On some characterizations of generalized quasi-cyclic codes over $ \mathbb{Z}_q $

Quasi-cyclic (QC) codes are a natural and remarkable generalization of cyclic codes. QC codes are further generalized into what are known as generalized quasi-cyclic (GQC) codes. In this paper we study GQC codes of index $ 2 $ over $ \mathbb{Z}_q $, where $ q = p^m $ is a prime power, by considering them as a special case of mixed alphabet codes. Following some recent works on codes over mixed alphabets, GQC codes of index $ 2 $ over $ \mathbb{Z}_q $ can be seen as $ \mathbb{Z}_q $-double cyclic codes. Such codes can be viewed as $ \mathbb{Z}_q[x] $-submodules of $ \mathbb{Z}_q[x]/(x^r-1)\times \mathbb{Z}_q[x]/(x^s-1) $. We determine explicitly the generator polynomials of this family of codes for two separate cases $ (r,q) = 1,(s,q) = 1 $ and $ (r,q)\neq1,(s,q)\neq1 $, and have given different characterizations of these codes. We have also obtained a minimal generating set for these codes. A new Gray map has been defined over $ \mathbb{Z}_q $ and some optimal $ p $-ary linear and nonlinear codes have been obtained through this Gray map.