351 research outputs found
A feasibility approach for constructing combinatorial designs of circulant type
In this work, we propose an optimization approach for constructing various
classes of circulant combinatorial designs that can be defined in terms of
autocorrelations. The problem is formulated as a so-called feasibility problem
having three sets, to which the Douglas-Rachford projection algorithm is
applied. The approach is illustrated on three different classes of circulant
combinatorial designs: circulant weighing matrices, D-optimal matrices, and
Hadamard matrices with two circulant cores. Furthermore, we explicitly
construct two new circulant weighing matrices, a and a
, whose existence was previously marked as unresolved in the most
recent version of Strassler's table
On the Existance of Certain Circulant Weighing Matrices
We prove nonexistence of circulant weighing matrices with parameters from ten previously open entries of the updated Strassler\u27s table. The method of proof utilizes some modular constraints on circulant weighing matrices with multipliers
On circulant and two-circulant weighing matrices
We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W(148, 144), W(152, 144), W(156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100
When the necessary conditions are not sufficient: sequences with zero autocorrelation function
Recently K. T. Arasu (personal communication) and Yoseph Strassler, in his PhD thesis, The Classification of Circulant Weighing Matrices of Weight 9, Bar-Ilan University, Ramat-Gan, 1997, have intensively studied circulant weighing matrices, or single sequences, with weight 9. They show many cases are non-existent. Here we give details of a search for two sequences with zero periodic autocorrelation and types (1,9), (1,16) and (4,9). We find some new cases but also many cases where the known necessary conditions are not sufficient. We instance a number of occasions when the known necessary conditions are not sufficient for the existence of weighing matrices and orthogonal de-signs constructed using sequences with zero autocorrelation function leading to intriguing new questions
Circulant weighing matrices
Circulant weighing matrices are matrices with entries in {-1,0,1}
where the rows are pairwise orthogonal and each successive row is obtained
from the previous row by a fixed cyclic permutation. They are useful in
solving problems where it is necessary to determine as accurately as
possible, the "weight" of n "objects" in n "weighings". They have
also been successfully used to improve the performance of certain optical
instruments such as spectrometers and image scanners.
In this thesis I discuss the basic properties of circulant weighing
matrices, prove most of the known existence results known to me at the time
of writing this thesis and classify the circulant weighing matrices with
precisely four nonzero entries in each row. The problem of classifying all
circulant weighing matrices is related to the "cyclic projective plane
problem". This relationship is established and I have devoted the final
chapter of this thesis to cyclic projective planes and their relationship
to circulant weighing matrices. The final theorem in this thesis yields
information about equations of the kind xy¯²=a in cyclic projective
planes
New Nonexistence Results on Circulant Weighing Matrices
A circulant weighing matrix is a square matrix of order
and entries in such that . In his thesis,
Strassler gave a table of existence results for such matrices with
and .
In the latest version of Strassler's table given by Tan
\cite{arXiv:1610.01914} there are 34 open cases remaining. In this paper we
give nonexistence proofs for 12 of these cases, report on preliminary searches
outside Strassler's table, and characterize the known proper circulant weighing
matrices.Comment: 15 page
Some non-existence and asymptotic existence results for weighing matrices
Orthogonal designs and weighing matrices have many applications in areas such
as coding theory, cryptography, wireless networking and communication. In this
paper, we first show that if positive integer cannot be written as the sum
of three integer squares, then there does not exist any skew-symmetric weighing
matrix of order and weight , where is an odd positive integer. Then
we show that for any square , there is an integer such that for each
, there is a symmetric weighing matrix of order and weight .
Moreover, we improve some of the asymptotic existence results for weighing
matrices obtained by Eades, Geramita and Seberry.Comment: To appear in International Journal of Combinatorics (Hindawi). in
Int. J. Combin. (Feb 2016
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