This paper provides a general method for finding a critical set for any Latin square of order \(n\). This method is used to prove the existence of critical sets of various sizes. It has also been applied to all main classes of Latin squares of order seven, thereby producing a critical set for each Latin square of order seven.

Anne Street wrote an expository article about de Bruijn graphs in the 1970’s. We review some subsequent lines of research, at least one of which was inspired by her article.

No general algorithm is known for the functional decomposition of wild polynomials over a finite field. However, partial solutions exist. In particular, a fast functional decomposition algorithm for linearised polynomials has been developed using factoring methods in skew-polynomial rings. This algorithm is extended to a related class of wild polynomials, which are sub-linearised polynomials.

This paper presents a comparison of the performance of three optimisation heuristics in automated attacks on a simple classical cipher. The three optimisation heuristics considered are simulated annealing, the genetic algorithm, and the tabu search. Although similar attacks have been proposed previously, a comparison of multiple techniques has not been performed. Performance criteria such as efficiency and speed are investigated. The use of tabu search in the field of automated cryptanalysis is a largely unexplored area of research. A new attack on the simple substitution cipher utilizing tabu search is also presented in this paper.

Secret sharing schemes are one of the most important primitives in distributed systems. In perfect secret sharing schemes, collaboration between unauthorized participants cannot reduce their uncertainty about the secret. This paper presents a perfect secret sharing scheme arising from critical sets of Room squares.

Let \(L(n, k, p, t)\) denote the minimum number of subsets of size \(k\) (\(k\)-subsets) of a set of size \(n\) (\(n\)-set) such that any \(p\)-subset intersects at least one of these \(k\)-subsets in at least \(t\) elements. The value of \(L(n, 6, 6, 2)\) is determined for \(n \leq 54\).

The structure of cocyclic Hadamard matrices allows for a much faster and more systematic search for binary, self-dual codes. Here, we consider \(\mathbf{Z}_{2}^{2} \times \mathbf{Z}_{t}\)-cocyclic Hadamard matrices for \(t = 3, 5, 7,\) and \(9\) to yield binary self-dual codes of lengths \(24, 40, 56,\) and \(72\). We show that the extended Golay code cannot be obtained as a member of this class and also demonstrate the existence of four apparently new codes – a \([56, 28, 8]\) code and three \([72, 36, 8]\) codes.

Let \(A = (a_{ij})\) be an \(m \times n\) nonnegative matrix, with row-sums \(r_i\) and column-sums \(c_j\). We show that \[
mn \sum\limits_{i,j} a_{ij} f(r_i) f(c_j) \geq \sum\limits_{i,j} a_{ij}\sum\limits_{i} f(r_i)\sum\limits_{j} f(c_j)
\] providing the function \(f\) meets certain conditions. When \(f\) is the identity function, this inequality is one proven by Atkinson, Watterson, and Moran in 1960. We also prove another inequality, of similar type, that refines a result of Ajtai, Komlós, and Szemerédi (1981).