Difference-covers that are not <i>k</i>-sum-covers. I

Abstract

Abstract Various constructions are described for sets F = F(q, k, l) of residues modulo q such that F–F contains all residues while (k)F = F + F + … + F ( k summands) omits l consecutive residues, in the cases k = 2, 3, 4 and 5. Let Q(k, l) denote the set of moduli q for which such a set F(q, k, l) exists. The following results are proved: q∈Q (2, 1) if and only if q ≥ 6; q∈Q (2, l ) if q ≥ 4 l + 2; 29, 31 and 33∈ Q (3, 1) and if q∈Q (3, 1) if q ≥ 35; 24 l + 21∈ Q (3, l ) if q ≥ 44 l + 41. Upper bounds are also obtained for the least elements in Q (4, l ) and Q (5, l ).