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

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### 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.

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###### Journal
1601.00144

Numerical methods, Information systems & theory, Algebra