Given a regular multiset $M$ on $[n]=\{1,2,\ldots,n\}$, a partial order $R$ on $M$, and a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, we define a pomset block metric $d_{(Pm,\pi)}$ on the direct sum $ \mathbb{Z}_{m}^{k_1} \oplus \mathbb{Z}_{m}^{k_2} \oplus \ldots \oplus \mathbb{Z}_{m}^{k_n}$ of $\mathbb{Z}_{m}^{N}$ based on the pomset $\mathbb{P}=(M,R)$. The pomset block metric extends the classical pomset metric introduced by I. G. Sudha and R. S. Selvaraj and generalizes the poset block metric introduced by M. M. S. Alves et al over $\mathbb{Z}_m$. The space $ (\mathbb{Z}_{m}^N,~d_{(Pm,\pi)} ) $ is called the pomset block space and we determine the complete weight distribution of it. Further, $I$-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between MDS codes and $I$-perfect codes for any ideal $I$ is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

翻译：给定$[n]=\{1,2,\ldots,n\}$上的正则多重集$M$、$M$上的偏序关系$R$以及由$\pi(i) = k_i$且$\sum_{i=1}^{n}\pi (i) = N$定义的标号映射$\pi : [n] \rightarrow \mathbb{N}$，我们在$\mathbb{Z}_{m}^{N}$的直和$\mathbb{Z}_{m}^{k_1} \oplus \mathbb{Z}_{m}^{k_2} \oplus \ldots \oplus \mathbb{Z}_{m}^{k_n}$上，基于偏序多重集$\mathbb{P}=(M,R)$定义了一种偏序多重集分组度量$d_{(Pm,\pi)}$。该度量推广了I. G. Sudha和R. S. Selvaraj引入的经典偏序多重集度量，并推广了M. M. S. Alves等人在$\mathbb{Z}_m$上引入的偏序集分组度量。空间$ (\mathbb{Z}_{m}^N,~d_{(Pm,\pi)} ) $被称为偏序多重集分组空间，我们确定了其完整权重分布。进一步，描述了具有部分计数和完全计数的理想$I$-完全偏序多重集分组码。随后，对于链式偏序多重集的分组码，建立了打包半径和Singleton界。研究了任意理想$I$下MDS码与$I$-完全码之间的关系。此外，当所有分块具有相同大小时，建立了MDS偏序多重集分组码的对偶定理。