Abelian Difference Sets Without Self-conjugacy

Authors:
K. T. Arasu;S. L. Ma
Affiliations:
Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435, U.S.A.;Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore
Venue:
Designs, Codes and Cryptography
Year:
1998

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Abstract

Weobtain some results that are useful to the study of abelian differencesets and relative difference sets in cases where the self-conjugacyassumption does not hold. As applications we investigate McFarlanddifference sets, which have parameters of the form v=q^d+1\left( q^d+ q^d-1 +\cdots + q+2\right),k=q^d\left( q^d+q^d-1+\cdots +q+1\right), \lambda = q^d \left( q^{d-1}+q^{d-2}+\cdots+q+1\right) , where q is a prime power andd a positive integer. Using our results, we characterizethose abelian groups that admit a McFarland difference set oforder k-\lambda =81. We show that the Sylow 3-subgroupof the underlying abelian group must be elementary abelian. Ourresults fill two missing entries in Kopilovich‘s table with answer’’no‘‘.