Elsevier

Discrete Mathematics

Volume 103, Issue 1, 25 May 1992, Pages 75-90
Discrete Mathematics

Supplementary difference sets and Jacobi sums

Dedicated to Professor R.G. Stanton.
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Abstract

Let q=e∝+1 be an odd prime power and Ci, 1⩽ie-1, be cyclotomic classes of the eth power residues in F=GF(q). Let Ai with #Ai=ui, 1⩽in, be non-empty subsets of Ω={0,1,…,e−1} and let Di=∪lϵAiCl, 1⩽in. Here we prove that D1,…, D n become n−{q:u1∝, u2∝, …, un∝;λ} supplementary difference sets if and only if the following equations are satisfied:

  • 1.

    (i) Σni=1 u i(ui∝−1)≡0 (mod e), (ii) Σni=1 Σe−1m=0π(χm, χt)ωi,mωi,tm=0, for all t, 1 ⩽te−1, where π(χm, χt) is the Jacobi sum for the eth power residue characters and ωi,m=ΣlϵAiζlme, 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.

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Current address: Department of Mathematics, Kyushu University, Hakozaki, Higashiku, Fukuoka 812, Japan