THE PROBLEM ORDER 66

© Nickolay Balonin 1.05.2014

Conference matrix catalogue and on-line algorithms

QUASI CONFERENCE MATRIX 66



The orthogonal matrix and diagram of absolute values of its entries


The conference matrices (the same like Hadamard matrices) have maximal determinant on the class of orthogonal ones. Take attention that matrix of maximal determinant has the same projection on the orthogonal area that this structure of order n=2*3*11. There are very much reasons, that matrix having entries 0 (on the diagonal), a=1, b=–1, and square of 9c2–2c–5=0; c=(1±sqrt(46))/9 of the 6 flat-cells, order 11x11, has the maximal determinant among all orthogonal matrices of order 66. In such case C66 doesn't exist definitely. Look: vitrages.



Legendre approximations with discrepancy 2 and 4 (symmetry)


Matrix C66=[A,B;B',–A'] has block-circulant A, B with circulant cells of size 11, that allow to take in consideration the Legendre symbols L=[-1|1,-1,1,1,1,-1,-1,-1,1,-1] of skew conference matrix C12, the skew approximation has no chance to be a solution, but it provides the minimal C'C=(n–1)I discrepancy 2! Orthogonal construction with "the black-block" A2 has to have entries {a=1, –b=–1; c}, c=(1±sqrt(46))/9=0.864703<1. This structure is coincide with a projection of maximal determinant matrix X66 onto the orthogonal area and supposedly it has the maximal determinant here (in such case the C66 doesn't exist definitely).

THE MULTI-CIRCULANT STRUCTURES



Two-circulant and multicirculant conference matrices C26


The C6, C26, C46, C66, C86 go with the same distance 20: C6 is classical and well known case, C26 is the first special case with several solutions, C46 has no one- or two-circulant forms (in different to previous cases): the hidden circulant cell solution looks like a critical point of this line of orders.

The CORE C26=circul(A,B,C,C',B') has circulant cells: A=diag([0,1,1,1,1]), and B, C based on the set of direct [-1|1,-1,-1,1] and sign-conjugated [-1|-1,1,1,-1] Legendre symbols of C6.

APPROXIMATION BY 5x5 STRUCTURE



Legendre and numerical approximations with discrepancy 8


The CORE C66=circul(A,B,C,C',B') with circulant cells A=diag([0,1,...,1]) and B, C based on the set of direct [-1|1,-1,1,1,-1,-1,-1,-1,1,1,-1,1] and sign-conjugated [-1|-1,1,-1,-1,1,1,1,1,-1,-1,1,-1] Legendre symbols of C14 provides the C'C=(n–1)I discrepancy 8. Take attention, that the symmetry structure has a big distantion to the need result.


All orders 6, 26, 46, connected with sizes of cores of more little conference matrices, they have multicirculant solutions based on them. In the contrary, we see, that the problem orders 66, 86 (and so on) cannot be resolved by the two-circulant and 5x5 or 6x6 block-circulant matrices. Observed approximations have extremal properties among multi-circulant constructions.

THE WAY THROUGH MAX DET X66



Maximal determinant matrix: det(X)=0.816*1060


Hadi Kharaghani (following Young C.H., 1976) constructed a max det matrix of order 66 with 6x6 blocks of order 11 by the Legendre symbols (take a look on construction above).

This asymmetry matrix (with discrepancy 2!) has a different with C-matrices projection on the orthogonal area



The X66 projection on the orthogonal area


also (second method)



The second X66 projection on the orthogonal area

YANG C.H. 1976 | HADI'S PAPER | J. SEBERRY | TABLE OF MAX DET


THE SYLVESTER LINE OF CONFERENCE MATRICES


The finite Sylvester Line of orders n=2k+2 includes matrices C6, C10, C18 (order 34 stay on the way to the order 66) that have two borders and four blocks CORE [A B; B' –A']. Recursive formulae A=[A B;B' A'], where B is one or two circulant matrix, the latest version it is based on the two flip-inversed sequences B1=circul(g), B2=circul(flip(-g)), B=[B1 B2; circshiftback(B2) B1].



Conference matrices C10 and C18


Balonin-Sebbery 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

Such solutions becoming be imposible after the critical point with observed hidden circulant cells: inner permutations allow to reach a principal bound.

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