Multi-twisted codes over finite fields and their dual codes

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 $1\le i\le \ell ,$ 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.