Generalized twisted centralizer codes

Abstract

An important code of length $n^2$ is obtained by taking centralizer of a square matrix over a finite field $\mathbb{F}_q$. Twisted centralizer codes, twisted by an element $a \in \mathbb{F}_q$, are also similar type of codes but different in nature. The main results were embedded on dimension and minimum distance. In this paper, we have defined a new family of twisted centralizer codes namely generalized twisted centralizer (GTC) codes by $\mathcal{C}(A,D):= \lbrace B \in \mathbb{F}_q^{n \times n}|AB=BAD \rbrace$ twisted by a matrix $D$ and investigated results on dimension and minimum distance. Parity-check matrix and syndromes are also investigated. Length of the centralizer codes is $n^2$ by construction but in this paper, we have constructed centralizer codes of length $(n^2-i)$, where $i$ is a positive integer. In twisted centralizer codes, minimum distance can be at most $n$ when the field is binary whereas GTC codes can be constructed with minimum distance more than $n$.