Supplementary difference sets and Jacobi sums

Abstract

Let q=e∝+1 be an odd prime power and Ci, 1⩽i⩽e-1, be cyclotomic classes of the eth power residues in F=GF(q). Let Ai with #Ai=ui, 1⩽i⩽n, be non-empty subsets of Ω={0,1,…,e−1} and let Di=∪lϵAiCl, 1⩽i⩽n. Here we prove that D1,…, Dn become n−{q:u1∝, u2∝, …, un∝;λ} supplementary difference sets if and only if the following equations are satisfied: 1.(i) Σni=1ui(ui∝−1)≡0 (mode), (ii) Σni=1Σe−1m=0π(χm, χ−t)ωi,mωi,t−m=0, for all t, 1 ⩽t⩽e−1, where π(χm, χ−t) is the Jacobi sum for the eth power residue characters and ωi,m=ΣlϵAiζ−lme, where ζe is a p rimitive eth root of unity. Furthermore, we give numerical results for e=2,n=1,2 and for e=4,n=1, 2, 3, 4.