On generalizations of repeated-root cyclic codes

We first consider repeated-root cyclic codes, i.e., cyclic codes whose block length is divisible by the characteristic of the underlying field. It is well known that the formula for the minimum distance of repeated-root cyclic codes is similar to that for generalized concatenated codes. We show that indecomposable repeated-root cyclic codes are product codes and that the minimum weight of each repeated-root cyclic code is attained by one of its subcodes being equivalent to a product code. We then generalize the coding theoretical results on repeated-root cyclic codes to a larger class of left ideals in group algebra F_pm𝒢 defined on non-Abelian groups, namely, groups 𝒢 containing a normal cyclic Sylow p-subgroup. We show that a class of these codes compares reasonably to (shortened) generalized Reed-Muller codes over the primes and finally indicate by the special linear group SL₂(F_p ) how a further generalization may in principle be settled