Generalized twisted centralizer codes

TL;DR

This paper introduces generalized twisted centralizer (GTC) codes, expanding on existing codes by defining a new family with adjustable parameters, and investigates their dimension, minimum distance, and decoding properties.

Contribution

The paper defines a new family of GTC codes twisted by a matrix D, extending previous twisted centralizer codes, and explores their parameters and construction methods.

Findings

GTC codes can have minimum distance greater than n in binary fields.

Constructed codes of length n^2 - i, where i is a positive integer.

Analyzed dimension, minimum distance, parity-check matrix, and syndromes of GTC 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 $n$.