A binary code has bounded synchronization delay if there exists an integer \(s\) such that at most \(s\) consecutive bits are required to establish word synchronization in any message. The code whose words are lexicographically least in the non-periodic orbits determined by cyclic permutation of all words of length \(n\) is called the canonical bounded synchronization delay code. It has the maximal number of words possible in a synchronizable code of fixed word length. Any code of fixed word length \(n\) can be represented as a set of vertices in the \(n\)-cube. We prove that the canonical bounded synchronization delay code is a connected subset of the \(n\)-cube.

A degree sequence which is realisable by a \(3\)-uniform hypergraph is called \(3\)-graphic. Seven necessary conditions, one sufficient condition, and one equivalent condition for degree sequences to be \(3\)-graphic are derived. Moreover, four special classes of degree sequences are examined. For each class an equivalent condition for the sequences in this class to be \(3\)-graphic is derived. Using these conditions, all the \(3\)-graphic degree sequences of length at most seven have been determined.

We prove that if \(m\) is even then a partial \(m\)-cycle system on \(n\) vertices can be embedded in an \(m\)-cycle system on \(2mn+1\) vertices.

Ideas are described that speed up the lattice basis reduction algorithm of Lenstra, Lenstra, and Lovász \([11]\) in practice. The resulting lattice basis reduction algorithm reduces the multiprecision operations needed in previous approaches. This paper describes these ideas in detail for lattices of the particular form arising from the subset sum (exact knapsack) problem. The idea of applying the \(L^3\) algorithm to the subset sum problem is due to Lagarias and Odlyzko \([8]\). The algorithm of this paper also uses a direct search for short vectors simultaneously with the basis reduction algorithm. Extensive computational tests show that this algorithm solves, with high probability, instances of low-density subset sum problems and has two major advantages over the method of Lagarias and Odlyzko: running time is an order of magnitude smaller and higher-density subset sum problems are solved.

It is shown that the collection of all \(\binom{9}{4}\) distinct quadruples chosen from a set of nine points can be partitioned into nine mutually disjoint \(3-(8,4,1)\) designs in just two non-isomorphic ways. Two proofs of this result are given: one by direct construction, the other by extending sets of eight mutually disjoint \(2-(7,3,1)\) designs based on a set of eight points.

Suppose that one is given \(v\) elements, and wishes to form a design that covers all \(t\)-sets from these elements exactly once. The design is to obey the further restriction that the longest block in the design has \(k\) elements in it; furthermore, we wish the design to contain as few blocks as possible.

The number of blocks in such a minimal design is denoted by the symbol \(\text{g}^{(\text{k})}(1,t;v)\); determination of this number is closely connected with the behaviour of Steiner Systems. Recently, much progress has been made in two important cases, namely, when \(t = 2\) (pairwise balanced designs) and \(t = 3\) (designs with balance on triples). This survey paper presents the subject from its inception up to current results.

Recent examples of perfect 1-factorizations arising from dicyclic groups have led to the question of whether or not dicyclic groups have symmetric sequencings. For every positive integer \(n\geq2\), there is a dicyclic group of order \(4n\). It is known that if \(n \geq 3\) is odd, then the dicyclic group of order \(4n\) has a symmetric sequencing. In this paper, a new proof is given for the odd case; a consequence being that in this situation sequencings abound. A generalization of the original proof is exploited to show that if \(n\geq 4\) is even and is not twice an odd number, then the dicyclic group of order \(4n\) has a symmetric sequencing.

Let \(L\) be an \(n \times m\) Latin rectangle on a set of \(v\) symbols with the property that each symbol occurs in precisely \(r\) cells of \(L\). Then \(L\) is said to have the row-column intersection property if each row and column of \(L\) have precisely \(r\) symbols in common. It is shown here that the trivial necessary conditions

1. \(rv = mn\) and
2. \(r \leq \min\{m,n\}\)

are sufficient to guarantee the existence of such a Latin rectangle.

For positive integers \(d\) and \(m\), let \(\text{P}_{\text{d,m}}(\text{G})\) denote the property that between each pair of vertices of the graph \(G\), there are m vertex disjoint (except for the endvertices) paths each of length at most \(d\). Minimal conditions involving various combinations of the connectivity, minimal degree, edge density, and size of a graph \(G\) to insure that \(\text{P}_{\text{d,m}}(\text{G})\) is satisfied are investigated. For example, if a graph \(G\) of order n has connectivity exceeding \((\text{n-m})/\text{d + m} – 1\), then \(\text{P}_{\text{d,m}}(\text{G})\) is satisfied. This result is the best possible in that there is a graph which has connectivity \((\text{n-m})/\text{d + m} – 1\) that does not satisfy \(\text{P}_{\text{d,m}}(\text{G})\). Also, if an \(m\)-connected graph \(G\) of order \(n\) has minimal degree at least \(\lfloor{(\text{n – m} + 2)}/\lfloor{(\text{d} + 4)}/3\rfloor\rfloor+\text{m}-2\), then \(G\) satisfies \(\text{P}_{\text{d,m}}(\text{G})\). Examples are given that show that this minimum degree requirement has the correct order of magnitude, and cannot be substantially weakened without losing Property \(\text{P}_{\text{d,m}}\).

A necessary and sufficient condition for a family of finite sets to possess a collection of \(n\) compatible systems of distinct representatives (SDR’s) is given. A decomposition of finite family of sets into partial SDR’s is also studied.