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 René 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.
It is proven that any Dembowski-Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar DembowskiOstrom polynomials. A direct connection between a planar Dembowski-Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition, all commutative semifields of order 3 5 are classified.
In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v 1 and v 2 with |v 1 − v 2 | = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte's theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3 5 or 3 7 , and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3 9 . Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3 11 .
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