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Existence of (q, 7, 1) difference families with q a prime power
Author(s) -
Chen K.,
Wei R.,
Zhu L.
Publication year - 2002
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.998
Subject(s) - mathematics , prime power , combinatorics , prime (order theory) , character (mathematics) , discrete mathematics , arithmetic , geometry
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