On minimum distance bounds for abelian codes

This paper is an exposition of two methods of formulating a lower bound for the minimum distance of a code which is an ideal in an abelian group ring. The first, a generalization of the cyclic BCH (Bose-Chaudhuri-Hoquenghem) bound, was proposed by Camion [2]. The second method, presented by Jensen [4], allows the application of the BCH bound or any of its improvements by viewing an abelian code as a direct sum of concatenations of cyclic codes. This second method avoids the mathematical analysis required for a direct generalization of a cyclic bound to the abelian case. It can produce a lower bound that improves the generalized BCH bound. We present simple algorithms for 1) deriving the generalized BCH bound for an abelian code 2) determining direct sum decompositions of an abelian code to concatenated codes and 3) deriving a bound on an abelian code, viewed as a direct sum of concatenated codes, by applying the cyclic BCH bound to the inner and outer code of each concatenation. Finally, we point out the applicability of these methods to codes that are not ideals in abelian group rings.