We describe a class of codes over GF(q), where q is a power of an odd prime. These codes are analogs of the binary Reed-Muller codes and share several features in common with them. We determine the minimum weight and properties of these codes. For a subclass of codes we find the weight distribution and prove that the minimum nonzero weight codewords give 1-designs.

We construct a class of codes of length n such that the minimum distance d outside of a certain subcode is, up to a constant factor, bounded below by the square root of n, a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizations as special cases. We show that most of the special properties of these codes have analogues for split group codes, and present examples of new classes of codes obtained by this construction.