有限域$\boldsymbol{\textsf{F}}_q$上的自对偶双循环码
Self-dual double cyclic codes over $\mathbb{F}_q$
AI总结：
本文研究有限域$\boldsymbol{\textsf{F}}_q$上的自对偶双循环码，给出其生成元的充要条件，针对特定长度考察存在性并给出构造方法，还梳理了其与其他自对偶码的关联。
AI中文摘要：
本文专门研究有限域$\boldsymbol{\textsf{F}}_q$上的自对偶双循环码，自对偶双循环码是与其对偶码相等的双循环码。结构上，有限域$\boldsymbol{\textsf{F}}_q$上长度为$(r,s)$的双循环码是$\boldsymbol{\textsf{F}}_{q,r,s}:=\boldsymbol{\textsf{F}}_q[x]/\boldsymbol{\text{理想}}\boldsymbol{\text{生成元}}\boldsymbol{\text{集}}\boldsymbol{\text{包含}}\boldsymbol{x^r-1}\boldsymbol{\text{的理想}}\times\boldsymbol{\textsf{F}}_q[x]/\boldsymbol{\text{理想}}\boldsymbol{\text{生成元}}\boldsymbol{\text{集}}\boldsymbol{\text{包含}}\boldsymbol{x^s-1}\boldsymbol{\text{的理想}}$的$\boldsymbol{\textsf{F}}_q[x]$子模，且有限域$\boldsymbol{\textsf{F}}_q$上任意长度为$(r,s)$的双循环码都由$\boldsymbol{\textsf{F}}_{q,r,s}$中的两对多项式生成。本文基于生成元的性质，给出$\boldsymbol{\textsf{F}}_{q,r,s}$中两对多项式生成自对偶码的充要条件，还针对特定长度$(r,r)$、$(r,2r)$与$(2r,r)$、以及$\boldsymbol{\text{最大公约数}}\boldsymbol{\text{gcd}}(r,s)=1$的$(r,s)$，考察自对偶双循环码的存在性，对每种情况给出构造方法并提供有限域上的具体实例，同时梳理自对偶双循环码与其他类自对偶码的关联。
英文摘要：
This article focuses specifically on the study of self-dual double cyclic codes over a finite field $\mathbb{F}_q$. A self-dual double cyclic code is a double cyclic code that is equal to its dual. Structurally, a double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is a $\mathbb{F}_q[x]$-submodule of $\mathbb{F}_{q,r,s}:=\mathbb{F}_q[x]/\langle x^r-1\rangle\times\mathbb{F}_q[x]/\langle x^s-1\rangle$. Moreover, any double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is generated by two pairs of polynomials in $\mathbb{F}_{q,r,s}$. From the properties of the generating elements, we provide the necessary and sufficient conditions such that two pairs of polynomials in $\mathbb{F}_{q,r,s}$ generate a self-dual code. Furthermore, we examine the existence of self-dual double cyclic codes for some specific lengths: $(r,r)$; $(r,2r)$ and $(2r,r)$; and $(r,s)$, where $\gcd(r,s)=1$. For each case, we provide a construction method with some explicit examples over various finite fields. We also observe some connections between self-dual double cyclic codes and other classes of self-dual codes.
