NEGACYCLIC WEIGHING MATRICES

© Nickolay A. Balonin, Dragomir Z. Djokovic, 22.04.2015

Hadamard-type matrix catalogue and on-line algorithms

PALEY MATRICES

WILLAMSON MATRICES


2-NEGACYCLIC WEIGHING MATRICES



2N-W(n,n–2) and 2N-W(n,n–4)

GOLAY-PALEY/GOLAY CODES MULTIPLIER

NUMERICAL ALGORITHM 2N W(n,nk)

GALOIS FIELD ALGORITHM 2N W(n,nk)

2C MATRICES: n=4+k*16, n=12+k*16

2-TOEPLITZ WEIGHING MATRICES LIBRARY



2N-W(n,n–3) and 2N-W(n,n–4)

1-NEGACYCLIC WEIGHING MATRICES


Specifical N1-construction W(n,n–2)=[A B;–B A], based on two non orthogonal (!) Toeplitz blocks A, B, where 1-column of B is reversed and shifted 1-row of A, 1-row of B is reversed and shifted the inversed 1-column of A.

N1-MATRICES WITH SYMMETRIES

WEIGHING MATRICES n=4+16*k


Weighing matrices W(n,n–2), n=4+16*k, 4, 20, 36!, 52, [68?], 84, 100?, .. with two zeros, giving two negacyclic Hadamard H2n=[A B;B' –A'] or zipper H*2n=[A B;B –A] matrices. Lenght v=n/2=34 can be crytical for these matrices, so order 68 is under doubts.

TWO NEGACYCLIC VERSIONS H2n=[A B;B' –A']



Weighing matrices W(4,2) and H8



Weighing matrices W(20,18) and H40



Weighing matrices W(36,34) and H72



Weighing matrices W(52,50) and H104



Weighing matrices W(52,50)



Weighing matrices W(84,82) and H168



Weighing matrices W(100,98) and H200

ZIPPER VERSION H*72=[A B;B –A]



Weighing matrices W(36,34) and H*72

WEIGHING MATRICES n=12+16*k


Weighing matrices W(n,n–2), n=12+16*k, 12, 28, [44?], 60, 76, [92?], 108?, .. with two zeros, giving two negacyclic Hadamard H2n=[A B;B' –A'] or zipper H*2n=[A B;B –A] matrices. Lenghts v=n/2=22 and 46 can be crytical for these matrices, so orders 44 and 92 are under doubts.



Weighing matrices W(12,10) and H24



Weighing matrices W(28,26) and H56



Weighing matrices W(60,58) and H120



Weighing matrices W(76,74) and H152

ZIPPER VERSIONS



Weighing matrices W(12,10) and H*12



Weighing matrices W(28,26) and H*28

GALOIS FIELD CALCULATIONS


Calculate N1 as [A B;–B A], A,B – Toeplitz blocks; we tie A-block with NPG-pair, then find B (GF(p2), p=4*t–1).

Calculate N1 as [A B;–B A], A,B – Toeplitz blocks; we tie A-block with NPG-pair, then find B (GF(p2), p=2*t–1).



Similar by sense W(n,n–2) matrix, based on two nega-circulant blocks.

Prove a root

By two C-matrices:

By shift

VITOLD (NEGA-CYCLIC C-MATRICES) | JANKO (BLOCKS OF BUSH-TYPE HM)


НОВЫЙ АЛГОРИТМ ПОИСКА

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