On perfect t-shift codes in abelian groups

Let G be a finite abelian group, t a positive integer. The t-shift sphere with center x ∈G is the set S_t(x)={±ix|i=1,..., t}. A t-shift code is a subset X of G such that the sets S_t(x) (x ∈X) have size 2 t and are disjoint. Clearly, the sphere packing bound: 2 t|X|+1≤|G| holds for any t-shift code X. A perfect t-shift code is a t-shift code X with 2 t|X|+1=|G|. A necessary and sufficient condition for the existence of a perfect t-shift code in a finite abelian group is known for t-1, 2. In this paper, we determine finite abelian groups in which there exists a perfect t-shift code for t=3, 4.