CONFERENCE MATRIX C_{442} Nickolay Balonin Conference matrix catalogue and on-line algorithms

VISUAL MATLAB PROGRAM

% VISUAL MATLAB PROGRAM
q=3, p=q+2, n=q*q*p+1, e=ones(1,n-1)
if q=3,
% FIND BELEVITCH M=C46
a=circul([0 1 1]), b=circul([-1 -1 1])
A=circul([a b b'])
% USUAL MATHON
% a=circul([-1 1 1]), b=circul([1 1 -1]')
% B=circulback([b a b'])
% NEW MATHON
a=circulback([1 1 -1]), b=circulback([-1 1 1])
B=circul([a b circshift(b)])
C=crossshift([1 -1 -1])
M=[0 e; e' circul([A B C C' B'])], M=?C
% M=[0 e; e' circul([A C B B' C'])], M=?C
end
if q=7,
% FIND BELEVITCH M=C442
matrices: B1, B2, B3
a=circul([0 1 1 1 1 1 1])
b=circul([-1 1 1 -1 1 -1 -1])
c=circul([-1 -1 -1 1 -1 1 1])
A=circul([a b b c b c c])
B1=circulback([1 1 1 -1 1 -1 -1]), B2=B1, B3=B1
B=B1, Do q-1, B=circshift(B), B1=[B1 B], end
B=B2, Do q-1, Do 2, B=circshift(B), end, B2=[B2 B], end
B=B3, Do q-1, Do 3, B=circshift(B), end, B3=[B3 B], end
B1=circul(B1), B2=circul(B2), B3=circul(B3)
C=crossshift([1 1 1 -1 1 -1 -1])
a=toeplitz([A B1 B2]), a=?C;
b=[-B3 C -C';-B2 -B3 C;B1 -B2 -B3], b=?C;
M=[0 e; e' circul([a b b'])], M=?C
end
r=M*M(2)', r(2)=0, r=max(abs(r)), r=?;

THE RICH CELL-STRUCTURE

Balonin-Seberry construction of Maton's type matrix C442=C(A,B1,B2,B3,C) has rich cell-structure: A is a circulant matrix of circulant cells (1-type), blocks B1, B2, B3 are circulant matrices of shifted 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 =7, cells A, C, B1, B2, B3 given by pictures below.

A of circulant cells (1-type), C is cross-matrix (0-type) Matrices of back-circulant cells: B1, B2, B3 (2-type) ILLUSTRATIONS TO THE PAPER

n=442; C=border(bsCore441(n));
{{X=C'*C}} puts(X);
// OUTPUT (DRAW MATRIX)
{{"C442V1.xml"=C}} plot("../files/C442V1.xml:C");
function bsCore441(n) {
var a,b,c,A,B,B1,B2,B3,C,U;
var T,M,MT,MT2,MT3,MT4,MT5,MT6,K1,K2,K3;
var B1T,B2T,B3T,MB1,MB2,MB3,MB1T,MB2T,MB3T,MC,MCT;
a=circul([0,1,1,1,1,1,1]);
b=circul([-1,1,1,-1,1,-1,-1]);
c=circul([-1,-1,-1,1,-1,1,1]);
A=circul(a,b,b,c,b,c,c);
M=circulback([1,1,1,-1,1,-1,-1]); T=circul([0,1,0,0,0,0,0]);
{{MT=M*T; MT2=MT*T; MT3=MT2*T; MT4=MT3*T; MT5=MT4*T; MT6=MT5*T;}}
B1=circul(M,MT,MT2,MT3,MT4,MT5,MT6);
B2=circul(M,MT2,MT4,MT6,MT,MT3,MT5);
B3=circul(M,MT3,MT6,MT2,MT5,MT,MT4);
C=[1,1,1,-1,1,-1,-1]; C=crossmatrix(C); CT=tr(C);
MC=mulp(-1,C); MCT=mulp(-1,CT);
B1T=tr(B1); MB1=mulp(-1,B1); MB1T=tr(MB1);
B2T=tr(B2); MB2=mulp(-1,B2); MB2T=tr(MB2);
B3T=tr(B3); MB3=mulp(-1,B3); MB3T=tr(MB3);
B1T=tr(B1); MC=mulp(-1,C); MCT=mulp(-1,CT);
B2T=tr(B2); MB2=mulp(-1,B2); MB2T=tr(MB2);
B3T=tr(B3); MB3=mulp(-1,B3); MB3T=tr(MB3);
K1=core3(A,B1,B2,B1T,B2T); // EXAMPLE 5 FORMULAE
K2=core3(MB3,C,MCT,MB2,B1);
K3=core3(MB3T,MB2T,B1T,CT,MC);
U=core3(K1,K2,K3,K3,K2);
return U;
}
function core3(A,B,C,D,E) { var H,K;
H=colline(A,B,C); K=tr(H);
H=colline(D,A,B); K=colline(K,tr(H));
H=colline(E,D,A); K=tr(colline(K,tr(H)));
return K;
}

VERSION WITH MORE SIMPLE CORE

n=442; C=border(bsCore441(n));
{{X=C'*C}} puts(X);
// OUTPUT (DRAW MATRIX)
{{"C442V2.xml"=C}} plot("../files/C442V2.xml:C");
function bsCore441(n) {
var a,b,c,A,B,B1,B2,B3,C,U;
var T,M,MT,MT2,MT3,MT4,MT5,MT6,K1,K2,K3;
var B1T,B2T,B3T,MB1,MB2,MB3,MB1T,MB2T,MB3T,MC,MCT;
a=circul([0,1,1,1,1,1,1]);
b=circul([-1,1,1,-1,1,-1,-1]);
c=circul([-1,-1,-1,1,-1,1,1]);
A=circul(a,b,b,c,b,c,c);
M=circulback([1,1,1,-1,1,-1,-1]); T=circul([0,1,0,0,0,0,0]);
{{MT=M*T; MT2=MT*T; MT3=MT2*T; MT4=MT3*T; MT5=MT4*T; MT6=MT5*T;}}
B1=circul(M,MT,MT2,MT3,MT4,MT5,MT6);
B2=circul(M,MT2,MT4,MT6,MT,MT3,MT5);
B3=circul(M,MT3,MT6,MT2,MT5,MT,MT4);
C=[1,1,1,-1,1,-1,-1]; C=crossmatrix(C); CT=tr(C);
// {{I=A*A'+B1*B1'+B1'*B1+B2*B2'+B2'*B2+B3*B3'+B3'*B3+C*C'+C'*C}}
// puts(I);
MC=mulp(-1,C); MCT=mulp(-1,CT);
B1T=tr(B1); MB1=mulp(-1,B1); MB1T=tr(MB1);
B2T=tr(B2); MB2=mulp(-1,B2); MB2T=tr(MB2);
B3T=tr(B3); MB3=mulp(-1,B3); MB3T=tr(MB3);
K1=core3(A,B1,B2,B1T,B2T); // EXAMPLE 5 FORMULAE
K2=core3(B3,MC,MCT,MB2,MB1);
K3=core3(B3T,MB2T,MB1T,MCT,MC);
U=core3(K1,K2,K3,K3,K2);
return U;
}
function core3(A,B,C,D,E) { var H,K;
H=colline(A,B,C); K=tr(H);
H=colline(D,A,B); K=colline(K,tr(H));
H=colline(E,D,A); K=tr(colline(K,tr(H)));
return K;
}
R. Mathon. Symmetric conference matrices of order pq^{2} +1 Canad. J. Math 30 (2), 321-331 Jennifer Seberry, Albert L. Whiteman New Hadamard matrices and conference matrices obtained via Mathon's construction, Graphs and Combinatorics, 4, 1988, 355-377.

ON LINE ALGORITHMS | HIDDEN CIRCULANT CELLS