Certain new M-matrices and their properties with applications

• Autorzy: Ratnakaram N. Mohan , Sanpei Kageyama , Moon H. Lee , G. Yang
• Języki publikacji: EN

Abstrakty

EN

The Mₙ-matrix was defined by Mohan [21] who has shown a method of constructing (1,-1)-matrices and studied some of their properties. The (1,-1)-matrices were constructed and studied by Cohn [6], Ehrlich [9], Ehrlich and Zeller [10], and Wang [34]. But in this paper, while giving some resemblances of this matrix with a Hadamard matrix, and by naming it as an M-matrix, we show how to construct partially balanced incomplete block designs and some regular graphs by it. Two types of these M-matrices have been considered. Also we will make a mention of certain applications of these M-matrices in signal and communication processing, and network systems and end with some open problems.

Słowa kluczowe

EN

M-matrices; non-orthogonality; orthogonal number; Hadamard matrix; partially balanced incomplete block (PBIB) design; regular graph

Kategorie tematyczne

• 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
• 05B05: Block designs

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae Probability and Statistics
• Rocznik: 2008
• Tom: 28
• Numer: 2
• Strony: 183-207

Daty

wydano

2008

otrzymano

2007-10-15

Twórcy

autor

Ratnakaram N. Mohan

• Sir CRR Institute of Mathematics, Eluru-534007, AP, India

autor

Sanpei Kageyama

• Hiroshima Institute of Technology, Hiroshima 731-5193, Japan

autor

Moon H. Lee

• Institute of Information & Communication, Chonbuk National University, Korea

autor

G. Yang

• Center for Combinatorics, Nankai University, Tianjin-300071, PR China

Bibliografia

• [1] B.E. Aupperle and J.F. Meyer, Fault-tolerant BIBD networks, Proc. Inter. Symp. (FTCS) 18, IEEE, (1988), 306-311.
• [2] L.N. Bhuyan and D.P. Agarwal, Design and performance of a general class of interconnection networks, IEEE Trans. Computers C-30 (1981), 587-590.
• [3] R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), 389-419.
• [4] R.C. Bose and K.R. Nair, Partially balanced incomplete block designs, Sankhyā 4 (1939), 337-372.
• [5] Y. Chang, Y. Hua, X.G. Xia and B. Sudler, An insight into space-time block codes using Hurwitz-Random families of matrices, (personal communication) 2005.
• [6] J.H.E. Cohn, Determinants with elements ± 1, J. London Math. Soc. 14 (1963), 581-588.
• [7] C. Colbourn, Applications of combinatorial designs in communications and networking, MSRI Project 2000.
• [8] C. Colbourn, J.H. Dinitz and D.R. Stinson, Applications of combinatorial designs in communications, cryptography and networking, Surveys in Combinatorics, 1993, Walker (Ed.), London Mathematical Society Lecture Note Series 187, Cambridge University Press 1993.
• [9] H. Ehlich, Determinantenabschätzungen für binäre matrizen, Math. Z. 83 (1964), 123-132.
• [10] H. Ehlich and K. Zeller, Binäre matrizen, Z. Angew. Math. Mech. 42 (1962), 20-21.
• [11] P. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press Ltd., Wiley, New York 1996.
• [12] A.V. Geramita and J. Seberry, Orthogonal Designs, Quadratic Forms and Hadamard Matrices, Marcel and Dekker, New York 1979.
• [13] J.M. Goethals and J.J. Seidel, Strongly regular graphs derived from combinatorial designs, Can. J. Math. 22 (1970), 597-614.
• [14] L.W. Hawkes, A regular fault-tolerant architecture for interconnection networks, IEEE Trans. Computers C-34 (1985), 677-680.
• [15] H. Jafarkhani, A quasi-orthogonal space-time block code, Preprint 2001.
• [16] S. Kageyama and R.N. Mohan, On μ-resolvable BIB designs, Discrete Math. 45 (1983), 113-121.
• [17] S. Kageyama and R.N. Mohan, On the construction of group divisible partially balanced incomplete block designs, Bull. Fac. Sch. Edu., Hrioshima Univ. 7 (1984), 57-60.
• [18] J. Kahn, J. Komlos and E. Szemeredi, On the probability that a random ± 1 matrix is singular, J. Amer. Math. Soc. 8 (1995), 223-240.
• [19] C.-W. Liu, A method of constructing certain symmetric partially balanced designs, Scientia Sinica 12 (1963), 1935-1936.
• [20] R.N. Mohan, A new series of μ-resolvable (d+1)-associate class GDPBIB designs, Indian J. Pure Appl. Math. 30 (1999), 89-98.
• [21] R.N. Mohan, Some classes of Mₙ-matrices and Mₙ-graphs and applications, JCISS 26 (2001), 51-78.
• [22] R.N. Mohan and P.T. Kulkarni, A new family of fault-tolerant M-networks, IEEE Trans. Comupters (under revision) (2006).
• [23] N.C. Nguyen, N.M.D. Vo and S.Y. Lee, Fault tolerant routing and broadcasting in deBruijn networks, Advanced Information Networking and Applications, AINA 2005, 19th International Conference, March 2005, 2 (2005), 35-40.
• [24] D. Raghavarao, A generalization of group divisible designs, Ann. Math. Statist. 31 (1960), 756-771.
• [25] D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York 1971.
• [26] C. Ramanujacharyulu, Non-linear spaces in the construction of symmetric PBIB designs, Sankhyā A27 (1965), 409-414.
• [27] J. Seberry and M. Yamada, Hadamard Matrices, Sequences and Block Designs, Contemporary Design Theory: A Collection of Surveys (edited by J.H. Dinitz and D.R. Stinson), Wiley, New York 1992, 431-560.
• [28] N. Seifer, Upper bounds for permanent of (1,-1)-matrices, Isreal J. Math. 48 (1984), 69-78.
• [29] S.S. Shrikhande, Cyclic solutions of symmetric group divisible designs, Calcutta Statist. Assoc. Bull. 5 (1953), 36-39.
• [30] D.B. Skillicorn, A new class of fault-tolerant static interconnection networks, IEEE Trans. Comupters 37 (1988), 1468-1470.
• [31] D.A. Sprott, A series of symmetric group divisible designs, Ann. Math. Statist. 30 (1959), 249-251.
• [32] M.R. Teague, Image analysis via the general theory of moments, J. Optical Soc. America 70 (1979), 920-930.
• [33] B. Vasic and O. Milenkovic, Combinatorial constructions of low-density parity-check codes for iterative decoding, IEEE Trans. on Information Theory 50 (2004), 1156-1176.
• [34] E.T. Wang, On permanents of (1,-1)-matrices, Isreal J. Math. 18 (1974), 353-361.

Identyfikatory

DOI

10.7151/dmps.1100