Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes

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Pseudocyclic and non-amorphic fusion schemes of the cyclotomic association schemes

We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov’s conjecture by Ikuta and Munemasa [13].

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Some implications on amorphic association schemes

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Davenport-Hasse theorem and cyclotomic association schemes

Definition. Let q be a prime power and e be a divisor of q − 1. Fix a generator α of the multiplicative group of GF (q). Then 〈α〉 is a subgroup of index e and its cosets are 〈α〉α, i = 0, . . . , e− 1. Define R0 = {(x, x)|x ∈ GF (q)} Ri = {(x, y)|x, y ∈ GF (q), x− y ∈ 〈αe〉αi−1}, (1 ≤ i ≤ e) R = {Ri|0 ≤ i ≤ e} Then (GF (q),R) forms an association scheme and is called the cyclotomic scheme of clas...

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Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a nonelementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer’s result. We use multiple distinct quadratic forms to constru...

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ژورنال

عنوان ژورنال: Designs, Codes and Cryptography

سال: 2011

ISSN: 0925-1022,1573-7586

DOI: 10.1007/s10623-011-9595-9