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

Ekkasit Sangwisut; Somphong Jitman; Patanee Udomkavanich

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arXiv · 1601.00144

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

Ekkasit Sangwisut·Somphong Jitman·Patanee Udomkavanich

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Abstract

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-λ$ over $\mathbb{F}_{q^2}$ is given, where $λ$ is a unit in $\mathbb{F}_{q^2}$. Based on this factorization, the dimensions of the Hermitian hulls of $λ$-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 $x^n-λ$ 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.

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Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. 2016-01-02. Constacyclic and Quasi-Twisted Hermitian Self-Dual Codes over Finite Fields. https://arxiv.org/abs/1601.00144

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