Quasi-symmetric 2-(31, 7, 7) designs and a revision of Hamada's conjecture
Journal of Combinatorial Theory Series A
Design theory
The binary self-dual codes of length up to 32: a revised enumeration
Journal of Combinatorial Theory Series A
Designs and their codes
Linear Perfect Codes and a Characterization of the ClassicalDesigns
Designs, Codes and Cryptography - Special issue on designs and codes—a memorial tribute to Ed Assmus
Bounds on the number of affine, symmetric, and Hadamard designs and matrices
Journal of Combinatorial Theory Series A
IEEE Transactions on Information Theory
All self-dual Z4 codes of length 15 or less are known
IEEE Transactions on Information Theory
Quantum error correction via codes over GF(4)
IEEE Transactions on Information Theory
Finite Fields and Their Applications
Polarities, quasi-symmetric designs, and Hamada's conjecture
Designs, Codes and Cryptography
On generalized Hadamard matrices of minimum rank
Finite Fields and Their Applications
A Hamada type characterization of the classical geometric designs
Designs, Codes and Cryptography
| Hi-index | 0.00 |
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 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 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 F4-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.