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Conference Matrices Nickolay Balonin and Jennifer Seberry To Hadi for your 70 th birthday

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Spot the Difference! Mathon C46 Balonin-Seberry C46

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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

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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

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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.

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Two-circulant Examples C6 C10

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Matrices C 14 and C 18 Examples

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Two-Circulant Matrix C 82

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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

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2 border and 2 circulant explained The 1 st row and the 1 st column are the same The 2 nd row and 2 nd column are the same Now the 4 circulants – take the form A B B -A

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2 Border 2 Circulant Examples C 6 C 10

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2 Border 2 Circulant Examples C 18 C 30

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2 Border 2 Circulant Examples C 42 C 54

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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 C 18 ) – This is a movement from mathematics to engineering concepts. C 18

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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 C 38 with circulated entries

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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).

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Curl Resolution C62 Example

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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.

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C30 – Two Versions

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C42 – Two Versions

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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 flip- inversed (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.

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Examples of Poor Matrices C42 C50

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Examples of Rich Matrices Rich structures use circulant and back-circulant matrices C50 C42

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Multi-circulant Conference Matrices C18 Another C18

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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.

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Multi-circulant Conference Matrix This example shows the method with the pair- sequences, 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

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Conference Matrices - Multiple Shifts Circulant Matrix C 42 – 3 shifts Circulant Matrix C 42 – 4 shifts

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Sylvester Method for Conference Matrices The Sylvester method, for orders n=2 k +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

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Sylvester Method Examples C 10 C 18

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The Challenge of Order 66 First try to find Max Det X 66 – 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*10 60

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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), 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), R. Mathon. Symmetric conference matrices of order pq 2 +1 Canad. J. Math 30 (2), Jennifer Seberry, Albert L. Whiteman New Hadamard matrices and conference matrices obtained via Mathon's construction, Graphs and Combinatorics, 4, 1988, Online:

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Thank You Happy Birthday Hadi Kharahani's decomposition of K40 into 2 of the 4 Siamese SRG

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