PROPUS



© Nickolay Balonin and Jennifer Seberry, 6.08.2014

Propus matrix catalogue and on-line algorithms


Definition. An Hadamard matrix H of order n is an nxn matrix with elements ±1 such that HTH=HHT=nI, where I – is the nxn identity matrix and T stands for transposition [1,2]. Propus [3] is a construction method for orthogonal ±1-matrices based on the array

P=
A
 B=С 
 C=B 
D
C
D
 –A 
 –B 
B
 –A 
 –D 
C
D
 –C 
B
 –A 


where cells A, B=C, and D, satisfy AAT+BBT+CCT+DDT=cI, c – constant. Symmetric A (minimum) gives symmetric Hadamard matrices. This construction, based on symmetric circulant ±1-matrices, – good Propus. Common Propus is based on the turn of B=C and D.

Common Propusi have a system of holes.



We observe three methods to fi nd propus-Hadamard matrices: using Williamson matrices, D-optimal designs, Dragomir's and Nick's computer algorithms. Many delightful visual images are included. The special orders n<100 are: n=92 inside the D-optimal design family (case m=23 has no symmetric A, there is other symmetric Propus: Dragomir's page) and the Common family with special cases: n=88 (non Propus?), 92 (Dragomir's page), 96 (non Propus?); n=16*k with cells of even orders resolvable as weighing matrices. Look: regular Propus matrices. There is also the possibility that this propus construction may lead to some insight into the existence or non-existence of symmetric conference matrices for some orders.



COMMON FORM



SKEW MATRICES



BY WILLIAMSON MATRICES



BY D-OPTIMAL DESIGNS



SYLVESTER FAMILY



EVEN ORDERS



BY CONFERENCE MATRICES



BY TWO CIRCULANT C-MATRICES



BY MULTI-CIRCULANT C-MATRICES



BY CONFERENCE MATRIX C14



SOME WEIGHING MATRICES



(a=1,–b)-MATRICES



DOUBLE PROPUS A=–D, B=C





GET MORE MATRICES

BY WILLIAMSON MATRICES



Propus H12 and H28



Propus H36 and H60

FAMILY: A=Q+I, C=B, D=Q–I



Propus H20 and H52

GOOD PROPUS BY D-OPTIMAL DESIGN



D-optimal design X6 and Propus H12



D-optimal design X14 and Propus H28



D-optimal design X38 and Propus H76

FINITE FAMILY



Propus H12 and H28 (B=C=D rich structure)

BY CONFERENCE MATRICES



Conference matrix C10 and Propus H20



Conference matrix C26 and Propus H52

WITH Q+I, Q–I AND MAX DET BLOCK B=X5 or B=X13



Conference matrix C10 and Propus H20



Conference matrix C26 and Propus H52

EVEN CELLS



Propus matrices H16 and H32


1. Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, eds. J. H. Dinitz and D. R. Stinson, John Wiley, New York, 1992, pp. 431–560 (PDF)
2. W.D. Wallis, A.P. Street and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematics, Springer-Verlag, Vol. 292, 1972.
3. N. A. Balonin and Jennifer Seberry, “Visualizing Hadamard Matrices: the Propus Construction”, Australasian Journal of Combinatorics (submitted 6 Aug 2014).
4. N. A. Balonin and Jennifer Seberry, A Review and New Symmetric Conference Matrices //Informatsionno-upravliaiushchie sistemy, 2014, № 4 (71), pp. 2–7. (PDF | WEB)

ELECTRONIC EDITIONS

Balonin N. A., Seberry, Jennifer. Visualizing Hadamard Matrices: the Propus Construction, Electronic edition, 2014.
N. A. Balonin, Jennifer Seberry Conference Matrices with Two Borders and Four Circulants, Electronic edition, 2014.

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