CONFERENCE MATRIX C46

Nickolay Balonin and Jennifer Seberry

Conference matrix catalogue and on-line algorithms


THE NEW FORM OF CONFERENCE MATRIX C46





Compare with maximum detherminant matrix X21



Compare with maximum detherminant matrix X25

ON LINE ALGORITHMS | HIDDEN CIRCULANT CELLS






THE CELL STRUCTURE



Block A with circulant 1-type cells



Block B with back-circulant 2-type cells



Block C is cross-shifted matrix with 0-type cells


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).

THE RICH CELL-STRUCTURE



Balonin-Seberry construction of Maton's type CORE C46=circul(A,B,C,C',B') has a rich cell-structure: A is a circulant matrix of circulant cells (1-type), B is a circulant matrix of back-circulant cells (2-type), C is a cross-matrix (0-type), the core of order n=q2(q+2), where q+2 is order of a core, q=3.



ILLUSTRATIONS TO THE PAPER | HIDDEN CIRCULANT CELLS




CORE C46=circul(A,C,B,B',C') reflects the Fourier phases: right shift, up, left shift, ..




CORE C46=circul(A,B,C,C',B') with b=circshift(-a) (!) c=circshift(b), a[0]=0


THE POOR CELL-STRUCTURE


Maton's type CORE C=circul(A,B,C,C',B') can have the poor-structure of cells: A is a circulant matrix of circulant cells (1-type), B is a back-circulant matrix of circulant cells (1-type), C is a cross-matrix (0-type)). The structural invariant 5 us equal to quanity of cell-and-block types: 2 types of cells but 3 types of A, B, C.




Old version of algorithm based on the Mathon's paper (poor structure)


R. Mathon. Symmetric conference matrices of order pq2+1 Canad. J. Math 30 (2), 321-331
Jennifer Seberry, Albert L. Whiteman New Hadamard matrices and conference matrices obtained via Mathon's construction, Graphs and Combinatorics, 4, 1988, 355-377.



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