Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4

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 ≅ ℤ₂ × ℤ₂, and 13 nets with G ≅ ℤ₄. Using a (4, 4)-net with full automorphism group of smallest order, the lower bound on the number of pair-wise 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 ℤ₄-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 double struck F sign₄-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.