Difference matrices with five rows over finite abelian groups

Let G be a finite group and k ⩾ 2 be an integer. A ( G , k , 1)-difference matrix (DM) is a k × | G | matrix D = ( d ij ) with entries from G , such that for all distinct rows x and y , the multiset of differences { d xi d yi - 1 : 1 ⩽ i ⩽ | G | } contains each element of G exactly once. This paper examines the existence of difference matrices with five rows over a finite abelian group. It is proved that if G is a finite abelian group and the Sylow 2-subgroup of G is trivial or noncyclic, then a ( G , 5, 1)-DM exists, except for G ∈ { Z 3 , Z 2 ⊕ Z 2 , Z 4 ⊕ Z 2 , Z 9 } and possibly for some groups whose Sylow 2-subgroup lies in { Z 2 ⊕ Z 2 , Z 4 ⊕ Z 2 , Z 32 ⊕ Z 2 , Z 16 ⊕ Z 4 } , and some cyclic groups of order 9 p with p prime.