MULTI-CIRCULANT CONFERENCE MATRICES



© Nickolay Balonin and Jennifer Seberry, 1.05.2014

Multi-circulant conference matrix catalogue and on-line algorithms


N. A. Balonin, Jennifer Seberry A Review and New Symmetric Conference Matrices //Informatsionno-upravliaiushchie sistemy, 2014, № 4 (71), pp. 2–7.

C=
 T1 
 T2 
 T2
 –T1
T1=
A
B
C
 B' 
A
B
 C' 
 B' 
A
T2=
D
E
F
 E' 
D
E
 F' 
 E' 
D


Main matrix consists the blocks with circulant matrices. The two circulant structure – it's too universal. The set of symmetry A, D and some tied pair-sequences of (B, C) and (E, F), it has enough big quantity of invariants to describe conference matrices iff n–1 is prime. The private line of orders, it is oriented on the C66. This matrix absence is a special question of theory. We observe a line of multi-circulant matrices with orders n=2·3·k, k – integer 1, 3, 5, 7, 9, 11, 13, 15.., it covers orders 6, 18, 34, 42, 54, (66!?), (70), 90, ...

THE SAMPLES OF CONFERENCE MATRICES


The solution has the pair-sequences (B, C=flip(±B)) and (E, F=flip(±E)). It leads to the following samples of multi-circulant matrices:



Circulant matrices C6 and C18



Circulant matrices C30 and C42



Circulant matrices C54


THE SAMPLES OF CONFERENCE MATRICES


The solution has the pair-sequences, C-secuence is shifted accordingly B to the left in (B, C) and F-secuence is shifted accordingly E to the right in (E, F). It could be any shifts to any sides, need they were different. Matrix C54 looks like non reachable.



Circulant matrix C6



Circulant matrices C18 and C30



Circulant matrices C42 (3 shifts) and C42 (4 shifts)


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