Conference matrix catalogue and on-line algorithms

N. A. Balonin, Jennifer SeberryConference Matrices with Two Borders and Four Circulants, Electronic edition, 2014. N. A. Balonin, Jennifer SeberryA 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 AA^{T}+BB^{T}+CC^{T}+DD^{T}=(n–1)I:

A

B

C

D

B^{T}

A

F

E

C^{T}

F^{T}

–A

–B

D^{T}

E^{T}

–B^{T}

–A

Two borders and four A,B,C,D-cells CORE [S G;G^{T} –S], S=[A B;B^{T} 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 C_{18}).

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).

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 C_{26} 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).

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.