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 




Синие P, красные GP, желтые GP двоякосимметричные по A, D


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 [2], [3]. The first unknown case for symmetric Hadamard matrices is 4x39 was found at 2016-2017 (Dragomir).


ЖОЗЕФ ЛИУВИЛЛ



Joseph Liouville


Кроме академических достижений, он был очень талантливым организатором. Лиувилль основал «Журнал чистой и прикладной математики» (фр. Journal de mathématiques pures et appliquées), который поддерживает свою репутацию до настоящего времени, для продвижения математических работ. Он первым прочитал неопубликованные работы Галуа и осознал их важность, они были опубликованы в журнале в 1846 г. В 1973 г. Международный астрономический союз присвоил имя Жозефа Лиувилля кратеру на видимой стороне Луны.


In 1796, Gauss observed that the sum of three triangular numbers, Tx+Ty+Tz, represents all natural numbers, a property which we call universal. In 1862 Liouville examined this problem further, by asking for which positive integer triples (α, β, γ) the weighted ternary sum αTx+βTy+γTz is universal. He determined that only seven such triples exist, namely (1, 1, 1), (1, 1, 2), (1, 1, 4), (1, 1, 5), (1, 2, 2), (1, 2, 3) and (1, 2, 4) (look Anna Haensch).

[1] 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
[2] 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.
[3] 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.
[4] Balonin N. A., Balonin Y. N., Djokovic D. Z., Karbovskiy D. A., Sergeev M. B. Construction of symmetric Hadamard matrices // Informatsionno-upravliaiushchie sistemy, 2017, № 5, pp. 2–11. (16 Aug 2017: arXiv:1708.05098 | PDF)
[5] Balonin N. A., Djokovic D. Z., Karbovskiy D. A. Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113 // Special matrices, 2018. Vol.6 pp. 11–22. It was accepted on Dec 22, 2017, Web of Science/Scopus PDF | HTML (9 Oct 2017: arXiv:1710.03037 | PDF).
[6] Balonin N. A., Djocovic D. Z. Negaperiodic Golay pairs and Hadamard matrices. // Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2015, no. 5, pp. 2–17. doi:10.15217/issn1684-8853.2015.5.2 (негациклические бициклы)

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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