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Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

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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.

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Guobiao Weng & Weisheng Qiu

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Zeying Wang & Qing Xiang

Authors
  1. Guobiao Weng
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  2. Weisheng Qiu
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  3. Zeying Wang
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  4. Qing Xiang
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Correspondence to Guobiao Weng.

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Dedicated to Dan Hughes on the occasion of his 80th birthday.

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Weng, G., Qiu, W., Wang, Z. et al. Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007). https://doi.org/10.1007/s10623-007-9057-6

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  • DOI: https://doi.org/10.1007/s10623-007-9057-6

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