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Gauss sums, Jacobi sums, and \(p\)-ranks of cyclic difference sets. (English) Zbl 0943.05021
Recently, new cyclic difference sets with the classical parameters \((2^d-1,2^{d-1},2^{d-2})\) have been constructed using hyperovals. It was not clear whether these constructions yield inequivalent difference sets. The authors show that, with only a few exceptions, the Singer difference sets, the GMW difference sets and the recently constructed differences arising from hyperovals [see A. Maschietti, Difference sets and hyperovals, Des. Codes Cryptography 14, No. 1, 89-98 (1998; Zbl 0887.05010)] are inequivalent. In fact, the authors prove a stronger result: They show that the \(2\)-ranks of the difference set codes are different. In order to do this, they relate the ranks to the prime factorization of certain Gauss sums. They use Stickelberger’s theorem to translate the problem into a combinatorial question on binary strings. The results of the paper as well as the proofs are interesting and important. It should be noted that even more constructions of difference sets with the classical parameters have been found recently by J. F. Dillon and H. Dobbertin. Using \(2\)-ranks, Dillon and Dobbertin can also show that their new difference sets are inequivalent.

MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B25 Combinatorial aspects of finite geometries
51E21 Blocking sets, ovals, \(k\)-arcs
05A15 Exact enumeration problems, generating functions
11L05 Gauss and Kloosterman sums; generalizations
11T24 Other character sums and Gauss sums
94B15 Cyclic codes
51E30 Other finite incidence structures (geometric aspects)
Software:
gfun
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