VITRAGES



© Nickolay Balonin, 1.01.2015

Vitrage matrix catalogue and on-line algorithms

Collection of vitrages – matrices, based on insertions of one matrix into the other one. Resulting matrix: hadamard matrix, jacobsthal matric, conference matrix: main operation is plug into. Main idea of vitrage: level b of two-level {a=1, –b} quasi-orthogonal matrix (Cretan matrix) arises to the value 1 or similar to 1 (look 668 order, etc.). The character of solution depends on the "distantion" D – difference of orders of matrices to plug in and to plug into.

General distantion is D=2: we will note such matrices of odd orders as S (1 mod 4) and M (3 mod 4). Generalised pairs (or triads, or tetrads) can be postulated and classified for the distance D>2 too: they will have cell–compensators different with J and J–2I, used instead border in vitrage constructions. Tetrads are the most known as Williamson matrices. The other idea is to use M or S instead Williamson array: for small distantions it gives good results. Distantion D=0 looks to be a trivial one.

Mersenne matrix M(a=1,–b) with rounded to 1 (by modulus) entries a=1, –b; b=1, is the core aI+Q(a,–b) of a normalized skew-Hadamard matrix of order 4t

H=
1
1
 ... 
1
 –1 
 –1 
 aI+Q(a,–b
 –1 


The same matrix with zero diagonal is named Jacobsthal matrix Q, it produces modifications: a skew symmetric type – Mersenne matrix M, and symmetric type – Seidel matrix S. Let us note: we say about matrices with entries different to 1, –1, in common.

Vitrages: the simpliest way to get all regular Hadamard matrices R=1+S°M. Besides: it is the way to close all famous problem orders [428], 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964.. by Propusi approximation.

HADAMARD MATRIX PLUGGED INTO C-MATRIX

TRIVIAL DISTANTION D=0: M OR S PLUGS INTO ITSELF



Seidel matrices: vitrages of orders 9, 49


Mersenne or Seidel (Jacobstal) matrix M/S, order q, can be plugged into itself, with levels arised to a=1, –b; b=1. It gives Jacobsthal (Seidel) matrices, vitrages of orders p2: 9 (q=3), 25 (q=5), 49 (q=7), 81 (q=9), .. etc. or conference matrices (with border), vitrages of orders 1+p2.

DISTANTION D=2, M PLUGS INTO S



Hadamard matrices: vitrages of orders 16, 36


Mersenne (Jacobstal) matrix M, order q, can be plugged into Seidel matrix S, orders p=q+2, p=q–2, with levels arised to a=1, –b; b=1. With border, it gives Hadamard matrices, vitrages of orders 1+qp: 16/4 (q=3), 64/36 (q=7), 256/196 (q=15), [non 400]/324 (q=19), 576/[non 484] (q=23), 784/676 (q=27), [non 1024]/900 (q=31), 1296?/[non 1156] (q=35?), 1600/1444 (q=39), .. etc.

DISTANTION D=2, S PLUGS INTO M



Hadamard matrices: vitrages of orders 16, 36


Seidel (Jacobstal) matrix S, order q, can be plugged into Mersenne matrix M, orders p=q+2, p=q–2, with levels arised to a=1, –b; b=1. With border, it gives Hadamard matrices, vitrages of orders 1+qp: 16/4 (q=3), 64/36 (q=7), 256/196 (q=15), [non 400]/324 (q=19), 576/[non 484] (q=23), 784/676 (q=27), [non 1024]/900 (q=31), 1296?/[non 1156] (q=35?), 1600/1444 (q=39), .. etc.

DISTANTION D=1: H (M-BASED) PLUGS INTO M



Hadamard matrix: vitrage of orders 56, Nick's and Scarpis versions


Famous historical case. Pictures of Hadamard matrix, order n=q(q+1)=56, q=7, say: Mersenne matrix M, order q, can be plugged with the border and a cell rotation into itself, with levels arised to a=1, –b; b=1. Given versions of Nick's and Scarpis algorithms use similar ideas to plug Mersenne matrix into itself. It generates Hadamard matrices, vitrages of orders: 12 (q=3), 56 (q=7), 132 (q=11), .. etc. Orders 12, 56, and, possible, 132, were found by Scarpis, 1898, that allowed to exceeed orders 12, 20, given by Hadamard: examples. Nick's algorithm (plugin based, written with Yura) given here is new.

Скарпи сто лет назад вставил матрицу Адамара размера 4 в ее блок, без каймы, размера 3×3. Тогда еще не было представлений о нормальной форме и блок не назывался core. По моему, он даже отсекал не первую строку. Метод работал, но требовал смещения этого самого ядра пропорционально отстоянию замещаемого элемента от левого верхнего угла шпигуемой матрицы. Проверил на вставке матрицы Адамара порядка 8 в основу 7×7, работает. Опубликовал статью, появились метод Скарпи и зомбирование сознания, что смещения нужны.


Смещения не всегда нужны. Можно назвать обобщение метода Скарпи методом снежинки. Смещение есть и его, как бы, нет. Ведь если поворот совпадает с симметрией, снежинки, то после поворота она будет такой же. В методе Скарпи надо антисимметрировать матрицу (кососимметрия), и поменять порядок вставки. Вставлять блоки размеров 3×3, 7×7 в матрицы Адамара размеров 4, 8, т. п. После чего смещение вставлямых блоков исчезнет. Проскочим сложное сочетание 27 на 28, не заметив проблем, получив матрицу 756. Вот и все обобщение метода Скарпи. Еще на диагональ, J, как обычно. Об этом подробнее далее.

Так можно расправиться и с ядром порядка 13, получив взвешенную W(14*13=182,169)!

DISTANTION D=1: S PLUGS INTO H



Conference matrix C30


Conference matrix, order n=q(q+1)=30. Seidel matrix S, order q=5, can be plugged with a cell rotation into symmetric conference matrix, order p=q+1=6=2×3, with levels arised to a=1, –b; b=1, look block two-circulant matrices. Case n=q(q+1), q is odd, q+1=2×(q+1)/2 generalises Scarpis one (he observed q=3 (mod 4)).

It is a method common with Scarpis one.

DISTANTION D=3: M PLUGS INTO H, SKEW HADAMARD MATRIX



Hadamard matrices: vitrages of orders 28, 88


Mersenne matrix M, order q, can be plugged into skew Hadamard matrix (skew conference matrix), order p=q–3, with levels arised to a=1, –b; b=1. It gives Hadamard matrices, vitrages of orders: 28 (q=7), 88 (q=11), 180 (q=15), 304 (q=19), 460 (q=23), 648 (q=27), 868 (q=31), 1120? (q=35?), 1404 (q=39), .. etc.

DISTANTION D=3: M PLUGS INTO C, SYMMETRIC CONFERENCE MATRIX



Circulant matrices C6 and C18


Mersenne matrix M, order q=3, can be plugged with a cell rotation into symmetric conference matrix, order p=q+3=6, with levels arised to a=1, –b; b=1. The black square playes the role of Mathon's constructions well known for order 45. Seidel matrix S, order q=5, can be plugged with a cell rotation into symmetric conference matrix, order p=q+1=6, look block two-circulant matrices.



Conference matrix C6 and vitrage of order 18



Similar to Bush-type construction


DISTANTION D=2: REGULAR HADAMARD MATRICES



Symmetric Nick's regular Hadamard matrix H36 based on 36=1+5×7


or

or


Symmetric regular Hadamard matrix H100 based on 100=1+9×11


or

SAILING DISTANTION TO 4: M 167 AND PROPUS 4: 4×167=668

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


A=M(a=1,–b), B=C=D=M(1,–1)


Calculate Cretan Propus 668 with level b=(p+1–sqrt(4*p+1))/p, where p=(4*n–12)/16, n=167 (order of Mersenne matix), P'*P=ωI.



Cretan Propus 668 and hystogramm of its moduli of entries


It is a class of Cretan Propisi ..., [428], 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964..



Mersenne 167

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