On higher dimensional cocyclic Hadamard matrices. (English) Zbl 1317.15028
Summary: Provided that a cohomological model for \(G\) is known, we describe a method for constructing a basis for \(n\)-cocycles over \(G\), from which the whole set of \(n\)-dimensional \(n\)-cocyclic matrices over \(G\) may be straightforwardly calculated. Focusing in the case \(n=2\) (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative \(2\)-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When \(n>2\), this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for \(n=2,3\). In particular, we give some examples of improper 3-dimensional \(3\)-cocyclic Hadamard matrices.
MSC:
| 15B34 | Boolean and Hadamard matrices |
References:
| [1] | Álvarez, V., Armario, J.A., Frau, M.D., Gudiel, F.: The maximal determinant of cocyclic \[(-1,1)(-1,1)\]-matrices over \[D_{2t}\] D2t. Linear Algebra Appl. 436, 858-873 (2012) · Zbl 1236.05043 · doi:10.1016/j.laa.2011.05.018 |
| [2] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: A genetic algorithm for cocyclic Hadamard matrices. AAECC-16 Proc. LNCS 3857, 144-153 (2006) · Zbl 1125.05304 |
| [3] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: Calculating cocyclic Hadamard matrices in mathematica: exhaustive and heuristic searches. ICMS-2 Proc. LNCS 4151, 422 (2006) · Zbl 1229.05053 |
| [4] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: A Mathematica notebook for computing the homology of iterated products of groups. ICMS-2 Proc. LNCS 4151, 57 (2006) · Zbl 1230.20050 |
| [5] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: http://library.wolfram.com/infocenter/MathSource/6516/ · Zbl 0475.05021 |
| [6] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: http://library.wolfram.com/infocenter/MathSource/6384/ |
| [7] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: http://library.wolfram.com/infocenter/MathSource/6621/ |
| [8] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: A system of equations for describing cocyclic Hadamard matrices. J. Comb. Des. 16, 276-290 (2008) · Zbl 1182.05019 · doi:10.1002/jcd.20191 |
| [9] | 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 · doi:10.1016/j.jsc.2007.06.009 |
| [10] | Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: (Co)homology of iterated semidirect products of abelian groups. Appl. Algebra Eng. Commun. Comput. 23, 101-127 (2012) · Zbl 1271.20063 · doi:10.1007/s00200-012-0163-y |
| [11] | Baliga, A., Horadam, K.J.: Cocyclic Hadamard matrices over \[{\bf Z}_t \,\times \,{\bf Z}_2^2\] Zt×Z22. Australas. J. Comb. 11, 123-134 (1995) · Zbl 0838.05017 |
| [12] | de Launey, \[W.: (0, G)\](0,G)-Designs with applications. Ph.D. Thesis. University of Sydney, Sydney, Australia (1987) · Zbl 0659.05034 |
| [13] | de Launey, W.: On the construction of \[n\] n-dimensional designs from 2-dimensional desgins. Australas. J. Comb. 1, 67-81 (1990) · Zbl 0758.05034 |
| [14] | de Launey, W., Flannery, D.: Algebraic Design Theory. Mathematical Surveys and Monographs, 175. American Mathematical Society, Providence (2011) · Zbl 1235.05001 |
| [15] | 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 · doi:10.1007/BF01389357 |
| [16] | Eilenberg, S., Mac Lane, S.: On the groups \[H(\pi, n)H\](π,n) II. Ann. Math. 66, 49-139 (1954) · Zbl 0055.41704 · doi:10.2307/1969702 |
| [17] | Flannery, D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192, 749-779 (1997) · Zbl 0889.05032 · doi:10.1006/jabr.1996.6949 |
| [18] | Grabmeier, J.; Lambe, LA; Betten, A. (ed.); Kohnert, A. (ed.); Lave, R. (ed.); Wassermann, A. (ed.), Computing resolutions over finite \[p\] p-groups (2000), New York |
| [19] | Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading (1989) · Zbl 0668.00003 |
| [20] | Hadamard, J.: Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17, 240-246 (1893) · JFM 25.0221.02 |
| [21] | Hammer, J., Seberry, J.: Higher dimensional orthogonal designs and applications. IEEE Trans. Inf. Theory 27(6), 772-779 (1981) · Zbl 0475.05021 · doi:10.1109/TIT.1981.1056426 |
| [22] | Hedayat, A., Wallis, W.D.: Hadamard matrices and their applications. Ann. Stat. 6, 1184-1238 (1978) · Zbl 0401.62061 · doi:10.1214/aos/1176344370 |
| [23] | Horadam, K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton (2007) · Zbl 1145.05014 |
| [24] | Horadam, K.J., Lin, C.: Construction of proper higher dimensional Hadamard matrices from perfect binary arrays. JCMCC 28, 237-248 (1998) · Zbl 0917.05016 |
| [25] | Horadam, K.J., de Launey, W.: Generation of cocyclic Hadamard matrices. Computational algebra and number theory (Sydney, 1992). volume 325 of Math. Appl., 279-290. Kluwer Acad. Publ., Dordrecht (1995) · Zbl 0838.05018 |
| [26] | Kotsireas, IS; Borwein, JM (ed.); etal., Structured Hadamard conjecture, 215-227 (2013), New York · Zbl 1268.05037 |
| [27] | Shlichta, P.J.: Three and four-dimensional Hadamard matrices. Bull. Am. Phys. Soc. 1 16, 825-826 (1971) |
| [28] | Shlichta, P.J.: Higher dimensional Hadamard matrices. IEEE Trans. Inf. Theory IT-25, 566-572 (1979) · Zbl 0436.05009 · doi:10.1109/TIT.1979.1056083 |
| [29] | Veblen, O.: Analisis Situs, 5th edn. A.M.S. Publications, Providence (1931) · JFM 57.0712.01 |
| [30] | Yang, Y.X.: The proofs of some conjectures on higher dimensional Hadamard matrices. Kexue Tongbao (English translation) 31, 1662-1667 (1986) |
| [31] | Yang, Y.X.: Theory and Applications of Higher-Dimensional Hadamard Matrices. Combinatorics and Computer Science Series. Science Press/Kluwer, Beijing/Dordrecht (2001) · Zbl 1044.05001 |
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