Existence of Abelian Group Code Partitions

Duadic codes were introduced by Leon et al. [5] as cyclic codes generalizing quadratic residue codes. Brualdi and Pless generalized them to polyadic cyclic codes [2] and Rushanan did so to duadic Abelian group codes [9]. Theorems concerning the existence of these codes in terms of field and group restrictions have been proved during this development, beginning with the one of Smid for cyclic duadic codes [10]. In this note we shall deal with a generalization of duadic codes for Abelian groups analogous to the cyclic m-adic residue codes in [2]. The goal is the necessary and sufficient conditions for existence presented in the Proposition and Theorem. The next section summarizes background for the lines we shall follow. Such background can be found in [8] and the book by Blake and Mullin [1], and it is surveyed in [12].