Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works connecting lattice theory, anticodes, and coding-theoretic invariants, we investigate ring-linear codes endowed with the Lee metric. We introduce and characterize optimal Lee-metric anticodes over the ring $\mathbb{Z}/p^s\mathbb{Z}$. We show that the family of such anticodes admits a natural partition into subtypes and forms a lattice under inclusion. We establish a bijection between this lattice and a lattice of weak compositions ordered by dominance. As an application, we use this correspondence to introduce new invariants for Lee-metric codes via an anticode approach.

翻译：格与偏序集在编码理论中扮演着日益重要的角色，为研究纠错码的结构与代数性质提供了组合框架。受近期连接格理论、反码与编码理论不变量研究工作的启发，我们研究了赋予Lee度量的环线性码。我们引入并刻画了环$\mathbb{Z}/p^s\mathbb{Z}$上的最优Lee度量反码。我们证明了此类反码族可按子类型自然划分，并在包含关系下构成一个格。我们建立了该格与一个按优势序排列的弱合成格之间的双射。作为应用，我们利用这一对应关系，通过反码方法为Lee度量码引入了新的不变量。