This paper studies the algebraic structure of linear, cyclic, and constacyclic codes over the finite semi-local non-chain Frobenius ring , where . A Gray map from to is constructed. Beyond basic linearity, we prove that cyclic and constacyclic codes over have a unique structural decomposition into two codes, making their construction and analysis straightforward. Furthermore, the study establishes a connection between these algebraic structures and DNA computing by introducing a mapping based on the -adic system. Also, a specific generator set is defined, enabling the construction of DNA codes that satisfy reversibility and reversible-complement constraints, which are essential for reliable biological applications. The results provide a combined algebraic framework for designing efficient DNA codes with desirable combinatorial properties.
BCH codes are an important subclass of cyclic codes with interesting properties and widespread applications. Although BCH codes are often optimal cyclic codes in many cases and have been extensively studied, determining their parameters remains challenging. Throughout the paper, we investigate the parameters of the BCH codes , where and is an odd prime power. A notable property of these BCH codes is that they are LCD codes. For and odd , we establish lower bounds on the minimum distances of the dual codes of . Furthermore, we provide necessary and sufficient conditions on the designed distances for the LCD BCH codes to be dually-BCH codes.
This paper investigates the resilient synchronization of multiple inertial neural networks (INNs) with matrix coupling under denial-of-service (DoS) attacks. By employing matrix weights to capture the multi-dimensional coupling among INNs, a resilient controller that adapts to DoS attacks is introduced. Sufficient conditions for resilient synchronization are established under the constraint that the total attack duration is bounded. Furthermore, the relationship between the attack duty cycle and the convergence performance is analyzed. Compared with existing works focusing on synchronization of INNs based on the assumption that the communication topology is scalar coupling, this paper considers the more general case of matrix coupling, and further extends the synchronization analysis of multiple INNs under DoS attacks. A numerical example validates the effectiveness of the theoretical results.
This article is concerned with the solvability of the following Caputo fractional three-point boundary value problem with a convection term at resonance {[ - ^CD_0^ + ^α u(t) + σ u'(t) = g(t,u(t),u'(t)), 0 < t < 1,; u(0) = 0, u(1) = β u(η ), ]. The resonance condition for this boundary value problem is characterized by an auxiliary function involving the two-parameter Mittag-Leffler function. The existence result is established via Mawhin’s coincidence degree theorem. Additionally, a parameterized projector is constructed directly from an integral identity of the two-parameter Mittag-Leffler function, thereby avoiding the tedious verification of well-definedness for the relevant projector and substantially streamlining the analysis.
This paper studies practical semi-global uniform exponential stability in mean square for nonlinear parameterized stochastic impulsive systems under an average dwell-time condition. Using Itô’s formula and a direct Lyapunov approach, we derive a comparison lemma that captures the effects of stochastic diffusion, impulsive perturbations, and impulse frequency. We then establish sufficient conditions for practical semi-global mean-square exponential stability with a residual bound independent of the prescribed initial-state radius. An illustrative nonlinear system with bounded network-induced perturbations is presented, and Monte Carlo simulations support the theoretical estimates.
