Symmetric (4,4)-Nets and Generalized Hadamard Matrices Over Groups of Order 4

abstract

The symmetric class-regular (4,4)-nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with $G \cong {\mathbb Z}_2 \times {\mathbb Z}_2$ , and 13 nets with $G \cong {\mathbb Z}_4$ . Using a (4,4)-net with full automorphism group of smallest order, the lower bound on the number of pairwise non-isomorphic affine 2-(64,16,5) designs is improved to 21,621,600. The classification of class-regular (4,4)-nets implies the classification of all generalized Hadamard matrices (or difference matrices) of order 16 over a group of order four up to monomial equivalence. The binary linear codes spanned by the incidence matrices of the nets, as well as the quaternary and ${\mathbb Z}_4$ -codes spanned by the generalized Hadamard matrices are computed and classified. The binary codes include the affine geometry [64,16,16] code spanned by the planes in AG(3,4) and two other inequivalent codes with the same weight distribution.These codes support non-isomorphic affine 2-(64,16,5) designs that have the same 2-rank as the classical affine design in AG(3,4), hence provide counter-examples to Hamada’s conjecture. Many of the ${\mathbb F}_4$ -codes spanned by generalized Hadamard matrices are self-orthogonal with respect to the Hermitian inner product and yield quantum error-correcting codes, including some codes with optimal parameters.