A \((v, k, \lambda)\) covering design of order \(v\), block size \(k\), and index \(\lambda\) is a collection of \(k\)-element subsets, called blocks of a set \(V\) such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks in a covering design. In this paper we solve the covering problem with \(k = 5\) and \(\lambda = 4\) and all positive integers \(v\) with the possible exception of \(v = 17, 18, 19, 22, 24, 27, 28, 78, 98\).

Let \(\rho\) be an \(h\)-ary areflexive relation. We study the complexity of finding a strong \(h\)-coloring for \(\rho\), which is defined in the same way for \(h\)-uniform hypergraphs.In particular we reduce the \(H\)-coloring problem for graphs to this problem, where \(H\) is a graph depending on \(\rho\). We also discuss the links of this problem with the problem of
finding a completeness criterion for finite algebras.

Let \( {Z}_k\) be the cyclic group of order \( k\). Let \( H\) be a graph. A function \( c: E(H) \to {Z}_k\) is called a \( {Z}_k\)-coloring of the edge set \( E(H)\) of \(H\). A subgraph \( G \subseteq H\) is called zero-sum (with respect to a \( {Z}_k\)-coloring) if \( \sum_{e \in E(G)} c(e) \equiv 0 \pmod{k}\). Define the zero-sum Turán numbers as follows. \( T(n, G, {Z}_k)\) is the maximum number of edges in a \( {Z}_k\)-colored graph on \( n\) vertices, not containing a zero-sum copy of \( G\). Extending a result of [BCR], we prove:

THEOREM.
Let \( m \geq k \geq 2\) be integers, \( k | m\). Suppose \( n > 2(m-1)(k-1)\), then
\[T(n,K_{1,m},{Z}_k) =
\left\{
\begin{array}{ll}
\frac{(m+k-2)-n}{2}-1, & if \;\; n-1 \equiv m \equiv k \equiv 0 \pmod{2}; \\
\lfloor \frac{(m+k-2)-n}{2} \rfloor, & otherwise.
\end{array}
\right.\]

A tricover of pairs by quintuples on a \(v\)-element set \(V\) is a family of 5-element subsets of \(V\), called blocks, with the property that every pair of distinct elements of \(V\) occurs in at least three blocks. If no other such tricover has fewer blocks, the tricover is said to be minimum, and the number of blocks in a minimum tricover is the tricovering number \(C_3(v,5,2)\), or simply \(C_3(v)\). It is well known that \(C_3(v) \geq \lceil \frac{{v} \lceil \frac {3(v-1)}{4} \rceil} {5} \rceil = B_3(v)\), where \(\lceil x\rceil\) is the smallest integer that is at least \(x\). It is shown here that if \(v \equiv 1 \pmod{4}\), then \(C_3(v) = B_3(v) + 1\) for \(v \equiv 9\) or \(17 \pmod{20}\), and \(C_3(v) = B_3(v)\) otherwise.

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 \(\tau_1(G) = s\).