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/spl sub/G such that every nonzero element g/spl isin/G can be uniquely written as m/spl middot/h for some m/spl isin/M and h/spl isin/S. Splittings M={/spl plusmn/1,/spl middot//spl middot//spl middot/,/spl plusmn/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/sub p/, p prime, by M={1,a,/spl middot//spl middot//spl middot/,a/sup r/,b,/spl middot//spl middot//spl middot/,b/sup s/} and M={/spl plusmn/1,/spl plusmn/a,/spl middot//spl middot//spl middot/,/spl plusmn/a/sup r/,/spl plusmn/b,/spl middot//spl middot//spl middot/,/spl plusmn/b/sup s/}. This yields new conditions on the existence of perfect 3- and 4-shift codes. Further, it can be shown that splittings of Z/sub p/ by {/spl plusmn/1,/spl plusmn/2,/spl plusmn/3} exist exactly if Z/sub p/ is also split by {1,2,3}.