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; (M_{3} ⊗ M_{3})

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=q^{2}(q+2), where q+2 is order of a core, q=3.

Matrices A, B, C of rich-structure C_{46}

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 C_{10} (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=L_{1}AR_{1}, C=–L_{2}BR_{2} built by permutation matrices:

Permutation matrices L_{1}, R_{1}, L_{2}, R_{2} 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.