CONFERENCE MATRICES



© Nickolay Balonin, 1.05.2014

Conference matrix catalogue and on-line algorithms


Definition. The conference matrix C is nxn matrix С'С=(n–1)I with zero diagonal and other ±1 units. The necessary condition of its existence: n–1 is sum of two squares. The problem orders are 22, 34, 58, 70, 78, 94, n<100.


Go to the two-circulant, two-borders two circulant cells, two-borders four circulant cells, and multi-circulant samples including Sylvester Line, C66-problem, С3646-line, matrix C10
, Gregory matrix, more colours of C14, two borders four circulant cells, paper illustrations, hidden circulant cells.



SOME BEGINNING MATRICES



TWO CIRCULANT C-MATRICES



TWO BORDERS TWO CIRCULANT CELLS



TWO BORDERS FOUR CIRCULANT CELLS



MULTI-CIRCULANT C-MATRICES



CONFERENCE MATRIX C14



WEIGHING MATRICES



WEIGHING MATRIX W(22,20)



ORTHOGONAL DESIGN



SKEW-HADAMARD MATRICES



Table of two squares sum


ON LINE ALGORITHMS

ILLUSTRATIONS TO MATHON'S-TYPE MATRICES


Let's observe three types of cells. Type 0) 0-circulant type (zero shift, every row is equal to the other). Type 1) circulant cells (circulant shift every row right). Type 2) back-circulant cells (circulant shift every row left). We say matrix has a rich structure, if it consists several types of cells (all three types).

CONFERENCE MATRIX C(46)



POOR CELLS of {1,1,0}-TYPES (CIRCULANT AND CROSS-SHIFTED)

BALONIN-SEBERRY CONSTRUCTION



RICH CELLS of {1,2,0}-TYPES (CIRCULANT, BACK-CIRCULANT AND CROSS-SHIFTED)

COMPILATION TO C-MATRIX WITH INVARIANT 5



OLD AND NEW STRUCTURES (5 IS QUANTITY OF CELL-and-BLOCK TYPES)

N. A. Balonin, Jennifer Seberry A Review and New Symmetric Conference Matrices //Informatsionno-upravliaiushchie sistemy, 2014, № 4 (71), pp. 2–7.

ON LINE ALGORITHMS | ILLUSTRATIONS FOR THE PAPER ABOVE

REFERENCES


[1] V. Belevitch, Conference networks and Hadamard matrices, Ann. Soc. Sci. Brux. T. 82 (1968), 13-32.
[2] J.-M. Goethals and J.J.Seidel, Orthogonal matrices with zero diagonal, Canad, J, Math, 19 (1967), 1001-1010.
[3] R. Mathon, Symmetric conference matrices of order pq2 +1, Canad. J. Math., 30 (1978), 321-331.
[4] R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys., 12 (1933), 311-320.
[5] Jennifer Seberry Wallis, Combinatorial Matrices, PhD Thesis, La Trobe University, 1971.
[6] Jennifer Seberry and A. L. Whiteman, New Hadamard matrices and conference matrices obtained via Mathon's construction, Graphs Combin. 4 (1988), 355-377.
[7] R. J. Turyn, On C-matrices of arbitrary powers, Bull. Canad. Math. Soc. 23, 1971, 531-535. 4


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