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Nickolay Balonin
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
• The conference matrix C is n × n matrix
С'С =( n -1)I with zero diagonal and other ±1
• The necessary condition for its existence: n-1
is sum of two squares.
• These orders don’t exist 22, 34, 58, 70, 78, 94,
• The known problem orders are 46 and 66
Two-Circulant Matrices
• Two circulant A,B-matrices are
interesting due their universal
– 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
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
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
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
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
• 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
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
• Example C30
Conference Matrices - Multiple Shifts
• Circulant Matrix C42
– 3 shifts
• Circulant Matrix C42
– 4 shifts
Sylvester Method for Conference
• 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
• 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:
Thank You
Happy Birthday
decomposition of
K40 into 2 of the
4 Siamese SRG