Improving Wilson’s bound on difference families - Chen K; Zhu L. (Utilitas Mathematica, 1999)
For general k, Wilson's bound shows that there exists a (q, k, 1) difference family in GF(q) for any prime power q ≡ 1 (mod k(k - 1)) and q > mk(k-1) where m = k(k-1)/2. In this article, we use Weil's theorem on character sums to improve this bound. It can be lowered to be q > D(k) = [E+√E2+4F/2]2, where (i) when k = 2a + 1, E = 2[((a2 - 3a + 4)aa - 1/2 (a-1)3aa-1 + (4a-3)(a - 1)a-1)ma + (k -4)aa+1(m - 1)ma-1] - 5aa + 1 and F = [dk + 2(k - 5)a + 4]ma-1aa (ii) when k = 2a, E = 2[((a - 2)(a - 1)aa - 1/2(a - 1)3aa-1 + (4a -3)(a - 1)a-1)ma-1 + (k - 3)(a - 1)aa(m - 1)ma-2] - aa + 1 and F = [dk + 2(k - 3)(a - 1)]ma-2aa, where dk = 0 if k ≤ 6, or dk = m(a - 3) otherwise.