The maximal determinant of cocyclic (-1,1)-matrices over \(D_{2t}\). (English) Zbl 1236.05043
Summary: Cocyclic construction has been successfully used for Hadamard matrices of order \(n\). These (-1,1)-matrices satisfy that \(HH^{T}=H^{T}H=nI\) and give the solution to the maximal determinant problem when \(n=1,2\) or a multiple of 4. In this paper, we approach the maximal determinant problem using cocyclic matrices when \(n\equiv 2(\mod 4)\). More concretely, we give a reformulation of the criterion to decide whether or not the \(2t\times 2t\) determinant with entries \(\pm 1\) attains the Ehlich-Wojtas’ bound in the \(D_{2t}\)-cocyclic framework. We also provide some algorithms for constructing \(D_{2t}\)-cocyclic matrices with large determinants and some explicit calculations up to \(t=19\).
MSC:
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
Online Encyclopedia of Integer Sequences:
Numerator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.Denominator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.
Number of cocyclic matrices D_{2n+2} of order 2n+2 with maximal determinant.
References:
| [1] | V. Álvarez, J.A. Armario, M.D. Frau, P. Real, A Mathematica Notebook for Computing the Homology of Iterated Products of Groups, in: A. Iglesias, N. Takayama (Eds.), ICMS 2006, LNCS, vol. 4151, Springer Verlag, Heidelberg, 2006, pp. 47-57.; V. Álvarez, J.A. Armario, M.D. Frau, P. Real, A Mathematica Notebook for Computing the Homology of Iterated Products of Groups, in: A. Iglesias, N. Takayama (Eds.), ICMS 2006, LNCS, vol. 4151, Springer Verlag, Heidelberg, 2006, pp. 47-57. · Zbl 1230.20050 |
| [2] | Álvarez, V.; Armario, J. A.; Frau, M. D.; Real, P., A system of equations for describing cocyclic Hadamard matrices, J. Comb. Des., 16, 4, 276-290 (2008) · Zbl 1182.05019 |
| [3] | Álvarez, V.; Armario, J. A.; Frau, M. D.; Real, P., The homological reduction method for computing cocyclic Hadamard matrices, J. Symb. Comput., 44, 558-570 (2009) · Zbl 1163.05004 |
| [4] | Armario, J. A., On an inequivalence criterion for cocyclic Hadamard matrices, Cryptogr. Commun., 2, 247-259 (2010) · Zbl 1225.05053 |
| [5] | Barba, G., Intorno al teorema di Hadamard sui determinanti a volre massimo, Giornale di Matematiche di Battaglini, 71, 70-86 (1933) · JFM 59.0900.04 |
| [6] | Chadjipantelis, T.; Kounias, S.; Moyssiadis, C., The maximum determinant of \(21 \times 21(1, - 1)\)-matrices and D-optimal designs, J. Stat. Plann. Inference, 16, 167-178 (1987) · Zbl 0625.62062 |
| [7] | Cohn, J. H.E., On determinants with elements \(\pm 1\), II, Bull. London Math. Soc., 21, 36-42 (1989) · Zbl 0725.05025 |
| [8] | Cohn, J. H.E., On the number of D-optimal designs, J. Combin. Theory Ser. A, 66, 214-225 (1994) · Zbl 0802.05018 |
| [9] | Cohn, J. H.E., Almost D-optimal designs, Util. Math., 57, 121-128 (2000) · Zbl 0951.62063 |
| [10] | de Launey, W.; Horadam, K. J., A weak difference set construction for higher dimensional designs, Des. Codes Cryptogr., 3, 75-87 (1993) · Zbl 0838.05019 |
| [11] | Ehlich, H., Determiantenabschätzungen für binäre Matrizen, Math. Z., 83, 123-132 (1964) · Zbl 0115.24704 |
| [12] | Ehlich, H., Determiantenabschätzung für binäre Matrizen mit \(n \equiv 3 mod 4)\), Math. Z., 84, 438-447 (1964) · Zbl 0208.39803 |
| [13] | Fletcher, R. J.; Koukouvinos, C.; Seberry, J., New skew-Hadamard matrices of order \(4 \cdot 59\) and new \(D\)-optimal designs of order \(2 \cdot 59\), Discrete Matehmatics, 286, 252-253 (2004) · Zbl 1054.62081 |
| [14] | Galil, Z.; Kiefer, J., D-optimum weighing designs, Ann. Stat., 8, 1293-1306 (1980) · Zbl 0466.62066 |
| [15] | Hadamard, J., Résolution d’une question relative aux déterminants, Bull. Sci. Math., 2, 17, 240-246 (1893) · JFM 25.0221.02 |
| [16] | Holland, J. H., Adaptation in Natural and Artificial Systems (1975), University of Michigan Press: University of Michigan Press Ann Arbor · Zbl 0317.68006 |
| [17] | K.J. Horadam, W. de Launey, Cocyclic development of designs, J. Algebraic Combin. 2 (3) (1993) 267-290 (Erratum: J. Algebraic Combin. 3 (1) (1994) 129).; K.J. Horadam, W. de Launey, Cocyclic development of designs, J. Algebraic Combin. 2 (3) (1993) 267-290 (Erratum: J. Algebraic Combin. 3 (1) (1994) 129). · Zbl 0785.05019 |
| [18] | Horadam, K. J.; de Launey, W., Generation of Cocyclic Hadamard matrices, (Computational Algebra and Number Theory, Math. Appl. (1995), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 279-290 · Zbl 0838.05018 |
| [19] | Horadam, K. J., Hadamard Matrices and Their Applications (2007), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1198.15001 |
| [20] | Kharaghani, H.; Orrick, W. P., D-optimal designs, (Colbourn, C.; Dinitz, J., The CRC Handbook of Combinatorial Designs (2006), Taylor and Francis: Taylor and Francis Boca Raton) |
| [21] | Kharaghani, H.; Tayfeh-Rezaie, B., On the classification of Hadamard matrices of order 32, J. Comb. Des., 18, 328-336 (2010) · Zbl 1209.05037 |
| [22] | C. Koukouvinos, <http://math.ntua.gr/ ckoukouv/>; C. Koukouvinos, <http://math.ntua.gr/ ckoukouv/> |
| [23] | Koukouvinos, C.; Mitroulli, M.; Seberry, J., Bounds on the maximum determinant for \((- 1, 1)\) matrices, Bull. ICA, 29, 39-48 (2000) · Zbl 0960.05031 |
| [24] | MacLane, S., Homology. Homology, Classics in Mathematics (1995), Springer-Verlang: Springer-Verlang Berlin, (Reprint of the 1975 edition) |
| [25] | Michalewicz, Z., Genetic Algorithms+Data Structures=Evolution Programs (1992), Springer-Verlag · Zbl 0763.68054 |
| [26] | Moyssiadis, C.; Kounias, S., The exact D-optimal first order saturated design with 17 observations, J. Statist. Plann. Inference, 7, 13-27 (1982) · Zbl 0515.62072 |
| [27] | Ó Catháin, P.; Röder, M., The Cocyclic Hadamard matrices of order less than 40, Des. Codes Cryptogr., 58, 1, 73-88 (2011) · Zbl 1246.05033 |
| [28] | Orrick, W. P., The maximal \(\{- 1, 1 \}\)-determinant of order 15, Metrika, 62, 195-219 (2005) · Zbl 1078.62080 |
| [29] | Orrick, W. P., On the enumeration of some D-optimal designs, J. Statist. Plann. Inference, 138, 286-293 (2008) · Zbl 1133.05016 |
| [30] | W. Orrick, B. Solomon, The Hadamard Maximal Determinant Problem (website), <http://www.indiana.edu/maxdet/>; W. Orrick, B. Solomon, The Hadamard Maximal Determinant Problem (website), <http://www.indiana.edu/maxdet/> |
| [31] | Seberry, J.; Xia, T.; Koukouvinos, C.; Mitrouli, M., The maximal determinant and subdeterminants of \(\pm 1\) matrices, Linear Algebra Appl., 373, 297-310 (2003) · Zbl 1048.15008 |
| [32] | N.J.A. Sloane, A Library of Hadamard Matrices (website), <http://www2.research.att.com/ njas/hadamard/>; N.J.A. Sloane, A Library of Hadamard Matrices (website), <http://www2.research.att.com/ njas/hadamard/> |
| [33] | N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences (website), <http://oeis.org/>; N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences (website), <http://oeis.org/> · Zbl 1274.11001 |
| [34] | Tamura, H., D-optimal designs and group divisible designs, J. Comb. Des., 14, 451-462 (2006) · Zbl 1104.62087 |
| [35] | Wojtas, W., On Hadamard’s inequallity for the determinants of order non-divisible by 4, Colloq. Math., 12, 73-83 (1964) · Zbl 0126.02604 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
