Related papers: New Nonexistence Results on Circulant Weighing Mat…

In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one…

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 $k$ cannot be written as the sum of…

Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no…

With the help of a computer, we prove the assertion made in the title.

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a…

We study the local geometry of the zero pattern of a weighing matrix $W(23,16)$. The geometry consists of $23$ lines and $23$ points where each line contains $7$ points. The incidence rules are that every two lines intersect in an odd…

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…

Assume that $H$ is a circulant Hadamard matrix of order $n\geq 4$. We consider an appropriate stochastic matrix $S$ of order $n$ depending on $H$. This allows us to prove that $n = 4$. Thus, there are only $10$ circulant Hadamard matrices.

A Hadamard matrix is a scaled orthogonal matrix with $\pm 1$ entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when $n$ is a multiple of 4. A conjecture attributed to Ryser is…

We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and…

Integer weighing matrices (IW-matrices for short) are integer valued orthogonal square matrices. One usecase of these is to create classical weighing matrices with various block structures. In this paper we study and classify the space…

This work presents closed formulas for determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under a condition about a sum of scalar products of rows of two other circulant matrices of size $n/2$ associated to $H.$

We study the structure of unit weighing matrices of order n and weights 2, 3 and 4. We show that the number of inequivalent unit weighing matrices UW(n,4) depends on the number of decompositions of n into sums of non-negative multiples of…

The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search lead to some infinite classes of disjoint weighing matrices, which in turn are shown to…

Let n\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\circ(j_{0},j_{1},...,j_{n-1}) be the n\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The…

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good matrices of order 57. We also find that…

For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $\omega(h) =k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result…

We describe an approach to the circulant Hadamard conjecture based on Walsh-Fourier analysis. We show that the existence of a circulant Hadamard matrix of order $n$ is equivalent to the existence of a non-trivial solution of a certain…

Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant…