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Abelian Difference Sets Without Self-conjugacy

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Abstract

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=qd+1( qd+ qd-1 +...+ q+2) ,k=qd( qd+qd-1+...+q+1) , λ = qd ( q(d-1)+q(d-2)+...+q+1), where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k-λ = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich's table with answer “no”.

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Arasu, K.T., Ma, S.L. Abelian Difference Sets Without Self-conjugacy. Designs, Codes and Cryptography 15, 223–230 (1998). https://doi.org/10.1023/A:1008323907194

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  • DOI: https://doi.org/10.1023/A:1008323907194

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