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Williamson matrices of orders 4·29 and 4·31

Dragomir Ž Doković
1992 Journal of combinatorial theory. Series A  
It is established that there is only one, up to equivalence, Williamson matrix of order 4.29 and there are only two non-equivalent such matrices of order 4.31.  ...  There exist five non-equivalent Williamson matrices of order 4.33 while only four of them are listed in [4] .  ...  For more details see our forthcoming paper "Williamson Matrices of Order 4n for n = 33, 35, 39" in Discrete Mathematics. is equivalent to the solution found byBaumert [ 11.  ... 
doi:10.1016/0097-3165(92)90071-2 fatcat:ypnjih7kvzb53fkjeu5ruuhr4y

Applying Computer Algebra Systems with SAT Solvers to the Williamson Conjecture [article]

Curtis Bright, Ilias Kotsireas, Vijay Ganesh
2019 arXiv   pre-print
We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70.  ...  As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35.  ...  The authors thank Dragomir Ðoković informing us about the construction of Turyn (1970) using complex Hadamard matrices.  ... 
arXiv:1804.01172v2 fatcat:nj2um7a7qzadbaa2om2ec3mr2a

Supplementary difference sets with symmetry for Hadamard matrices [article]

Dragomir Z. Djokovic
2009 arXiv   pre-print
Five new Williamson matrices over the elementary abelian groups of order 25, 27 and 49 are constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127 are presented.  ...  An old non-published example of G-matrices of order 37 is also included.  ...  We refer to [11] for the up-to-date information on cyclic Williamson matrices, including the complete listing of all nonequivalent such matrices of odd order59.  ... 
arXiv:0903.5053v2 fatcat:nupvjb5ihfe5lcaiybp7kd7wve

The SAT+CAS method for combinatorial search with applications to best matrices

Curtis Bright, Dragomir Ž. Đoković, Ilias Kotsireas, Vijay Ganesh
2019 Annals of Mathematics and Artificial Intelligence  
We show the best matrix conjecture (that best matrices exist in all orders of the form r^2+r+1) which was previously known to hold for r≤6 also holds for r=7.  ...  Additionally, we apply the method to construct the largest best matrices yet known and present new skew Hadamard matrices constructed from best matrices.  ...  However, this approach was only able to find Williamson matrices for orders up to n = 30.  ... 
doi:10.1007/s10472-019-09681-3 fatcat:paxe7oruzfeypf7cd2d6b75slu

Page 603 of Mathematical Reviews Vol. 50, Issue 3 [page]

1975 Mathematical Reviews  
Currently, Williamson matrices are known to exist for all orders less than 100 except: 35, 39, 47, 53, 59, 65, 67, 70, 71, 73, 76, 77, 83, 89, 94.  ...  Hadamard matrices of order less than 16 are unique up to equivalence. There are just 5 equivalence classes for order 16, 3 for order 20 and at least 6 for order 2" (n= 5).  ... 

Page 5350 of Mathematical Reviews Vol. , Issue 92j [page]

1992 Mathematical Reviews  
Summary: “It is established that there is only one, up to equiv- alence, Williamson matrix of order 4-29 and there are only two nonequivalent such matrices of order 4-31.” 92j:05035 05B20 15A51 Hwang,  ...  ) Williamson matrices of orders 4 - 29 and 4-31.  ... 

Page 1945 of Mathematical Reviews Vol. 49, Issue 6 [page]

1975 Mathematical Reviews  
Hence Hadamard matrices of order 4wt and 20wt exist for the values of w and ¢ mentioned 49 #4/10575-10579 above, in particular for the orders 4-59 and 4-101, which were previously unknown. J. M.  ...  From a Williamson-type Hadamard matrix of order 4w and a set of Baumert-Hall units of order 4¢, a Hadamard matrix of order 4wt can be constructed.  ... 

On Z_t ×Z_2^2-cocyclic Hadamard matrices [article]

Victor Alvarez, Felix Gudiel, Maria Belen Guemes
2014 arXiv   pre-print
In particular, these notions lead to the establishment of some bounds on the number and distribution of 2-coboundaries over Z_t ×Z _2^2 to use and the way in which they have to be combined in order to  ...  Let H be the set of cocyclic Hadamard matrices over Z_t ×Z_2^2 having a symmetric diagram. We also prove that the set of Williamson type matrices is a subset of H of size | H|/t.  ...  ACKNOWLEDGMENTS The authors want to express their gratitude to the anonymous referees for their comments and suggestions, which have permitted to improve the readability and understandability of the paper  ... 
arXiv:1112.4296v2 fatcat:p6ifxhdq7jfbvncv3cuhcdevjq

Symmetric Hadamard matrices of orders 268, 412, 436 and 604 [article]

N.A. Balonin, D.Z. Djokovic
2018 arXiv   pre-print
We construct many symmetric Hadamard matrices of small order by using the so called propus construction.  ...  Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604.  ...  Acknowledgements The research of the first author leading to these results has received funding from the Ministry of Education and Science of the Russian Federation according to the project part of the  ... 
arXiv:1803.08787v1 fatcat:gc6evadp6zcebojd23xop3njr4


N. A. Balonina, D. Z. Djokovicb
2015 Information and Control Systems  
We investigate the construction of Hadamard (and weighing matrices) from two negacyclic blocks (2N-type). The Hadamard matrices of 2N-type are equivalent to negaperiodic Golay pairs.  ...  We show that the Ito's conjecture about Hadamard matrices is equivalent to the conjecture that negaperiodic Golay pairs exist for all even lengths.  ...  The known series of four Williamson matrices of odd order give rise to the series of NG-pairs. As an example, we have computed four quasi-Williamson matrices of order 35.  ... 
doi:10.15217/issn1684-8853.2015.5.2 fatcat:pqwnhupsxjat7hvjdmvwwywp4u

Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings

R.J Turyn
1974 Journal of combinatorial theory. Series A  
If a Williamson matrix of order 4w exists and a special type of design, a set of Baumert-Hall units of order 4t, exists, then there exists a Hadamard matrix of order 4tw.  ...  The constructions relate to various topics such as pulse compression and image encodings. 313  ...  It seems very probable that there is some construction of Hadamard matrices of all orders 4t which is very analogous to the matrices discussed here: Williamson matrices exist of all orders 4t fort,<31,  ... 
doi:10.1016/0097-3165(74)90056-9 fatcat:w4hnqh6fmvfrvgudj7vreqkdf4

Higher dimensional orthogonal designs and applications

J. Hammer, J. Seberry
1981 IEEE Transactions on Information Theory  
come to be known as Williamson matrices of side t.  ...  APPLYING ORTHOGONAL DESIGNS TO FORM HIGHER DIMENSIONAL HADAMARD MATRICES Williamson matrices of order t (or 4-suitable matrices in our present terminology) exist for the following orders (see [15[, [7  ... 
doi:10.1109/tit.1981.1056426 fatcat:bjmx4yar4vee3a5rbl2jz3ueyu

Williamson theorem in classical, quantum, and statistical physics [article]

F. Nicacio
2021 arXiv   pre-print
The objective of this text is to present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics.  ...  A modest introduction of the symplectic formalism in quantum mechanics is presented, which consequently opens up the use of the theorem to study quantum normal modes and quantum small oscillations, allowing  ...  WILLIAMSON THEOREM The question addressed by the Williamson theorem is the diagonalization of positive definite matrices through symplectic matrices.  ... 
arXiv:2106.11965v1 fatcat:t45cqhati5gkjhcryiaaxj7r3e

Page 5273 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
Stanley Gill Williamson (1-UCSD-CS; La Jolla, CA) 2001h:05015 05A15 33B10 Johnson, Warren P. (1-PAS; University Park, PA) Some polynomials associated with up-down permutations.  ...  The identity ch(F,) = ch(U,) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson.  ... 

Some new operations on Zt x Z2,2-cocyclic Hadamard matrices [article]

Victor Alvarez, Felix Gudiel, Maria Belen Guemes
2014 arXiv   pre-print
These operations will be defined on the set of coboundaries defining the matrix, preserve the Hadamard character of the cocyclic matrices, and allow us to obtain new Hadamard matrices from old ones.  ...  We split the set of Hadamard matrices into disjoint orbits, define representatives for them and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way.  ...  • The result about Williamson matrices allows us to go beyond t = 23.  ... 
arXiv:1112.4300v2 fatcat:ruyuuufb7jeqtgzfivk3afnec4
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