In this paper we examine the classical Williamson construction for Hadamard matrices, from the point of view of a striking analogy with isomorphisms of division algebras. By interpreting the 4 Williamson array as a matrix arising from the real quaternion division algebra, we construct Williamson arrays with 8 matrices, based on the real octonion division algebra. Using a Computational Algebra formalism we perform exhaustive searches for even-order 4-Williamson matrices up to 18 and odd- and even-order 8-Williamson matrices up to 9 and partial searches for even-order 4-Williamson matrices up to 22 and odd- and even-order 8-Williamson matrices for orders 10 — 13. Using Magma, we conduct searches for inequivalent Hadamard matrices within all the sets of matrices obtained by exhaustive and partial searches. In particular, we establish constructively ten new lower bounds for the number of inequivalent, Hadamard matrices of the consecutive orders 72, 76, 80, 84, 88, 92, 96, 100, 104 and 108.

This article continues the study of a class of non-terminating expansions of sin\( (m\alpha) \) (even \(m \geq 2 \)) which in each case possesses embedded Catalan numbers. A known series form of the sine function (said to be associated with Euler) is taken here as our basic representation, the coefficient of the general term being developed analytically in an interesting fashion and shown to be dependent on the Catalan sequence in the manner expected.
The work, which has a historical backdrop to it, is discussed in the context of prior results by the author and others.

A \((v,k,\lambda)\) covering design is a set of \(b\) blocks of size \(k\) such that each pair of points occurs in at least \(\lambda\) blocks, and the covering number \(C(v, k, \lambda)\) is the minimum value of \(b\) in any \((v, k, \lambda)\) covering design. For \(k = 5\) and \(v\) even, there are 24 open cases with \(2 \leq \lambda \leq 21\), each of which is the start of an open series for \(\lambda,\lambda + 20, \lambda + 40, \ldots\). In this article, we solve 22 of these cases with \(\lambda \leq 21\), leaving open \((v, 5, \lambda)=(44, 5, 13)\) and \((44, 5, 17)\) (and the series initiated for the former).

The basis number of a graph \( G \) is defined to be the least integer \( d \) such that there is a basis \( \mathcal{B} \) of the cycle space of \( G \) such that each edge of \( G \) is contained in at most \( d \) members of \( \mathcal{B} \). MacLane [16] proved that a graph, \( G \), is planar if and only if the basis number of \( G \) is less than or equal to 2. Ali and Marougi [3] proved that the basis number of the strong product of two cycles and a path with a star is less than or equal to 4. In this work, (1) we prove the basis number of the strong product of two cycles is 3. (2) We give the exact basis number of a path with a tree containing no subgraph isomorphic to a 3-special star of order 7. (3) We investigate the basis number of a cycle with a tree containing no subgraph isomorphic to a 3-special star of order 7. The results in (1) and (2) improve the upper bound of the basis number of the strong product of two cycles and a star with a path which were obtained by Ali and Marougi.

A set \( S \) of vertices is a total dominating set of a graph \( G \) if every vertex of \( G \) is adjacent to some vertex in \( S \). The minimum cardinality of a total dominating set is the total domination number \( \gamma_t(G) \). We show that for a nontrivial tree \( T \) of order \( n \) and with \( \ell \) leaves, \( \gamma_t(T) \geqslant \frac{n + 2 – \ell}{2} \), and we characterize the trees attaining this lower bound.

This paper presents a computationally efficient algorithm for solving the following well-known die problem: Consider a “crazy die” to be a die with \( n \) faces where each face has some “cost”. Costs need not be sequential. The problem is to determine the exact probability that the sum of costs from \( U \) throws of this die is \( \geq T \), \( T \in \mathbb{R} \). Our approach uses “slice” volume computation in \( U \)-dimensional space. Detailed algorithms, complexity analysis, and comparison with traditional generating functions approach are presented.

Difference systems of sets (DSS), introduced by Levenshtein, are used to design code synchronization in the presence of errors. The paper gives a new lower bound of DSS’s size.

In a loop transversal code, the set of errors is given the structure of a loop transversal to the linear code as a subgroup of the channel. A greedy algorithm for specifying the loop structure, and thus for the construction of loop transversal codes, was discussed by Hummer et al. Apart from some theoretical considerations, the focus was mainly on error correction, in the white noise case constructing codes with odd minimum distance. In this paper, an algorithm to compute loop transversal codes with even minimum distance is given. Some record-breaking codes over a 7-ary alphabet are presented.

Let \( a, b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) =
\begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a+\frac{q-2}{2})\} & \text{if } q \text{ is even} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/{2})\} & \text{if } q \text{ is odd}
\end{cases}
\]

\[
P(b) =
\begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/{2})\} & \text{if } p \text{ is even} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (\frac{p-3}{2})/2)\} & \text{if } p \text{ is odd}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+ : V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{f(u,v) : (u,v) \in E(G)\}.
\]

We investigate \( Q(a)P(b) \) super edge-graceful graphs.

Let \( A \) be a non-trivial abelian group. We call a graph \( G = (V,E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A \setminus \{0\} \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum f(u,v) \) where the sum is over all \( (u,v) \in E(G) \), is a constant map. In this paper, we show that \( K_{k_1,k_2,\ldots,k_n} \) (where \( K_{i} \geq 2 \)) is \( A \)-magic, for all \( A \) where \( |A| \geq 3 \).