It is known that there are at least 8784 non-isomorphic designs with parameters \(2-(64, 28, 12)\) whose derived \(2-(28, 12, 11)\) designs are quasi-symmetric. In this paper, we examine the binary codes related to a class of non-isomorphic designs with these parameters and invariant under the Frobenius group of order 21 for which the derived \(2-(28, 12, 11)\) designs are not quasi-symmetric. We show that up to equivalence, there are 30 non-isomorphic binary codes obtained from them. Moreover, we classify the self-orthogonal doubly-even codes of length 13 obtained from the non-fixed parts of orbit matrices of these \(2-(64, 28, 12)\) designs under an action of an automorphism group of order four having 12 fixed points. The subcodes of codimension 1 and minimum weight 8 in these codes are all optimal single weight codes.

Let \( N(n, t_1, \ldots, t_r) \) be the number of irreducible polynomials of degree \( n \) over the finite field \( \mathbb{F}_2 \) where the coefficients of the terms \( x^{n-1}, \ldots, x^{n-r} \) are prescribed. Finding the exact values for the numbers \( N(n, t_1, \ldots, t_r) \) for \( r \geq 4 \) seems difficult. In this paper, we give an approximation for these numbers. We treat in detail the case \( N(n, t_1, \ldots, t_4) \), and we state the approximation in the general case. We experimentally show how good our approximation is.

The degree sequence of a finite graph \( G \) is its list \( D = (d_1, \ldots, d_n) \) of vertex degrees in non-increasing order. The graph \( G \) is called a realization of \( D \). In this paper, we characterize the graphic degree sequences \( D \) such that no realization of \( D \) contains an induced four-cycle. Our characterization is stated in terms of the class of forcibly chordal graphs.

A dice family \( D(n, a, b, s) \) includes all lists \( (x_1, \ldots, x_n) \) of integers with \( n \geq 1 \), \( a \leq x_1 \leq \ldots \leq x_n \leq b \), and \( \sum x_i = s \). Given two dice \( X \) and \( Y \), we compare the number of pairs \( (i, j) \) with \( x_i y_j \). If the second number is larger, then \( X \) is \emph{stronger} than \( Y \), and if the two numbers are equal, then \( X \) and \( Y \) are \emph{tied}. In previous work, it has been observed that the density of ties in \( D(n, a, b, s) \) is generally lower than one might expect. In this note, we provide more information about this observation by calculating the asymptotic proportion of ties in certain kinds of dice families. Many other properties of dice families remain to be determined.

After introducing and discussing the notion of length two path centered surface area for general graphs, particularly for bipartite graphs, we derive a closed-form expression and an explicit expression for the length two path centered surface areas of the hypercube and the star graph, respectively.

The Ramsey numbers \( r(F, G) \) are investigated, where \( F \) is a non-tree graph of order \( 5 \) and minimum degree \( 1 \), and \( G \) is a connected graph of order \( 6 \). For all pairs \( (F, G) \) where \( F \neq K_5 – K_{1,3} \), the exact value of \( r(F, G) \) is determined. In order to settle \( F = K_5 – K_{1,3} \), we prove \( r(K_5 – K_{1,3}, G) = r(K_4, G) \).

A Stanton-type graph \( S(n, m) \) is a connected multigraph on \( n \) vertices such that for a fixed \( m \) with \( n-1 \leq m \leq \frac{n}{2} \), there is exactly one edge of multiplicity \( i \) (and no others) for each \( i = 1, 2, \ldots, m \). In this note, we show how to decompose \( \lambda K_n \) (for the appropriate minimal values of \( \lambda \)) into Stanton-type graphs \( S(4, 3) \) of the LOE and OLE types.

In this paper, we consider the use of balanced arrays (B-arrays) in constructing discrete fractional factorial designs (FFD) of resolution \((2u+1)\), with \(u=2\) and \(3\), in which each of the \(m\) factors is at two levels (say, \(0\) and \(1\)), denoted by factorial designs of \(2^m\) series. We make use of the well-known fact that such designs can be realized under certain conditions, by using balanced arrays of strength four and six (with two symbols), respectively. Here, we consider the existence of B-arrays of strength \(t=4\) and \(t=6\), and discuss how the results presented can be used to obtain the maximum value of \(m\) for a given set of treatment-combinations. Also, we provide some illustrative examples in which the currently available \(\max(m)\) values have been improved upon.

For integers \( k \geq 1 \), a \((p,q)\)-graph \( G = (V, E) \) is said to admit an AL(\(k\))-traversal if there exists a sequence of vertices \((v_1, v_2, \ldots, v_p)\) such that for each \( i = 1, 2, \ldots, p-1 \), the distance between \( v_i \) and \( v_{i+1} \) is \( k \). We call a graph \( k \)-step Hamiltonian (or say it admits a \( k \)-step Hamiltonian tour) if it has an AL(\(k\))-traversal and \( d(v_1, v_p) = k \). In this paper, we investigate the \( k \)-step Hamiltonicity of graphs. In particular, we show that every graph is an induced subgraph of a \( k \)-step Hamiltonian graph for all \( k \geq 2 \).