Splittings of Cyclic Groups and Perfect Shift Codes

A splitting of an additive Abelian group G is a pair (M; S), where M is a set of integers and S is a subset of G such that every nonzero element g 2 G can be uniquely written as m h for some m 2 M and h 2 S. Splittings of groups by the set M = ff1; : : :; kg are intimately related to tilings of the n{ dimensional Euclidean space. Further, such a splitting corresponds to a perfect shift code used in the analysis of run{length limited codes correcting single peak shifts. We shall give the structure of the splitting set S for splittings of cyclic groups Z p of prime order by sets of the form