On subsets of GF(q2) with dth power differences

Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730–737) conjectured that the only q-subset X of GF(q2), with the properties 0,1∈X and x−y is a square for all x,y∈X, is the set GF(q). Aart Blokhuis (Indag. Math. 46 (1984) 369–372) proved the conjecture for arbitrary odd q. In this paper we give a similar characterization of GF(q) in GF(q2), proving the analogue of Blokhuis’ theorem for dth powers (instead of squares), when d|(q+1). We also prove an embedding-type result, stating that if |S|>q−(1−1/d)q with the same properties as X above, then S⊆GF(q).