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 ≅ 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 language | English |
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Article number | PIPS5384195 |
Pages (from-to) | 71-87 |
Number of pages | 17 |
Journal | Designs, Codes, and Cryptography |
Volume | 34 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2005 Jan |
Externally published | Yes |
Keywords
- Affine design
- Generalized Hadamard matrix
- Hamada conjecture
- Quantum code
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Applied Mathematics