under which bounds are attained for ternary LCD codes.

Findings

Alternative proof of Massey's theorem for LCD codes

Characterization of maximum minimum distance for ternary LCD codes

Conditions for bounds attainment when q=3

Abstract

The main aim of this paper is to study $LCD$ codes. Linear code with complementary dual($LCD$) are those codes which have their intersection with their dual code as $\{0\}$. In this paper we will give rather alternative proof of Massey's theorem\cite{8}, which is one of the most important characterization of $LCD$ codes. Let $LCD[n,k]_3$ denote the maximum of possible values of $d$ among $[n,k,d]$ ternary $LCD$ codes. In \cite{4}, authors have given upper bound on $LCD[n,k]_2$ and extended this result for $LCD[n,k]_q$, for any $q$, where $q$ is some prime power. We will discuss cases when this bound is attained for $q=3$.