COMMON PROPUS A=AT, B=C, D



© Nickolay Balonin and Jennifer Seberry 6.08.2014

Propus-Hadamard matrices in common form


Common form CP (Goethals-Seidel, Seberry-Whiteman proposed, adapted to symmetry) describes some special even orders of cells m (or orders, where C-matrices do not exist: 2m=22, [case 34 is special]) by symmetric circulant cell A, circulant pair B=C, and circulant D.

P=
A
 B=С 
 C=B 
D
C
D
 –A 
 –B 
B
 –A 
 –D 
C
D
 –C 
B
 –A 
CP=
A
 BR 
 CR 
 DR 
 CR 
 DT
 –A 
 –BT
 BR 
 –A 
 –DT
 CT
 DR 
 –CT
 BT
 –A 


There are special cases 24, 56 and 88, n≤100: order 24 is resolvable only accordingly symmetric weighing matrices W(24,22) and case 92 has symmetric A (look news at Dragomir's list). The propus Hadamard matrices of orders 92 (v=23), 116 (v=29) and 172 (v=43) are presented in [1], [2]. The first unknown case for symmetric Hadamard matrices is 4x39 was found at 2016-2017 (Dragomir).


[1] Olivia Di Matteo, Dragomir Z. Djokovic, Ilias S. Kotsireas, Symmetric Hadamard matrices of order 116 and 172 exist, 2015 http://arxiv.org/abs/1503.04226.
[2] Olivia Di Matteo, Dragomir Djokovic, and Ilias S. Kotsireas, Symmetric Hadamard matrices of order 116 and 172 exist. Special Matrices, 3 (2015), pp. 227-234.

MATRICES A=AT, B=C, D



Dragomir's and Nick's common Propusi 68



Dragomir's and Nick's common Propusi 92

GALOIS FIELD PROCEDURE

GOETHAL SEIDEL ARRAY

PROPUS



Hadamard matrices H12 and H16 (even m)



Hadamard matrices H20 and H24 (non symmetry!)



Hadamard matrices H28 and H32 (even m)



Hadamard matrices H36 and H40 (even m)



Hadamard matrices H44 (m=11!) and H48 (even m)



Hadamard matrices H52 and H56 (even m, non Propus)



Hadamard matrices H60 and H64



Hadamard matrix H68 not Propus, non symmetry! and Dragomir's Propus)



Hadamard matrix H72 and H76



Hadamard matrix H80 and H84 GOODS



Hadamard matrix H88 (not Propus) and H88 (8x8 cells)



Hadamard matrix H92 and H92 (non-symmetry!)



Hadamard matrix H96 (not Propus) and H96 (8x8 cells)



H100

TWO SYLVESTER FORMS



Hadamard matrix H88 and H96

WEIGHING MATRICES



Weighing matrix W24



Two weighing matrices W40



Weighing matrix W68 and Hadamard matrix H68 (non-symmetry)


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