Let [Formula: see text] be a prime number and [Formula: see text] finite field. Let [Formula: see text] be a finite chain ring, and let [Formula: see text], where [Formula: see text]. and [Formula: see text]. In this paper, we study [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], where [Formula: see text]. First, we use the Discrete Fourier Transform to decompose these codes via a ring isomorphism into direct sums of principal ideals. Using this decomposition, we explicitly determine the generator polynomials of such codes. We then study their duals and establish necessary and sufficient conditions for the existence of self-dual [Formula: see text]-constacyclic codes in the case where [Formula: see text]. Under this condition, we also provide their generator polynomials explicitly. Finally, we investigate the [Formula: see text]-symbol distances of these codes and derive explicit formulas. As an application, we generalize the results obtained in [6].
In this paper, we study an approximation algorithm for the problem of Min-Sum Movement Target Coverage (MinSMTC). Given [Formula: see text] targets [Formula: see text] and [Formula: see text] mobile sensors [Formula: see text] on the plane, each sensor has a coverage radius [Formula: see text] and can cover those targets within distance [Formula: see text] from it. The sensors, at their original locations, might not be able to cover all targets. The goal of MinSMTC is to move sensors to new locations such that all targets can be covered and the total movement distance is minimized. We show that even determining the feasibility of the problem is NP-complete. Then, we present a [Formula: see text]-approximation algorithm as long as the algorithm can return a feasible solution, where [Formula: see text] is a parameter related to the number of targets that can be covered by a sensor at a candidate destination location.
We examined the distribution of domination behavior in the zero-divisor graphs associated with [Formula: see text] for all [Formula: see text]. For every non-zero zero-divisor [Formula: see text], we define its domination-degree [Formula: see text] as the number of minimal dominating sets in which [Formula: see text] appears. Incorporating these values into Zagreb-type constructions yields the following three descriptors: [Formula: see text] These indices capture the concentration of domination-degrees across vertices, the interaction of dominating contributions along adjacency relations, and the structural imbalance arising from their distribution within the graph, respectively. Our contributions are twofold. First, we provide a complete catalogue of [Formula: see text], trio [Formula: see text], and the total number of minimal dominating sets for every [Formula: see text], organized by the factorization pattern. Second, because enumerating minimal dominating sets is notoriously expensive, we present a deterministic algorithm tailored to [Formula: see text] that uses Constraint Programming and Boolean Satisfiability (CP-SAT) to compute the number of minimal dominating sets and thereby reliably compute [Formula: see text] within this range. We close with the application sketches in fault-tolerant sensor placement and emergency facility placement illustrating how these indices translate algebraic constraints into actionable design guidance.
Let [Formula: see text] be the [Formula: see text]-dimensional symplectic space and [Formula: see text] be the corresponding symplectic group of degree [Formula: see text] over a finite commutative ring [Formula: see text]. By using a matrix method, we describe the orbits of subspaces of [Formula: see text] under the action of [Formula: see text], and determine the lengths of these orbits, which is a generalization of similar results of the symplectic space over a finite field. All totally isotropic subspaces of [Formula: see text] form a poset ordered by inclusion, denoted by [Formula: see text]. For a fixed [Formula: see text]-totally isotropic subspace [Formula: see text] of [Formula: see text], each set [Formula: see text] is a subposet of [Formula: see text], where [Formula: see text]. We show that [Formula: see text] has the normalized matching (NM) property, which implies that [Formula: see text] has the strong Sperner property and the LYM property.
A signed graph [Formula: see text] consists of graph [Formula: see text] and signature [Formula: see text]. An incidence of [Formula: see text] is a pair [Formula: see text], where [Formula: see text] is one of the end vertices of edge [Formula: see text]. A proper [Formula: see text]-edge coloring [Formula: see text] of the signed graph [Formula: see text] is an assignment of colors to incidences satisfying that [Formula: see text] for every edge [Formula: see text] and for any two incidences [Formula: see text] and [Formula: see text], involving the same vertex, [Formula: see text]. The chromatic index of a signed graph [Formula: see text], denoted by [Formula: see text], is the minimum number [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-edge coloring. In this paper, we consider the edge coloring of signed generalized book graphs and signed complete graphs. We first classified the switching non-isomorphic signed generalized book graphs and then computed the chromatic index of all signed generalized book graphs. We also determined the chromatic index of the signed complete graphs of up to six order.
