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=C, F=circshift(D), 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).

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. 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).

Matrices C6 and C10

Matrices C14 and C18

Matrices C26 and C30

Matrices C38 and C42

Matriсes C50 and C54

Matrix C62 and C100

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. 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.

 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

VISUAL MATLAB PROGRAM

Common Propus gives no need matrix

RECONSTRUCTION C26

TWO CIRCULANT MATRICES | CONVERTOR PNG-EPS

 ..C18C26