标题： 有限域上的恒数循环和准扭 Hebert 自对偶码 显示英文标题

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

摘要： 研究了有限域上的恒同循环和拟扭 Hermitian 自对偶码。给出了在$\mathbb{F}_{q^2}$上分解$x^n-\lambda$的算法，其中$\lambda$是$\mathbb{F}_{q^2}$中的单位元。基于这种分解，确定了$\mathbb{F}_{q^2}$上长度为$n$的$\lambda$-恒同循环码的 Hermitian 壳的维数。 给出了长度为$n$且定义在$\mathbb{F}_{q^2}$上的常循环埃尔米特自对偶（相应地，互补对偶）码通过它们的埃尔米特 Hull 的刻画和计数。随后，引入了$\mathbb{F}_{q^2}$上的一类新的 MDS 常循环埃尔米特自对偶码。作为常循环码的推广，研究了拟循环埃尔米特自对偶码。利用$x^n-\lambda$的分解和中国剩余定理，拟循环码可以看作是较短长度线性码的乘积，其中一些定义在$\mathbb{F}_{q^2}$的扩张域上。给出了拟循环码成为埃尔米特自对偶的充要条件，并确定了此类自对偶码的计数。 显示英文摘要

摘要： 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.