We investigate collections \( {H} = \{H_1, H_2, \ldots, H_m\}\) of pairwise disjoint \(w\)-subsets \(H_i\) of an \(r\)-dimensional vector space \(V\) over \( {GF}(q)\) that arise in the construction of byte error control codes. The main problem is to maximize \(m\) for fixed \(w, r,\) and \(q\) when \({H}\) is required to satisfy a subset of the following properties: (i) each \(H_i\) is linearly independent; (ii) \(H_i \cap H_j = \{0\}\) if \(i \neq j\); (iii) \((H_i) \cap (H_j) = \{0\}\) if \(i \neq j\);( iv) any two elements of \(H_i \cup H_j\) are linearly independent;(v) any three elements of \(H_1 \cup H_2 \cup \cdots \cup H_m\) are linearly independent.
Here \((x)\) denotes the subspace of \(V\) spanned by \(X\). Solutions to these problems yield linear block codes which are useful in controlling various combinations of byte and single bit errors in computer memories. For \(r = w + 1\) and for small values of \(w\) the problem is solved or nearly solved. We list a variety of methods for constructing such partial partitions and give several bounds on \(m\).

There is a conjecture of Golomb and Taylor that asserts that the Welch construction for Costas sequences with length \(p-1\), \(p\) prime, is the only one with the property of single periodicity.

In the present paper we present and prove a weaker conjecture: the Welch construction is the only one with the property that its differences are a shift of the original sequence.

In this paper we give a necessary condition for the Steiner system \(S(3,4,16)\) obtained from a one-point extension of the points and lines of \( {PG}(3,2)\) to be further extendable to a Steiner system \(S(4,5,17)\).

The edge-toughness \(\tau_1(G)\) of a graph \(G\) is defined as

\[\tau_1(G) = \min\left\{\frac{|E(G)|}{w(G-X)} \mid X { is an edge-cutset of } G\right\},\]

where \(w(G-X)\) denotes the number of components of \(G-X\). Call a graph \(G\) balanced if \(\tau_1(G) = \frac{|E(G)|}{w(G-E(G))-1}\). It is known that for any graph \(G\) with edge-connectivity \(\lambda(G)\),
\(\frac{\lambda(G)}{2} < \tau_1(G) \leq \lambda(G).\) In this paper we prove that for any integer \(r\), \(r > 2\) and any rational number \(s\) with \(\frac{r}{2} < s \leq r\), there always exists a balanced graph \(G\) such that \(\lambda(G) = r\) and \(\tau_1(G) = s\).

Using the permutation action of the group \( {PSL}_2(2^m)\) on its dihedral subgroups of order \(2(2^m + 1)\) for the description of the class of designs \(W(2^m)\) derived from regular ovals in the Desarguesian projective plane of order \(2^m\), we construct a \(2\)-design of ovals for \(W(2^m)\) for \(m \geq 3\), and thus determine certain properties of the binary codes of these classes of designs.

We give a complete solution to the intersection problem for a pair of Steiner systems \(S(2,4,v)\), leaving a handful of exceptions when \(v = 25, 28,\) {and } \(37\).

A scheme for classifying hamiltonian cycles in \(P_m \times P_n\), is introduced.We then derive recurrence relations, exact and asymptotic values for the number of hamiltonian cycles in \(P_4 \times P_n\) and \(P_5 \times P_n\).

If each pair of vertices in a graph \(G\) is connected by a long path, then the cycle space of \(G\) has a basis consisting of long cycles. We propose a conjecture regarding the above relationship. A few results supporting the conjecture are given.

For any tree \(T\), let \(\gamma(T)\) represent the size of a minimum dominating set. Let \({E}_0\) represent the collection of trees with the property that, regardless of the choice of edge \(e\) belonging to the tree \(T\), \(\gamma(T – e) = \gamma(T)\). We present a constructive characterization of \({E}_0\).