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Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. (English) Zbl 1126.05026

Summary: 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. 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 which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant \(p\)-ranks.

MSC:

05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B25 Combinatorial aspects of finite geometries
17A35 Nonassociative division algebras
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