×

Construction of generalized Hadamard matrices \(D(r^m(r+1),r^m(r+1);p)\). (English) Zbl 0996.05020

A difference matrix \(D(\lambda p,m;p)\) is a \(\lambda p \times m\) matrix, with entries in \(G\), where \(G\) is a finite abelian group of order \(p\), such that the vector difference of any two columns contains each element of \(G\) \(\lambda \) times. A generalized Hadamard matrix is a difference matrix with \(\lambda p =m.\) The authors give a method for constructing generalized Hadamard matrices \(D(r^m(r+1),r^m(r+1);p)\) for \(n \geq 1\) from known \(D(r,r;r)\) and \(D(r+1,r+1;p)\) generalized Hadamard matrices. The orthogonal arrays \(L_{r^m(r+1)_p}((r^m(r+1))^1p^{r^m(r+1)})\) are equivalent to these constructed Hadamard matrices. These structures are then used to construct other difference matrices and orthogonal arrays using orthogonal decomposition of projection matrices.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B15 Orthogonal arrays, Latin squares, Room squares
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1986), Cambridge University Press: Cambridge University Press Cambridge
[2] Bose, R. C.; Bush, K. A., Orthogonal arrays of strength two and three, Ann. Math. Statist., 23, 508-524 (1952) · Zbl 0048.00803
[3] Dawson, J. E., A construction for generalized Hadamard matrices GH \((4q\),,EA \((q))\), J. Statist. Plann. Inference, 11, 103-110 (1985) · Zbl 0574.62073
[4] de Launey, W., A survey of generalized Hadamard matrices \(D(k,λ ; G)\) with large \(k\), Utilitas Math., 30, 5-29 (1986) · Zbl 0597.05016
[5] de Launey, W., 1988. (0,\(G\); de Launey, W., 1988. (0,\(G\)
[6] de Launey, W., GBRD’s: some new constructions for difference matrices, generalized Hadamard matrices and balanced weighing matrices, Graphs Combin., 5, 125-135 (1989) · Zbl 0687.05011
[7] de Launey, W.; Dawson, J. E., A note on the construction of GH(4t \(q\); EA \((q))\) for \(t=1,2\), Australas. J. Combin., 6, 177-186 (1992) · Zbl 0770.05023
[8] de Launey, W.; Dawson, J. E., An asymptotic result on the existence of generalised Hadamard matrices, J. Combin. Theory, 65, 158-163 (1994) · Zbl 0792.05033
[9] Jiang, S., A simple method for \(2p\)×\(2p\) difference sets, \(p\) any odd prime, Acta Math. Appl. Sinica, 2, 1, 179-184 (1979)
[10] Jungnickel, D., On difference matrices, resolvable transversal designs and generalized Hadamard matrices, Math. Z, 167, 49-60 (1979) · Zbl 0387.05003
[11] Liu, Z. W., For the odd prime \(p\) the construction of difference matrix for OA \((2p^2,2p+1; p,2)\), Acta Math. Appl. Sinica, 3, 35-45 (1977)
[12] Masuyama, M., On difference sets for constructing orthogonal arrays of index two and of strength two, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 5, 27-34 (1957)
[13] Masuyama, M., Construction of difference sets for OA \((2p^2,2p+1;p,2) p\) being an odd prime, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 16, 1-9 (1969) · Zbl 0204.52701
[14] Masuyama, M., Construction of difference sets for OA \((2p^2,2p+1;p,2), p\) being an odd prime, Rep. Stat. Appl. Res. JUSE, 16, 1-9 (1969) · Zbl 0204.52701
[15] Seberry, J., A construction for generalized Hadamard matrices, J. Statist. Plann. Inference, 4, 365-368 (1980) · Zbl 0458.62068
[16] Seiden, E., On the problem of construction of orthogonal arrays, Ann. Math. Statist., 25, 151-156 (1954) · Zbl 0055.00703
[17] Shrikhande, S., Generalized Hadamard matrices and orthogonal arrays strength two, Canad. J. Math., 16, 736-740 (1964) · Zbl 0123.00301
[18] Street, D. J., Generalized Hadamard matrices, Orthogonal arrays and \(F\)-squares, Ars Combin., 8, 131-141 (1979) · Zbl 0449.05009
[19] Street, D.J., 1979b. Cyclotomy and Designs. Ph.D. Thesis, University of Sydney, 1981.; Street, D.J., 1979b. Cyclotomy and Designs. Ph.D. Thesis, University of Sydney, 1981. · Zbl 0479.05008
[20] Xiang, K. F., The difference set table \(λ = 2\), Acta Math. Appl. Sinica, 5, 1, 160-166 (1983) · Zbl 0524.05013
[21] Xu, C. X., The construction of the \(OAL2p^u (p^{1+∑i=1^{u−1}p^u\)
[22] Zhang, Y. S., Asymmetrical orthogonal array with run size 100, Chinese Sci. Bull., 23, 1835-1836 (1989)
[23] Zhang, Y. S., Orthogonal arrays with run size 36, J. Henan Normal University, 4, 1-5 (1990)
[24] Zhang, Y. S., Orthogonal array \(L_{100}(20^15^{20})\), J. Henan Normal University, 4, 93 (1990)
[25] Zhang, Y. S., The orthogonal array \(L_{72}(24^13^{24})\), Chinese J. Appl. Statist. Manager, 3, 45 (1991)
[26] Zhang, Y. S., Asymmetrical orthogonal design by multi-matrix methods, J. Chinese Statist. Assoc., 29, 197-218 (1991)
[27] Zhang, Y. S., Orthogonal array and matrices, J. Math. Res. Exposition, 3, 438-440 (1992) · Zbl 0770.05022
[28] Zhang, Y.S., 1993. Theory of Multilateral Matrix. China Statistical Press.; Zhang, Y.S., 1993. Theory of Multilateral Matrix. China Statistical Press.
[29] Zhang, Y.S., Li, W.Q., Zheng, Z.G., 2000. Orthogonal arrays obtained by generalized Kronecker product. J. Statist. Plann., submitted for publication.; Zhang, Y.S., Li, W.Q., Zheng, Z.G., 2000. Orthogonal arrays obtained by generalized Kronecker product. J. Statist. Plann., submitted for publication.
[30] Zhang, Y.S., Lu, Y.Q., Duan, L., 1995. Construction of generalized Hadamard matrices \(Dr^mrr^{m\)
[31] Zhang, Y. S.; Lu, Y. Q.; Pang, S., Orthogonal arrays obtained by orthogonal decomposition of projection matrices, Statist. Sinica, 9, 595-604 (1999) · Zbl 0923.62084
[32] Zhang, Y. S.; Pang, S. Q.; Wang, Y. P., Orthogonal arrays obtained by generalized Hadamard product, Discrete Math., 238, 151-170 (2001) · Zbl 0981.62066
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.