Existence of (q, 7, 1) difference families with q a prime power

The existence of a ( q,k , 1) difference family in GF ( q ) has been completely solved for k  = 3,4,5,6. For k  = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a ( q ,7,1) difference family in GF ( q ), i.e. q  ≡ 1; (mod 42) is also sufficient except for q  = 43 and possibly except for q  = 127, q  = 211, q  = 31 6 and primes q ∈ [261239791, 1.236597 × 10 13 ] such that $(-3)^{q-1\over 14} = 1$ in GF ( q ). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998