Constacyclic and Quasi-Twisted Hermitian Self-Dual Codes over Finite Fields

TL;DR

This paper investigates the structure, enumeration, and construction of Hermitian self-dual constacyclic and quasi-twisted codes over finite fields, introducing new algorithms, characterizations, and a family of MDS codes.

Contribution

It provides a new factorization algorithm for $x^n-\lambda$, characterizes Hermitian self-dual codes, and introduces a new family of MDS codes over finite fields.

Findings

Factorization algorithm for $x^n-\lambda$ over $_{q^2}$

Characterization and enumeration of Hermitian self-dual codes

Introduction of a new family of MDS constacyclic Hermitian self-dual codes

Abstract

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-\lambda$ over $\mathbb{F}_{q^2}$ is given, where $\lambda$ is a unit in $\mathbb{F}_{q^2}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda$-constacyclic codes of length $n$ over $\mathbb{F}_{q^2}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over $\mathbb{F}_{q^2}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$ is introduced. As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda$ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length some over extension fields of $\mathbb{F}_{q^2}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.