Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

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 mod-ulo 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 pranks of these pseudo-Paley graphs when q = 3 4 , 3 6 , 3 8 , 3 10 , 5 4 , and 7 4. The prank 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. of nonprime order are uniquely determined by their parameters and the minimality of their relevant pranks .

DOI: 10.1007/s10623-007-9057-6

Extracted Key Phrases

1 Figure or Table

Showing 1-10 of 18 references

Finite semifields In: Finite geometries, groups, and computation

  • Wm Kantor
  • 2006

Finite geometries. Reprint of the 1968 original. Classics in mathematics

  • P Dembowski
  • 1997

Ranks and structure of graphs All self-complementary symmetric graphs

  • R Peeters
  • 1995

Uniqueness of strongly regular graphs having minimal p-rank

  • R Peeters
  • 1995

A survey of partial difference sets

  • Sl Ma
  • 1994