Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

  • Authors:

  • Guobiao Weng;Weisheng Qiu;Zeying Wang;Qing Xiang

  • Affiliations:

  • LMAM, School of Mathematical Sciences, Peking University, Beijing, People's Republic of China 100871;LMAM, School of Mathematical Sciences, Peking University, Beijing, People's Republic of China 100871;Department of Mathematical Sciences, University of Delaware, Newark, USA 19716;Department of Mathematical Sciences, University of Delaware, Newark, USA 19716

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2007

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Abstract

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 = 34, 36, 38, 310, 54, and 74. 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 of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.