Abstract
A weighing matrix of order n and weight m2 is a square matrix M of order n with entries from {-1,0,+1} such that MMT=m2I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.MT=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2pr and weight p2 where p is a prime and p≥ 5.
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References
M. H. Ang, Group Weighing Matrices, Ph.D. Thesis, National University of Singapore, (2003).
K. T. Arasu and J. F. Dillon, Perfect ternary arrays, in Difference sets, sequences and their correlation properties, NATO Adv.Sci.Inst.Ser.C Math.Phys.Sci., Vol.542, Kluwer Academic Publishers, Dordrecht, (1999) pp. 1–15.
K. T. Arasu S.L. Ma (2001) ArticleTitleSome new results on circulant weighing matrices J. Alg. Combin. 14 91–101 Occurrence Handle10.1023/A:1011903510338
K. T. Arasu J. Seberry (1996) ArticleTitleCirculant weighing matrices J. Combin. Designs 4 439–447 Occurrence Handle10.1002/(SICI)1520-6610(1996)4:6<439::AID-JCD4>3.0.CO;2-G
K. T. Arasu J. Seberry (1998) ArticleTitleOn circulant weighing matrices Australasian J. Combin. 17 21–37
K. T. Arasu D. Torban (1999) ArticleTitleNew weighing matrices of weight 25 J. Combin. Designs. 7 11–15 Occurrence Handle10.1002/(SICI)1520-6610(1999)7:1<11::AID-JCD2>3.0.CO;2-4
T. Beth D. Jungnickel H. Lenz (1999) Design Theory EditionNumber2 Cambridge University Press Cambridge
R. Craigen (1996) Weighing matrices and conference matrices C. J. Conlbourn J. H. Dinitz (Eds) The CRC Handbook of Combinatorial Designs. CRC Press Boca Raton 496–504
A. V. Geramita J. Seberry (1979) Orthogonal Designs: Quadratic Forms and Hadamard Matrices Marcel Dekker New York-Basel
X. D. Hou K. H. Leung S. L. Ma (2003) ArticleTitleOn the groups of units of finite commutative chain rings Finite Field Appl. 9 20–38 Occurrence Handle10.1016/S1071-5797(02)00003-5
K. H. Leung S. L. Ma (1990) ArticleTitleConstruction of partial difference sets and relative difference sets on p-groups Bull. London Math. Soc. 22 533–539
K. H. Leung S. L. Ma B. Schmidt (2002) ArticleTitleConstructions of relative difference sets with classical parameters and circulant weighing matrices J. Combin. Theory Series A 99 111–127 Occurrence Handle10.1006/jcta.2002.3262
B.R. McDonald (1974) Finite Rings with Identity Dekker Finite Rings with Identity
R. L. McFarland (1970) ArticleTitleA family of difference sets in non-cyclic groups J. Combin. Theory Ser. A 15 1–10 Occurrence Handle10.1016/0097-3165(73)90031-9
P. K. Menon (1962) ArticleTitleOn difference sets whose parameters satisfy a certain relation Proc. Amer. Math. Soc. 13 739–745
Y. Strassler (1997) The classification of circulant weighing matrices of weight EditionNumber9 Ph.D Thesis Bar-Ilan University
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Communicated by: K.T. Arasu
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Ang, M.H., Ma, S.L. Symmetric Weighing Matrices Constructed using Group Matrices. Des Codes Crypt 37, 195–210 (2005). https://doi.org/10.1007/s10623-004-3985-1
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DOI: https://doi.org/10.1007/s10623-004-3985-1