Hadamard difference sets and related combinatorial objects in groups of order 144

Abstract

In this paper we address an appealing and so far not completed combinatorial problem of difference set (DS) existence in groups of order 144. We apply our recently established method for DS construction which proves to be very efficient. The result is more than 5000 inequivalent (144, 66, 30) DSes obtained in 131 groups of order 144. The number of nonisomorphic symmetric designs rising from them is 1364. Using the obtained DSes as a source, new regular (144, 66, 30, 30) and (144, 65, 28, 30) partial difference sets are constructed, together with the corresponding strongly regular graphs. 43 non-isomorphic graphs of valency 66 are obtained and 78 of valency 65. The full automorphism groups of these graphs, as well as those of symmetric designs, are explored using the software package Magma.