The nonexistence of (18, 3, 18, 6) relative difference sets

Abstract

It is known that relative difference sets with parameters (18, 3, 18, 6) in a group of order 54 with normal subgroup N of order 3 do not exist in any abelian group of order 54. In this paper, using the recent classification of all generalized Hadamard matrices of order 18 over a group of order 3 [5], we show that such relative difference sets do not exist in any non-abelian group of order 54 as well. Our results are validated computationally using the computer algebra system Magma.