Splittings of cyclic groups and perfect shift codes

A splitting (M,S) of an additive Abelian group G consists of a set of integers M and a subset S⊂G such that every nonzero element g∈G can be uniquely written as m·h for some m∈M and h∈S. Splittings M={±1,···,±k} correspond to perfect k-shift codes used in the analysis of run-length-limited codes correcting single peak shifts. We shall determine the set S for splittings of cyclic groups Z_p, p prime, by M={1,a,···,a^r ,b,···,b^s} and M={±1,±a,···,±a^r ,±b,···,±b^s}. This yields new conditions on the existence of perfect 3- and 4-shift codes. Further, it can be shown that splittings of Z_p by {±1,±2,±3} exist exactly if Z_p is also split by {1,2,3}