Minimal Abelian Codes And Its G-Equivalence Classes In A Class Of Group Algebras

Let Fq be a finite field with q elements, Cm be a cyclic group with order m, where m is a positive integer. Let G=Cp2×Cp2+n be a finite abelian p-group, where p is an odd prime, and n is an non-negative integer.Let F2G be a group algebra over the binary field F2, where G=Cp2x Cp2+n is an abelian group. An arbitrary ideal in F2G is called an abelian code, and a minimal ideal is called a minimal abelian code. In this paper, we study minimal abelian codes over F2G, and we obtain generators of these minimal abelian codes. we also obtain the dimensions and the minimal Hamming weights of these minimal abelian codes.Furthermore, we study the G-equivalence classes of these minimal abelian codes, and we get G-equivalence classes of these minimal abelian codes except the G-equivalence classes of the minimal abelian codes which correspond to the co-cyclic subgroups of order p3over F2G.As we know if two minimal abelian codes are G-equivalent, then they have same Hamming weight distributions, we give an example to show that there exist two non G-equivalent minimal abelian codes over F2G which have same weight distributions.