Self-dual double cyclic codes over ${\mathbb{F}}_q$

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 ${\mathbb{F}}_{q,r,s}:={\mathbb{F}}_q[x]/⟨x^r-1⟩\times {\mathbb{F}}_q[x]/⟨x^s-1⟩$. 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.

Cryptography and Communications (accepted for publication)