Let \( {R} = (r_1, r_2, \ldots, r_m)\) and \( {S} = (s_1, s_2, \ldots, s_n)\) be nonnegative integral vectors. Denote by \( {A}( {R}, {S})\) the class of \((0,1)\) matrices with row sum vector \( {R}\) and column sum vector \( {S}\). We study a generalization of invariant positions called locally invariant positions of a class \( {A}( {R}, {S})\). For a normalized class, locally invariant positions have in common with invariant positions the property that they lie above and to the left of some simple rook path through the set of positions.

This paper examines the numbers of lattice paths of length \(n\) from the origin to integer points along the line \((a,b,c,d) + t(1,-1,1,-1)\). These numbers form a sequence which this paper shows is log concave, and for sufficiently large values of \(n\), the location of the maximum of this sequence is shown. This paper also shows unimodality of such sequences for other lines provided that \(n\) is sufficiently large.

A cover of a finite set \(N\) is a collection of subsets of \(N\) whose union is \(N\). We determine the number of such covers whose blocks all have distinct sizes. The cases of unordered and ordered blocks are each considered.

Let \(n(k)\) be the smallest number of vertices of a bipartite graph not being \(k\)-choosable. We show that \(n(3) = 14\) and moreover that \(n(k) \leq k. n(k-2)+2^k\). In particular, it follows that \(n(4) \leq 40\) and \(n(6) \leq 304\).

Eight new codes are presented which improve the bounds on maximum minimum distance for binary linear codes. They are rate \(\frac{m-r}{pm},r\geq 1\) , \(r\)-degenerate quasi-cyclic codes.

A method for synthesizing combinatorial structures which are members of an extended class of resolvable incomplete lattice designs is presented. Square and rectangular lattices both are realizable, yet designs in the extended class are not limited in number of treatments by the classically severe restriction \(v = s^2\) or \(v = s(s-1)\). Rather, the current restriction is the condition that there exist a finite closable set of \(k\)-permutations on the objects of some group or finite field, which is then used as the generating array for a \(L(0,1)\) lattice design. A connection to Hadamard matrices \(H(p,p)\) is considered.

Near-perfect protection is a useful extension of perfect protection which is a necessary condition for authentication systems that satisfy Pei-Rosenbaum’s bound. Near-perfect protection implies perfect protection for key strategies, defined in the paper, in which the enemy tries to guess the correct key. We prove a bound on the probability of deception for key strategies, characterize codes that satisfy the bound with equality and conclude the paper with a comparison of this bound and Pei-Rosenbaum’s bound.

This note gives what is believed to be the first published example of a symmetric \(11 \times 11\) Latin square which, although not cyclic, has the property that the permutation between any two rows is an \(11\)-cycle. The square has the further property that two subsets of its rows constitute \(5 \times 11\) Youden squares. The note shows how this \(11 \times 11\) Latin square can be obtained by a general construction for \(n \times n\) Latin squares where \(n\) is prime with \(n \geq 11\). The permutation between any two rows of any Latin square obtained by the general construction is an \(n\)-cycle; two subsets of \((n-1)/2\) rows from the Latin square constitute Youden squares if \(n \equiv 3 \pmod{8}\).

The twenty-five year old \(\lambda\)-design conjecture remains unsettled. Attempts to characterize these irregular, tight, \(2\)-designs have produced a great number of parametric and dual structure characterizations of the so-called Type-I Designs. We establish some new structural characterizations and establish the conjecture in the smallest unsettled case (\(\lambda = 14\)) of the \(2p\) family.

In this paper we consider a random walk in a plane in which a particle at any stage moves one unit in any one of the four directions, namely, north, south, east, and west with equal probability and derive the joint and marginal distributions of certain characteristics of this random walk by using combinatorial methods.