TWO BORDERS FOUR CIRCULANT MATRICES



© Nickolay Balonin and Jennifer Seberry, 1.05.2014

Conference matrix catalogue and on-line algorithms


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

Observe a construction method for orthogonal (±1, 0)-matrices based on the array (a core), taken with two borders, where cells A (symmetry), B, C, and D, satisfy AAT+BBT+CCT+DDT=(n–1)I:

A
B
C
D
 BT 
A
F
E
 CT 
 FT 
 –A 
 –B 
 DT 
 ET 
 –BT 
 –A 


Two borders and four A,B,C,D-cells CORE [S G;GT –S], S=[A B;BT A], G=[C D;F(D) E(C)]; we will call sequence of cells: A, B, C, D, E, F, situated as shown, the curl of Seberry (Lokon, Vichr). The solution depend on the curl resolvance: could be the poor and the rich (matrices with circulated entries) cell-construction. In different to column separation of Walsh-matrices we see a kind of cell separation motivated by sign-frequence (look C18).



MATRICES WITH CIRCULATED ENTRIES


The rich construction based on circulant and back-circulant cells leads to the matrix portraits with the two centers of circulation of entries. This form reflects a Fourier's-type basis of the orthogonal matrices (in some sense, these matrices reflect some gross-object taken in the tune details when we go to the big orders: something like the photos with the more big resolution).



COLOUR MATRIX C18



COLOUR MATRIX C26

The solution depend on the curl resolvance: A is circulant and symmetry matrix of the left square, the right square G=[C D;F C*] based on the two flip-inversed (or inversed or/and shifted) sequences for C and D, F=mirror(D), E=C* is a few times shifted (for orders 18, 26, 42, ..) back-circulant cell mirror(C). Matrix C26 is a special case, by it has symmetry accordingly both diagonals cell B (so it has a mirror symmetry of F=RDR or E=RCR, and it has a simple solution also).



Matrices C6 and C10



Matrices C14 and C18



Matrices C26, two versions



Matrices C30, two versions



Matrices C38, two versions



Matrices C42, two versions



Matrices C50 and C54



Matrix C62


Kernel of integral operator

THE STRUCTURAL SINGULAR POINTS | THE C66 PROBLEM

MATRICES OF THE POOR STRUCTURE


The solution depend on the curl resolvance: A is circulant and symmetry matrix of the left square, the right square G=[C D;D* C] based on the two flip-inversed (or shifted) sequences, D* is a few times shifted circulant cell. The poor and rich structures look like block-permutated to eath-other, but column- and row-permutations of B and C cells at S it does not answer the permutation of –B at –S.



Matrices C6 and C10



Matrices C14 and C18



Matrices C26 and C30



Matrices C38 and C42



Matriсes C50 and C54



Matrix C62



TWO CIRCULANT MATRICES | CONVERTOR PNG-EPS

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