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```Conference
Matrices
Nickolay Balonin
and
Jennifer Seberry
To Hadi
for your 70th birthday
Spot the Difference!
• Mathon C46
• Balonin-Seberry C46
In this presentation
•
•
•
•
•
•
•
•
Two Circulant Matrices
Two Border Two Circulant Matrices
Two Border Four Circulant Matrices
Curl Resolution
Poor and Rich Structure Matrices
Multi-Circulant Conference Matrices
Sylvester Inspired Matrices
References
Definition
• The conference matrix C is n × n matrix
С'С =( n -1)I with zero diagonal and other ±1
units.
• The necessary condition for its existence: n-1
is sum of two squares.
• These orders don’t exist 22, 34, 58, 70, 78, 94,
n<100.
• The known problem orders are 46 and 66
Two-Circulant Matrices
• Two circulant A,B-matrices are
interesting due their universal
structure:
– they exist for complicated cases
10, 26, 50 (not for 46, 66, 86!),
– A is circulant and symmetry, type
of matrix depends on circulant
block B.
– Symmetric versions are equivalent
to Paley constructions.
Two-circulant Examples
• C6
• C10
Matrices C14 and C18 Examples
Two-Circulant Matrix C82
Two Borders Two Circulant Matrices
• Two borders and two
circulant A,B-matrices are
interesting due to their
universal structure:
– they exist for prime power
plus 1 cases 10, 26, 50
(not for 46, 66, 86!),
– A is circulant and symmetric,
– block B is based on the two
flip-inversed sequences
2 border and 2 circulant explained
• The 1st row and the 1st
column are the same
• The 2nd row and 2nd
column are the same
• Now the 4 circulants
– take the form
A B
B -A
2 Border 2 Circulant Examples
• C6
• C10
2 Border 2 Circulant Examples
• C18
• C30
2 Border 2 Circulant Examples
• C42
• C54
Two Borders Four Circulant Matrices
• Two borders and four A,B,C,D-cells
core [S G;G' -S], S=[A B;B' A],
G=[C D;F(-D) E(C)] we will call
sequence of cells: A, B, C, D, E, F,
situated as shown, the curl of Seberry.
• The solution depends on the curl
resolution: it could be either poor or
rich cell-construction.
• In comparison to the column
separation of Walsh-matrices we see a
kind of cell separation motivated by
sign-frequency (look at C18) – This is a
movement from mathematics to
engineering concepts.
• C18
Curl Resolution
• A is circulant and symmetric
matrix for the left top corner
(excluding the two borders),
• the right square G=[C D;F C*]
based on the two flip-inversed
(or inversed or/and shifted)
sequences for C and D,
• F=mirror(-D), E=C* may be
shifted a few times (for orders
18, 26, 42, ..) the back-circulant
cell is mirror(C).
• Matrix C38
with circulated
entries
Curl Resolution
• A rich construction based on circulant and
back circulant cells leads to matrix portraits
with two curls. This form reflects a Fourier
type basis for the orthogonal matrices (in
some sense, these matrices reflect some
gross-object given in fine detail when we
consider big orders: something like the next
example but with higher resolution).
Curl Resolution
• C62 Example
Matrices C26 – two versions
Matrix C26 is a special case; it has symmetry given by both
diagonals of cell B (so it has a mirror symmetry of F=RDR or
E=RCR, R is the back diagonal matrix) and it has a simple solution
also.
C30 – Two Versions
C42 – Two Versions
Matrices of Poor and Rich Structure
• The solution depends on the curl resolution:
• A is circulant and symmetric matrix in the top left
corner (excluding the two borders),
• the right square G=[C D; D* C] is based on the two flipinversed (or shifted) sequences,
• D* is a circulant cell shifted a few times.
• The poor structures use only circulant matrices: rich
structures use circulant and back circulant matrices.
• They look like block permutations of each other, but
column and row permutations of one cannot be
equivalent to the other as the structure is not
preserved.
Examples of Poor Matrices
• C42
• C50
Examples of Rich Matrices
Rich structures use circulant and back-circulant matrices
• C42
• C50
Multi-circulant Conference Matrices
• C18
• Another C18
Multi-circulant Conference Matrix
• Main matrix consists of
circulant blocks of circulant
matrices.
• The set of symmetric A, D and
some tied pair-sequences of
(B, C) and (E, F), has enough
invariants to describe
conference matrices iff n–1 is
prime.
Multi-circulant Conference Matrix
• This example shows the
method with the pairsequences, one sequence
shifted to the left at (B, C)
and to the right at (E, F).
• It could be any shift to any
side, however they must be
different.
• Example C30
Conference Matrices - Multiple Shifts
• Circulant Matrix C42
– 3 shifts
• Circulant Matrix C42
– 4 shifts
Sylvester Method for Conference
Matrices
• The Sylvester method, for
orders n=2k+2 including
matrices C6, C10, C18 has
two borders and four
blocks core [A B; B' -A'].
• order 34 does not exist
• the next unsolved case is
order 66
• C18
Sylvester Method Examples
• C10
• C18
The Challenge of Order 66
• First try to find
Max Det X66
– Hadi Kharaghani
(following Young
C.H., 1976)
constructed a max
det matrix of order
66 with 6x6 blocks
of order 11 using
Legendre symbols
• Maximal determinant
matrix: det(X)=0.816*1060
References
• N. A. Balonin and Jennifer Seberry, A Review and New Symmetric
Conference Matrices. Informatsionno-upravliaiushchie sistemy,
2014, no. 4 (71), 2--7
• V. Belevitch, Conference networks and Hadamard matrices, Ann.
Soc. Sci. Brux. T. 82 (1968), 13-32.
• Christos Koukouvinos and Jennifer Seberry, New weighing matrices
constructed using two sequences with zero autocorrelation
function – a review, J. Stat. Planning and Inf., 81 (1999), 153-182
• R. Mathon. Symmetric conference matrices of order pq2+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.
• Online: http://www.mathscinet.ru/catalogue/conference/
Thank You
Happy Birthday
Hadi
Kharahani's
decomposition of
K40 into 2 of the
4 Siamese SRG
```
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