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 \(G \cong {\mathbb Z}_2 \times {\mathbb Z}_2\) , and 13 nets with \(G \cong {\mathbb Z}_4\) . 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 \({\mathbb Z}_4\) -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 \({\mathbb F}_4\) -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.
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References
E. F. Assmus J. D. Key (1992) Designs and their codes Cambridge University Press Cambridge
T. Beth D. Jungnickel H. Lenz (1999) Design Theory EditionNumber2 Cambridge University Press Cambridge
A. Bonnecaze P. Solé C. Bachoc B. Mourrain (1997) ArticleTitleType II codes over \({\mathbb Z}_4\) IEEE Trans. Inform. Theory. 43 969–976
W. Bosma J. Cannon (1999) Handbook of Magma Functions, School of Mathematics and Statistics University of Sydney Sydney
A. E. Brouwer (1998) Bounds on the size of linear codes V. S. Pless W. C. Huffman (Eds) Handbook of Coding Theory Elsevier Amsterdam 295–461
A. R. Calderbank E. M. Rains P. W. Shor N. J. A. Sloane (1998) ArticleTitleQuantum error correction via codes over GF(4) IEEE Trans. Inform. Theory. 44 1369–1387
C. J. Colbourn J. H. Dinitz (Eds) (1996) The CRC Handbook of Combinatorial Designs CRC Press New York
J. H. Conway V. Pless N. J. A. Sloane (1992) ArticleTitleThe binary self-dual codes of length up to 32: a revised enumeration J. Comb. Theory, Ser. A. 60 183–195
S. T. Dougherty M. Harada P. Solé (2001) ArticleTitleShadow codes over \({\mathbb Z}_4\) Finite Fields Appl. 7 507–529
J. Doyen X. Hubaut M. Vandensavel (1978) ArticleTitleRanks of incidence matrices of Steiner triple systems Math. Z. 163 251–259
J. Fields P. Gaborit J. S. Leon V. Pless (1998) ArticleTitleAll self-dual \({\mathbb Z}_4\) codes of length 15 or less are known IEEE Trans. Inform. Theory 44 311–322
N. Hamada (1973) ArticleTitleOn the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes Hiroshima Math. J. 3 153–226
N. Hamada H. Ohmori (1975) ArticleTitleOn the BIB designs having minimum p-rank, J. Comb. Theory, Ser. A. 18 131–140
A. S. Hedayat N. J. A. Sloane J. Stufken (1999) Orthogonal Arrays: Theory and Applications Springer Berlin
D. Jungnickel (1984) ArticleTitleThe number of designs with classical parameters grows exponentially Geom. Dedicata 16 167–178
C. Lam (1997) Computer Construction of block designs in Surveys in Combinatorics Bailey (Eds) London Mathematical Society Lecture Note Series, Vol. 241 Cambridge University Press Cambridge 51–66
C. Lam S. Lam V. Tonchev (2000) ArticleTitleBounds on the number of affine, symmetric and Hadamard designs and matrices J. Combin. Theory, Ser. A. 92 186–196
C. Lam and V. D. Tonchev, Classification of affine resolvable 2-(27,9,4) designs, J. Statist. Plann. Inference 56 (1996), 187--202, Corrigendum: J. Statist. Plann. Inference, Vol. 86 (2000) pp. 277--278.
V. C. Mavron (1977) ArticleTitleTranslations and parallel classes of lines in affine designs J. Combin. Theory, Ser. A. 22 322–330
V. C. Mavron (1984) ArticleTitleOn complete nets which have no symmetric subnets Mitt. Mathem. Sem. Giessen 165 83–91
V. C. Mavron V. D. Tonchev (2000) ArticleTitleOn symmetric nets and generalized Hadamard matrices from affine designs J. Geometry 67 180–187
V. D. Tonchev (1986) ArticleTitleQuasi-symmetric 2-(31,7,7) designs and a revision of Hamada’s conjecture J. Combin. Theory, Ser. A. 42 104–110
V. D. Tonchev (1999) ArticleTitleLinear perfect codes and a characterization of the classical designs Designs, Codes and Cryptography 17 121–128
L. Teirlinck (1980) ArticleTitleOn projective and affine hyperplanes J. Combin. Theory, Ser. A. 28 290–306
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communicated by D. Jungnickel
Vladimir D. Tonchev-Research of this author sponsored by the National Security Agency under Grant MDA904-03-1-0088.
classification 5B, 51E, 94B
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Harada, M., Lam, C. & Tonchev, V.D. Symmetric (4,4)-Nets and Generalized Hadamard Matrices Over Groups of Order 4. Des Codes Crypt 34, 71–87 (2005). https://doi.org/10.1007/s10623-003-4195-y
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DOI: https://doi.org/10.1007/s10623-003-4195-y