Improving Wilson's bound on difference families

Abstract

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 > m^k(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)a^a - 1/2 (a-1)³a^a-1 + (4a-3)(a - 1)^a-1)m^a + (k -4)a^a+1(m - 1)m^a-1] - 5a^a + 1 and F = [d_k + 2(k - 5)a + 4]m^a-1a^a (ii) when k = 2a, E = 2[((a - 2)(a - 1)a^a - 1/2(a - 1)³a^a-1 + (4a -3)(a - 1)^a-1)m^a-1 + (k - 3)(a - 1)a^a(m - 1)m^a-2] - a^a + 1 and F = [d_k + 2(k - 3)(a - 1)]m^a-2a^a, where d_k = 0 if k ≤ 6, or d_k = m(a - 3) otherwise.