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Although Hadamard matrices have been investigated since the nineteenth century, relatively little is known about their higher-dimensional analogues. In this paper, we introduce two constructions of Hadamard hypercubes. The first…
In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this…
Using the ideas of concatenation construction of codes over the $q$-ary alphabet, we modify the known generalized Sylvester-type construction of the Hadamard matrices. The new construction is based on two collections of the Hadamard…
Cretan matrices are orthogonal matrices with elements $\leq 1$. These may have application in forming some new materials. There is a search for Cretan matrices, especially with high determinant, for all orders. These have been found by both…
Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show…
We construct several cyclic $(v;k_1,k_2,k_3;\lambda)$ difference families with $v\equiv3 \pmod{4}$ a prime and $\lambda=k_1+k_2+k_3-(3v-1)/4$. Such families can be used in conjunction with the well-known Paley-Todd difference sets to…
A family of two-unitary complex Hadamard matrices (CHM) stemming from a particular matrix, of size $36$ is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a…
It is known that a real symmetric circulant matrix with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we…
Building on the results of our previous work on Euclidean leaper tours, considering all integers $k>1$ and $h>0$, we study the existence of Hamiltonian cycles in the vertex set $C(2,k):=\{0,1\}^k$ of the $k$-dimensional hypercube when the…
Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first…
An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering…
We construct new symmetric Hadamard matrices of orders $92,116$, and $172$. While the existence of those of order $92$ was known since 1978, the orders $116$ and $172$ are new. Our construction is based on a recent new combinatorial array…
The cycle set of a graph $G$ is the set consisting of all sizes of cycles in $G$. Answering a conjecture of Erd\H{o}s and Faudree, Verstra\"{e}te showed that there are at most $2^{n - n^{1/10}}$ different cycle sets of graphs with $n$…
A spin model (for link invariants) is a square matrix $W$ which satisfies certain axioms. For a spin model $W$, it is known that $W^TW^{-1}$ is a permutation matrix, and its order is called the index of $W$. F. Jaeger and K. Nomura found…
A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by $2J$ toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real but non-Hermitian $2K$ by $2K$…
Recently, binary cyclic codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$ have been a hot topic since their minimum distances have a square-root bound. In this paper, we construct four classes of binary cyclic codes…
In this paper, we study the distribution of the number of occurrences of the simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$ pattern if $i <…
Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The…
It can be shown that any symmetric $(0,1)$-matrix $A$ with $\tr A = 0$ can be interpreted as the adjacency matrix of a simple, finite graph. The square of an adjacency matrix $A^2=(s_{ij})$ has the property that $s_{ij}$ represents the…
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 $H.$