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

Masaaki Harada, Clement Lam, Vladimir D. Tonchev

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27 Citations (Scopus)


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.

Original languageEnglish
Article numberPIPS5384195
Pages (from-to)71-87
Number of pages17
JournalDesigns, Codes, and Cryptography
Issue number1
Publication statusPublished - 2005 Jan


  • Affine design
  • Generalized Hadamard matrix
  • Hamada conjecture
  • Quantum code


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