Pseudo-Paley graphs and skew Hadamard difference sets from presemifields
Pseudo-Paley graphs and skew Hadamard difference sets from presemifields
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DOI：
10.1007/s10623-007-9057-6
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发表时间：
2007-09
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影响因子：
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通讯作者：
Guobiao Weng;W. Qiu;Zeying Wang;Qing Xiang
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文献类型：
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作者：
Guobiao Weng;W. Qiu;Zeying Wang;Qing Xiang
设（K，+，*）是具有交换乘法的奇阶预半域.证明了（K，*）的非零平方集是（K，+）中的斜Hadamard差集或Paley型偏差集，条件是q全等于3模4或q全等于1模4.将这个结果应用于Coulter-Matthews预半域及其Ding-Yuan变差，我们恢复了Ding和Yuan [7]最近构造的斜Hadamard差集.另一方面，将这一结果应用于已知的具有交换乘法且阶q同余于1模4的预半域，我们构造了几类伪Paley图.当q = 34，36，38，310，54和74时，我们计算这些伪Paley图的p-秩。p-秩结果表明这些图似乎是新的。沿着这条路，我们还反驳了René Peeters [17，p.47]的一个猜想，即非素序的Paley图是由它们的参数和它们的相关p-秩的最小性唯一确定的。
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 asqis congruent to 3 modulo 4 orqis 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 orderqcongruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute thep-ranks of these pseudo-Paley graphs whenq= 34, 36, 38, 310, 54, and 74. Thep-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 relevantp-ranks.