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

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.