57 research outputs found
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Four-class Skew-symmetric Association Schemes
An association scheme is called skew-symmetric if it has no symmetric
adjacency relations other than the diagonal one. In this paper, we study
4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of
skew-symmetric amorphous association schemes, submitted for publication], we
discovered that their character tables fall into three types. We now determine
their intersection matrices. We then determine the character tables and
intersection numbers for 4-class skew-symmetric pseudocyclic association
schemes, the only known examples of which are cyclotomic schemes. As a result,
we answer a question raised by S. Y. Song [Commutative association schemes
whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996].
We characterize and classify 4-class imprimitive skew-symmetric association
schemes. We also prove that no 2-class Johnson scheme can admit a 4-class
skew-symmetric fission scheme. Based on three types of character tables above,
a short list of feasible parameters is generated.Comment: 12 page
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