A construction of generalized quasi-cyclic codes over finite field using gray map

Cyclic code is one of the important type of linear codes. This type of codes has interesting algebraic sturctures and important applications. One of the generalization of cyclic codes is quasi cyclic codes which also could be generalized further to be Generalized Quasi-Cyclic (GQC) code. The latter codes use arbitrary permutation instead of cyclic shift as in cyclic codes. The GQC code has been an interesting topic to study until now and has an important application in Post-Quantum Cryptography. For instance, it has been used as keys in McEliece cryptosystem. In this paper, we give a construction of q-ary GQC code using Gray Map from GQC code over the ring B1. The main result in this paper is if we have a GQC code over B1, then the image of such code under certain Gray map is a q-ary GQC code. Also, some examples of optimal codes produced by computer search based on the main result.