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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, 45435, U.S.A

    K. T. Arasu

  2. Department of Mathematics, National University of Singapore, Kent Ridge, Singapore, 119260, Republic of Singapore

    S. L. Ma

Authors
  1. K. T. Arasu
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  2. S. L. Ma
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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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