In this paper, we study further bounds of constant dimension codes in Grassmannian space \(\mathcal{G}_q(n,k)\). There is increasing interest in subspace codes since they are essential for error-correction in networks. Additionally, there is a connection to the theory over finite fields. By revising the specific construction methods of the constant dimension codes in [1], [2], we improve some bounds on \(q\)-ary constant dimension codes in certain cases.

In this paper, we use a recent result of Bryant, Horsley, and Pettersson in [1] to provide an alternate and more straightforward proof of results concerning neighborhood graphs in maximum packings of \(2K_n\) with triples, some of which were only recently obtained.

To set the stage, consider any partial triple system \((V,B)\) of \(2K_n\). In this system, the neighborhood of a vertex \(v\) is defined as the subgraph induced by the set \(\{\{x,y\} \mid \{v,x,y\} \in B\}\). This concept plays a crucial role in the results initially obtained by Colbourn and Rosa for \(n \equiv 0,1 \pmod{3}\) and by Chaffee and Rodger for \(n \equiv 2 \pmod{3}\). These results offer a complete characterization of the possible neighborhoods in a maximum packing of \(2K_n\).

In both of these original papers, the authors employed difference methods—a combinatorial technique that often involves selecting pairs of elements from a group and studying their differences—and a pull-up technique, which is used to modify the neighborhood of a vertex. However, despite the effectiveness of these methods, neither approach seems to lend itself easily to deriving the results of the other.

In our paper, we present a more unified and simplified proof that brings both of these results together. By leveraging the recent findings of Bryant, Horsley, and Pettersson, we can bypass the need for the more complex difference methods and pull-up techniques, instead relying on the underlying principles elucidated in their work. This approach not only simplifies the proof process but also provides a clearer and more direct route to understanding the structure of neighborhood graphs in these maximum packings.

Compressed sensing (CS) has broken through the traditional Nyquist sampling theory as it is a new technique in signal processing. According to CS theory, compressed sensing makes full use of sparsity so that a sparse signal can be reconstructed from very few measurements. It is well known that the construction of CS matrices is the central problem. In this paper, we provide one kind of deterministic sensing matrix by describing a combinatorial design. Then, we obtain two cases by instantiating the RIP framework with the obtained design, with the latter case being the majorization of the former. Finally, we show that our construction has better properties than DeVore’s construction using polynomials over finite fields.

In this paper, with the help of the residue method, we find some interesting formulas relating residue and ordinary Bell polynomials, \(\hat{B}_{n,k}(x_1,x_2,\ldots)\). Further, we prove identities involving some combinatorial numbers to demonstrate the application of the formulas.

We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at least one pebble to the target. A pebbling move of \(p\) pebbles across an edge with weight \(w\) leaves \(\lfloor pw \rfloor\) pebbles on the next vertex. We find the weighted pebbling numbers of stars, graphs with at least \(2|V|-1\) edges, and trees with given targets. We give an explicit formula for the minimum total weight required on the edges of a length-2 path, solvable with \(p\) pebbles, and exhibit a graph that requires an edge with weight \(1/3\) in order to achieve its weighted pebbling number.

We examine two particular constructions of Costas arrays known as the Taylor variant of the Lempel construction, or the \(T_4\) construction, and the variant of the Golomb construction, or the \(G_4\) construction. We connect these with Fibonacci primitive roots, and show that under the Extended Riemann Hypothesis, the \(T_4\) and \(G_4\) constructions are valid infinitely often.

The authentication codes with arbitration are said to be $A^2$-codes. Two constructions of $A^2$-codes with secrecy from polynomials over finite fields are constructed to prevent communication systems from attacks which come from the opponent, the transmitter and the receiver. Parameters of the codes and probabilities of successful attacks are also computed. At last, two constructions are compared with a known one. It is important that a source state can’t be recovered from the message without the knowledge of the transmitter’s encoding rule or the receiver’s decoding rule. It must be decoded before verification.

For \(n \geq 1\), we let \(a_n\) count the number of nonempty subsets \(S\) of \(\{1,2,3,\ldots,n\} = [n]\), where the size of \(S\) equals the minimal element of \(S\). Such a subset is called an extraordinary subset of \([n]\), and we find that \(a_n = F_n\), the \(n\)th Fibonacci number. Then, for \(n \geq k \geq 1\), we let \(a(n, k)\) count the number of times the integer \(k\) appears among these \(a_n\) extraordinary subsets of \(n\). Here we have \(a(n, k) = a(n-1, k) + a(n-2, k-1)\), for \(n \geq 3\) and \(n > k \geq 2\). Formulas and properties for \(t_n = \sum_{k=1}^n a(n, k)\) and \(s_n = \sum_{k=1}^n ka(n, k)\) are given for \(n \geq 1\). Finally, for fixed \(n \geq 1\), we find that the sequence \(a(n, k)\) is unimodal and examine the maximum element for the sequence. In this context, the Catalan numbers make an entrance.

The cycle length distribution (CLD) of a graph of order \(n\) is \((c_1, c_2, \ldots, c_n)\), where \(c_i\) is the number of cycles of length \(i\), for \(i = 1, 2, \ldots, n\). For an integer sequence \((a_1, a_2, \ldots, a_n)\), we consider the problem of characterizing those graphs \(G\) with the minimum possible edge number and with \(\text{CLD}(G) = (c_1, c_2, \ldots, c_n)\) such that \(c_i \geq a_i\) for \(i = 1, 2, \ldots, n\). The number of edges in such a graph is denoted by \(g(a_1, a_2, \ldots, a_n)\). In this paper, we give the lower and upper bounds of \(g(0, 0, k, \ldots, k)\) for \(k = 2, 3, 4\).

Two-fold automorphisms (or “TF-isomorphisms”) of graphs are a generalisation of automorphisms. Suppose \(\alpha, \beta\) are two permutations of \(V = V(G)\) such that for any pair \((u,v)\), \(u, v \in V\), \((u,v)\) is an arc of \(G\) if and only if \((\alpha(u), \beta(v))\) is an arc of \(G\). Such a pair of permutations is called a two-fold automorphism of \(G\). These pairs form a group that is called the two-fold automorphism group. Clearly, it contains all the pairs \((\alpha, \alpha)\) where \(\alpha\) is an automorphism of \(G\). The two-fold automorphism group of \(G\) can be larger than \(\text{Aut}(G)\) since it may contain pairs \((\alpha, \beta)\) with \(\alpha \neq \beta\). It is known that when this happens, \(\text{Aut}(G) \times \mathbb{Z}_2\) is strictly contained in \(\text{Aut}(G \times K_2)\). In the literature, when this inclusion is strict, the graph \(G\) is called unstable.

Now let \(\Gamma \leq S_V \times S_V\). A two-fold orbital (or “TF-orbital”) of \(F\) is an orbit of the action \((\alpha, \beta) : (u,v) \mapsto (\alpha(u), \beta(v))\) for \((\alpha, \beta) \in \Gamma\) and \(u,v \in V\). Clearly, \(\Gamma\) is a subgroup of the TF-automorphism group of any of its TF-orbitals. We give a short proof of a characterization of TF-orbitals which are disconnected graphs and prove that a similar characterization of TF-orbitals which are digraphs might not be possible. We shall also show that the TF-rank of \(\Gamma\), that is the number of its TF-orbitals, can be equal to \(1\) and we shall obtain necessary and sufficient conditions on I for this to happen.