For \( n \in \mathbb{N} \), we interpret the vertex set \( W_n \) of the \( n \)-cube as a vector space over the field \( \mathbb{F}_2 \) and prove that a regular \( n \)-simplex can be inscribed into the \( n \)-cube such that its vertices constitute a subgroup of \( W_n \) if and only if \( n+1 \) is a power of 2. Furthermore, a connection to the theory of Hamming Codes will be established.

An \( (n,k) \) binary self-orthogonal code is an \( (n,k) \) binary linear code \( C \) that is contained in its orthogonal complement \( C^\bot \). A self-orthogonal code \( C \) is self-dual if \( C = C^\bot \). Two codes, \( C_1 \) and \( C_2 \), are \emph{equivalent} if and only if there exists a coordinate permutation of \( C_1 \) that takes \( C_1 \) into \( C_2 \). The automorphism group of a code \( C \) is the set of all coordinate permutations of \( C \) that takes \( C \) into itself.

This paper is a continuation of the work presented in [2], in which we described an algorithm for enumerating inequivalent binary self-dual codes. We used our algorithm to enumerate the self-dual codes of length up to and including 32. Our algorithm also found the size of the automorphism group of each code.

We have since made several improvements to our algorithm. It now generally runs faster. It also now finds generators for the automorphism group of each code. We have used our improved algorithm to enumerate the self-dual codes of length 34. We have also found the automorphism groups for each of our self-dual codes of length less than or equal to 34. The list of length 34 codes are new, as are the lists of automorphism groups for the length 32 and length 34 codes. We have found there are 19914 inequivalent length 34 codes with distance 4 and 938 length 34 codes with distance 6.

A graph is claw-free if it has no induced \( K_{1,3} \) subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edge-connected claw-free graph has a spanning Eulerian subgraph with maximum degree at most 4.

A labeling \( f \) of a graph \( G \) is called semi-H-cordial if for each vertex \( v \), \( |f(v)| \leq 1 \), \( |e_f(1) – e_f(-1)| \leq 1 \) and \( |v_f(1) – v_f(-1)| \leq 1 \). In this paper we study the forcing semi-H-cordial numbers of paths, cycles, stars, trees, Dutch-windmill graphs, wheels, grids and cylinders.

A three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) is a three-fold resolvable packing with maximum possible number of parallel classes, each containing one block of size 3 and all other blocks of size 4. This article investigates the spectra of three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) for \( s = 5 \) and \( 6 \), and we show that it contains all positive integers \( v \equiv s – 4 \pmod{4} \) with \( v \geq 17 \) if \( s = 5 \), and \( v \geq 26 \) if \( s = 6 \).

Let \( G \) be a \( (p,q) \)-graph in which the edges are labeled \( 1, 2, 3, \ldots, q \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If \( G \) can be labeled so that the vertex sums are distinct, mod \( p \), then \( G \) is said to be edge-graceful. If the edges of \( G \) can be labeled \( 1, 2, 3, \ldots, q \) so that the vertex sums are constant, mod \( p \), then \( G \) is said to be edge-magic. It is conjectured by Lee [9] that any connected simple \( (p,q) \)-graph with \( q(q+1) \equiv p(p-1)/2 \pmod{p} \) vertices is edge-graceful. We show that the conjecture is true for maximal outerplanar graphs. We also completely determine the edge-magic maximal outerplanar graphs.

We enumerate the self-orthogonal Latin squares of orders 1 through 9 and discuss the nature of the isomorphism classes of each order. Furthermore, we consider the possibility of enlarging sets of self-orthogonal Latin squares to produce complete sets.

A vertex-magic total labeling on a graph with \( v \) vertices and \( e \) edges is a one-to-one map taking the vertices and edges onto the integers \( 1, 2, \ldots, v+e \) with the property that the sum of the label on a vertex and the labels of its incident edges is constant, independent of the choice of vertex. We give vertex-magic total labelings for several classes of regular graphs. The paper concludes with several conjectures and open problems in the area.

In this paper, we complete the classification of optimal binary linear self-orthogonal codes up to length 25. Optimal self-orthogonal codes are also classified for parameters up to length 40 and dimension 10. The results were obtained via two independent computer searches.

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.