Self-dual Quasi-cyclic Codes Of Index 3

Let l be a nonzero natural number, and let R=Fq[x]/(xm-1), where Fq is a finite field. In this paper, we regard a quasicyclic code as a subrnodule of the algebra Re. We use Grobner bases of modules and algebraic structure of quasicyclic codes as tools to do some research on self-dual quasicyclic codes of index 3, and then we get two main theorems as follows:Theorem 3.1 elucidates how to have a formula, in the index 3 case, for the POT Grobner bases generating set in terms of an rPOT Grobner bases generating set; Theorem 3.5 elucidates the necessary and sufficient condition on which quasicyclic codes of index 3 are self-dua', which provides a complete characterisation of self-dual quasicyclic codes of index 3.