We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of Fq of order N = 2p m 1 , where p1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.
Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2K!1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C R of length 2K!1 with de"ning zeros and R, where is a primitive element in GF(2K). There are many pairs (m, t) for which C, R is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases:m"2r#1 odd, t"2P#3, and (d) m odd, 4r,!1 mod m, t"2P#2P!1. We present a method of proving many of these known or conjectured results, including all of the above cases, in a uni"ed way.2001 Academic Press
We give two constructions of strongly regular Cayley graphs on finite fields F q by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.
Let (K, +, * ) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, * ) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter-Matthews presemifield and the Ding-Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 3 4 , 3 6 , 3 8 , 3 10 , 5 4 , and 7 4 . The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of Ren茅 Peeters [17, p. 47] which says that the Paley graphs Dedicated to Dan Hughes on the occasion of his 80th birthday. 50 Des. Codes Cryptogr. (2007) 44:49-62of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.
In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2 ). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2 ) admitting C (q 3 +1)/4 : C 3 .2010 Mathematics Subject Classification. 05B25 (primary), 05E30, 51E12 (secondary).
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