A construction of authentication codes with arbitration from singular symplectic geometry over finite fields is given, and the parameters of the codes are computed. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of success for different types of deceptions are also computed.

Let \(M\) be a simple connected binary matroid with corank at least two such that \(M\) has no connected hyperplane. Seymour proved that \(M\) has a non-trivial series class. We improve this result by proving that \(M\) has at least two disjoint non-trivial series classes \(L_1\) and \(L_2\) such that both \(M \backslash L_1\) and \(M \backslash L_2\) are connected. Our result extends the corresponding result of Kriesell regarding critically \(2\)-connected graphs.

For a non-complete graph \(\Gamma\), a vertex triple \((u,v,w)\) with \(v\) adjacent to both \(u\) and \(w\) is called a \(2\)-geodesic if \(u \neq w\) and \(u,w\) are not adjacent. Then \(\Gamma\) is said to be \(2\)-geodesic transitive if its automorphism group is transitive on both arcs and \(2\)-geodesics. In this paper, we classify the family of connected \(2\)-geodesic transitive graphs of valency \(3p\), where \(p\) is an odd prime.

We generalize the well known congruence Lucas\(^1\) Theorem for binomial coefficient to the bi\(^s\)nomial coefficients.

The linear arboricity \(la(G)\) of a graph \(G\) is the minimum number of linear forests that partition the edges of \(G\). In this paper, it is proved that if \(G\) is a planar graph with maximum degree \(\Delta \geq 7\) and every \(7\)-cycle of \(G\) contains at most two chords, then \(la(G) = \left\lceil \frac{\Delta(G)}{2} \right\rceil\).

In this paper, we study the generalized Pell \(p\)-sequences modulo \(m\). Additionally, we define the generalized Pell \(p\)-sequences and the basic generalized Pell sequences in groups, and then examine these sequences in finite groups. Furthermore, we obtain the periods of the generalized Pell \(p\)-sequences and the basic periods of the basic generalized Pell sequences in the binary polyhedral groups \(\langle n,2,2\rangle\), \(\langle2,n,2\rangle\), and \(\langle2,2,n\rangle\).

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined as the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the star graphs, one of the most popular interconnection networks.