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

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.