Name
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 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. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1007/s10623-003-4195-y | Des. Codes Cryptography |
Keywords | Field | DocType |
binary code,generalized hadamard matrix,binary linear code,smallest order,non-isomorphic affine,affine geometry,full automorphism group,quantum code,bitranslations g,hamada conjecture,pairwise non-isomorphic affine,classical affine design,affine design,inner product,lower bound,linear code,weight distribution,hadamard matrix | Affine geometry,Hadamard's maximal determinant problem,Discrete mathematics,Combinatorics,Hadamard matrix,Hadamard product,Hadamard three-lines theorem,Linear code,Hadamard's inequality,Complex Hadamard matrix,Mathematics | Journal |
Volume | Issue | ISSN |
34 | 1 | 0925-1022 |
Citations | PageRank | References |
15 | 1.89 | 11 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Masaaki Harada | 1 | 367 | 69.47 |
| Clement Lam | 2 | 59 | 9.25 |
| Vladimir D. Tonchev | 3 | 407 | 76.90 |