SEIDEL MATRICES OF ORDER 4t–3
© Nickolay Balonin 19.10.2014
Three level Seidel matrices, order n=4t–3; a=1, –b, δ; b=1–2δ, δ=1/(1+sqrt(v), based on cores of symmetrical conference matrics with non zero diagonal element.
Matrix n=5; levels 1,–0.3819,0.3090
Matrix n=9; levels 1,–0.5,0.25; (M3 ⊗ M3)
Matrices n=13; levels 1,–0.5657,0.2171 and n=17; levels 1,–0.6096,0.1952
THE LEVEL OPTIMISATION
THE HIDDEN CIRCULANT MATRIX
Balonin-Seberry CM(45;1,–0.7405,0.1297) of hidden circulant type.
Balonin-Seberry construction of Mathon's type CORE C46=circul(A,B,C,C',B') has 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.
Matrices A, B, C of rich-structure C46
Block A=circul(a,b,c) has cells a=circul([0 1 1]), b=circul([-1 -1 1]), c=b'=circul([-1 1 -1]) based on the Legendre symbols of core C10 (Paley case).
Observe A=circul(a,b,c) with cells a=a|0→1, b=–a, c=–c based on the sign-conjugated Legendre symbols and B, C as blocks with hidden circulant cells: B=L1AR1, C=–L2BR2 built by permutation matrices:
Permutation matrices L1, R1, L2, R2 of the hidden circulant structure
OTHER MATRICES OF ORDER 4t–3
Two level matrices CM(v=4t–3;a=1, –b; b=((k–λ+sqrt(k–λ))/(v–2*k+λ)) and histograms of moduli of their elements.
CM(5;1,–0.6666) by SBIBD(5,1,0)
CM(13;1,–0.7887) by SBIBD(13,4,1) *
CM(21;1,–0.5) by SBIBD(21,5,1) *
CM(37;1,–0.4593) by SBIBD(37,9,2) *
CM(57;1,–0.2297) by SBIBD(57,8,1) *
CM(73;1,–0.1934) by SBIBD(73,9,1) *
CM(85;1,–0.4167) by SBIBD(85,21,5) *
CM(109;1,–0.4264) by SBIBD(109,28,7) *
* found by v,k,λ from La Jola catalogue