Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Let (K,+, ∗) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, ∗) is a skewHadamard difference set or a Paley type partial difference set in (K,+) according as q is congruent to 3modulo 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 1modulo 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 Dedicated to Dan Hughes on the occasion of his 80th birthday. G. Weng (B) ·W. Qiu LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China e-mail: [email protected] Z. Wang ·Q. Xiang Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA W. Qiu e-mail: [email protected] Z. Wang e-mail: [email protected] Q. Xiang e-mail: [email protected] 50 Des. Codes Cryptogr. (2007) 44:49–62 of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.