Let q = mt + 1 be a prime power, and let v( m, t) be the ( m + 1)-vector ( b 1, b 2, …, b m + 1 ) of elements of GF( q) such that for each k, 1 ⩽ k ⩽ m + 1, the set { b i − b j : i∈{1,2,… m+1} − { m + 2 − k}, j i + k(mod m + 2) and 1⩽ j⩽ m+1} forms a system of representatives for the cyclotomic classes of index m in GF( q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v( m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.