Construction Methods for Galois LCD codes over Finite Fields

TL;DR

This paper introduces new construction methods for Galois LCD codes over finite fields, enabling the creation of codes with improved parameters and extending existing results to the $\sigma$-inner product.

Contribution

It presents novel methods for constructing Galois LCD codes from existing codes and extends known results to the $\sigma$-inner product setting.

Findings

Constructed new ternary LCD codes with better parameters for lengths 26 to 40.

Obtained optimal 2-Galois LCD codes over $ ext{F}_{2^3}$ for lengths up to 15.

Extended results to the $\sigma$-inner product from Euclidean inner product.

Abstract

In this article, first we present a method for constructing many Hermitian LCD codes from a given Hermitian LCD code, and then provide several methods which utilize either a given [n, k, d] linear code or a given [n, k, d] Galois LCD code to construct new Galois LCD codes with different parameters. Using these construction methods, we construct several new [n, k, d] ternary LCD codes with better parameters for $26\leq n \leq 40$, and $21 \leq k \leq 30$. Also, optimal 2-Galois LCD codes over $\mathbb{F}_{2^3}$ for code length, $1 \leq n \leq 15$ have been obtained. Finally, we extend some previously known results to the $\sigma$-inner product from Euclidean inner product.