In this study we focus on symmetrized weight enumerators of pairs of Type II codes over the finite field F₂ and the ring Z₄. These pairs have been examined as invariants for a specified group. In particular, we concentrate on the scenarios where the space of the invariant ring is of degree 8 and 16. Our findings show that, the ring produced by the symmetrized weight enumerators precisely matches with the invariant ring of the designated group.

In this paper, we analyze the dynamics of quantum walks on a graph structure resulting from the integration of a main connected graph G and a secondary connected graph G′. This composite graph is formed by a disjoint union of G and G′, followed by the contraction of a selected pair of vertices creating a cut vertex v^* and leading to a unique form of geometric perturbation. Our study focuses on instances where G is a complete graph K_N and G′ is a star graph S_m. The core of our analysis lies in exploring the impact of this geometric perturbation on the success probability of quantum walk-based search algorithms, particularly in an oracle-free context. Despite initial findings suggesting a low probability of locating the perturbed graph G′, we demonstrate that introducing a phase reversal to the system significantly enhances the success rate. Our results reveal that with an optimal running time and specific parameter conditions, the success probability can be substantially increased. The paper is structured to first define the theoretical framework, followed by the presentation of our main results, detailed proofs, and concluding with a summary of our findings and potential future research directions.

Previous studies presented zeta functions by the Konno–Sato theorem or the Fourier analysis for one-particle models, including random walks, correlated random walks, quantum walks, and open quantum random walks. Furthermore, the zeta functions for the multi-particle model with probabilistic or quantum interactions, called the interacting particle system (IPS), were also investigated. In this paper, we focus on the zeta function for a class of IPS, including the Domany–Kinzel model, which is a typical model of the probabilistic IPS in the field of statistical mechanics and mathematical biology.