🤖 AI Summary
This study addresses the existence and explicit construction of self-dual bicyclic codes over finite fields. By viewing bicyclic codes as submodules of a polynomial ring, the authors characterize their structure via pairs of generator polynomials and establish necessary and sufficient conditions for such codes to be self-dual. For the first time, systematic construction methods are proposed for specific code lengths—including $(r,r)$, $(r,2r)$, $(2r,r)$, and cases where $\gcd(r,s)=1$—by integrating finite field theory, the algebraic structure of cyclic codes, and properties of dual codes. Concrete examples are provided to demonstrate feasibility, and the work reveals intrinsic connections between these codes and other classes of self-dual codes, thereby filling a notable gap in the existing theoretical framework.
📝 Abstract
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.