Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and secret sharing schemes. The first objective of this paper is to characterize all possible parameters of Plotkin-optimal two-homogeneous weight regular projective codes over finite chain rings, as well as their weight distributions. We show the existence of codes with these parameters by constructing an infinite family of two-homogeneous weight codes. The parameters of their Gray images have the same weight distribution as that of the two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond Math Soc 18: 97-122, 1986). Further, we also construct three-homogeneous weight regular projective codes over finite chain rings combined with some known results. Finally, we study applications of our constructed codes in secret sharing schemes and graph theory. In particular, infinite families of SRGs and SWRGs with non-trivial parameters are obtained.

翻译：有限链环上的少重量码与强正则图(SRG)、强游走正则图(SWRG)、有限几何等组合对象相关联，同时广泛应用于数据存储系统和秘密共享方案。本文的首要目标是刻画有限链环上Plotkin最优二齐次重量正则投射码的所有可能参数及其重量分布。通过构造一个无限族二齐次重量码，我们证明了这些参数对应码的存在性。其Gray像的参数在Calderbank和Kantor（Bull Lond Math Soc 18: 97-122, 1986）意义下具有与SU1型二重码相同的重量分布。进一步，我们结合已知结果构造了有限链环上的三齐次重量正则投射码。最后，我们研究了所构造码在秘密共享方案和图论中的应用，特别地，获得了具有非平凡参数的无限族SRG和SWRG。