Explicit Determination of Certain Minimal Abelian Codes and Their Minimum Distances

Abstract

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.

Essential idempotents and simplex codes

“…Since in the case when char(F) | |A| the group algebra FA is semisimple and all ideals are direct sums of the minimal ones, it is only natural to study minimal abelian code -or -equivalently, primitive idempotents -and these has been done by several authors (see, for example [6], [5], [4] [9] [12] [14]). Also, Sabin and Lomonaco [15] have shown that central codes in metacyclic group algebras are equivalent to abelian codes.…”