A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes

Abstract

A new family of distance-regular graphs is constructed. They are antipodal 22t−1-fold covers of the complete graph on 22t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matricess and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.