Block $LS$–poset codes over ${\mathbb{Z}}_m^n$: Perfect codes, Singleton bound, and MDS characterization

We introduce a block $LS$–poset (partially ordered set) metric on ${\mathbb{Z}}_m^n$ constructed from a block decomposition of ${\mathbb{Z}}_m^n$, a poset structure on the block indices, and the lattice of subgroups of ${\mathbb{Z}}_m$ arising from the prime factorization of $m$. Using a multiset representation associated with this subgroup lattice, we define the block $LS$–poset weight and show that the induced distance is a metric on ${\mathbb{Z}}_m^n$. We investigate the geometry of $r$-balls and $\mathcal{I}$-balls and establish their fundamental properties, including linearity, translation invariance, and duality. These structural results lead to characterizations of $\mathcal{I}$-perfect block $LS$–poset codes for ideals with full count and partial count. We further derive a Singleton-type bound for block $LS$–poset codes and introduce the notions of maximum distance separable (MDS) and partial-MDS block $LS$–poset codes. Connections among perfect codes, MDS codes, and $r$-perfect codes are also examined for certain classes of posets.