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  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 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

Common Propusi have a system of holes.

Mersenne line: v=4k-1, all n=4v=12, 28, 44, 60, .... do exists ! 2*n=24, 56, 88, 120, ... do not exist: main holes line are orders m=8(4*k-1)=24, 56, 88, 120, 152....

Odin line: q=4k-3, all n=2(q+1)=4, 12, 20, 28, .. do exist, for 2*n=8, 24!, 40, 56!, 72, 96!, 104, 120!.. every second order 24, 56, 96, 120.. do not exist: main holes line are orders m=8(4*k-1)=24, 56, 88, 120, 152....

Finite Golay lines: 1) Simple 2, 4, 8, 16, 32, 64 do exist, 128=32x4 Common! 2) 20, 40 (Common), 80 do exist, 160=40x4 Common! 3) Simple 52 and 104=26x4, 208 Common !

D-optimal lines: D-optimal design: 86 (158?) is the last symmetric solution, so 172 is the last Propus with it.  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; 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.

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

MULTICIRCULANT PROPUS 364  Seberry, Jennifer and Balonin N.A. Two infinite families of symmetric Hadamard matrices // Australian Journal of Combinatorics 2017. Vol. 69 (3) pp. 349–357 AJC 2017 | Abstract

To get Williamson matrix, v=3 mod 4, need C-matrix v–1, and core Q of C-matrix (v–1)/2.

Example gives multicirculant Propus 364.

Theorem 3, Collorary 1: example gives multicirculant Propus 364, while there is array 364 based on circulant blocks made by C-matrix 182. For n=364, v=3*30+1=91=7*13, 90=3*30=5*19 we have conference matrix C182; k1=40; k2=41.

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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