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.
