Multi-twisted codes over finite fields and their dual codes

TL;DR

This paper explores the algebraic structure, duality properties, and bounds of multi-twisted codes over finite fields, providing new criteria for self-duality, orthogonality, and code construction methods.

Contribution

It introduces a comprehensive analysis of $ ext{Lambda}$-multi-twisted codes, including conditions for self-duality, enumeration formulas, and a trace-based construction approach.

Findings

Necessary and sufficient conditions for self-dual codes.

Enumeration formulas for self-dual and self-orthogonal codes.

A BCH-type bound on minimum Hamming distance.

Abstract

Let $\mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,\cdots,m_{\ell}$ be positive integers satisfying $\gcd(m_i,q)=1$ for $1 \leq i \leq \ell,$ and let $n=m_1+m_2+\cdots+m_{\ell}.$ Let $\Lambda=(\lambda_1,\lambda_2,\cdots,\lambda_{\ell})$ be fixed, where $\lambda_1,\lambda_2,\cdots,\lambda_{\ell}$ are non-zero elements of $\mathbb{F}_{q}.$ In this paper, we study the algebraic structure of $\Lambda$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ and their dual codes with respect to the standard inner product on $\mathbb{F}_{q}^n.$ We provide necessary and sufficient conditions for the existence of a self-dual $\Lambda$-multi-twisted code of length $n$ over $\mathbb{F}_{q},$ and obtain enumeration formulae for all self-dual and self-orthogonal $\Lambda$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}.$ We also derive some sufficient conditions under which a $\Lambda$-multi-twisted code is LCD. We determine the parity-check polynomial of all $\Lambda$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some $\Lambda$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ from the generating sets of the codes. Besides this, we provide a trace description for all $\Lambda$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.