# Group developed weighing matrices

@article{Arasu2013GroupDW, title={Group developed weighing matrices}, author={K. Arasu and J. Hollon}, journal={Australas. J Comb.}, year={2013}, volume={55}, pages={205-234} }

A weighing matrix is a square matrix whose entries are 1, 0 or −1, such that the matrix times its transpose is some integer multiple of the identity matrix. We examine the case where these matrices are said to be developed by an abelian group. Through a combination of extending previous results and by giving explicit constructions we will answer the question of existence for 318 such matrices of order and weight both below 100. At the end, we are left with 98 open cases out of a possible 1,022… Expand

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#### 6 Citations

Structure of group invariant weighing matrices of small weight

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The aim of this work is to construct families of weighing matrices via automorphisms and cohomology. We study some well known families such as Payley's conference and Hadamard matrices and Projective… Expand

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A circulant weighing matrix $W = (w_{i,j}) \in CW(n,k)$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{-1, 0, +1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave tables of known… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

A note on balanced weighing matrices

- Mathematics
- 1975

A balanced weighing matrix is a square orthogonal matrix of 0’s, 1’s and −1’s such that the matrix obtained by squaring entries is the incidence matrix of a (v, k, λ) configuration. Properties of… Expand

Some New Results on Circulant Weighing Matrices

- Mathematics
- 2001

We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish the… Expand

Perfect Ternary Arrays

- Physics
- 1999

A perfect ternary array is an r-dimensional array with entries 0, +1 and —1 such that all of its out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a group… Expand

Study of proper circulant weighing matrices with weight 9

- Computer Science, Mathematics
- Discret. Math.
- 2008

The first theoretical proof of the spectrum of orders n for which circulant weighing matrices with weight 9 exist is provided, which consists of those positive integers n, which are multiples of 13 or 24. Expand

On circulant and two-circulant weighing matrices

- Computer Science
- Australas. J Comb.
- 2010

New weighing matrices are constructed which are listed as open in the second edition of the Handbook of Combinatorial Designs and fill a missing entry in Strassler’s table with answer “YES”. Expand

Hadamard matrices of order 764 exist

- Computer Science
- Comb.
- 2008

Two Hadamard matrices of order 764 of Goethals– Seidel type are constructed and it is shown that among the remaining 14 integers n only four are less than 1000, and the revised list now includes these four integers. Expand

Circulant weighing matrices

- Mathematics, Computer Science
- Cryptography and Communications
- 2010

The results fill in 52 missing entries in Strassler’s table of circulant weighing matrices (Strassler 1997), which considers matrices of order 1–200 with weight k ≤ 100. Expand

Circulant weighing designs

- Mathematics
- 1996

Algebraic techniques are employed to obtain necessary conditions for the existence of certain families of circulant weighing designs. As an application we rule out the existence of many circulant… Expand

Wieferich pairs and Barker sequences

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 2009

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 1030. This improves the lower bound on the length of a long Barker… Expand

Determination of all possible orders of weight 16 circulant weighing matrices

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2006

We show that a circulant weighing matrix of order n and weight 16 exists if and only if n>=21 and n is a multiple of 14,21 or 31.