Block Codes on Pomset Metric

TL;DR

This paper introduces a new class of block codes based on pomset metrics, extending existing poset and pomset metrics, and explores their weight distribution, perfect codes, bounds, and duality properties.

Contribution

It defines the pomset block metric on direct sums of modules, generalizes previous metrics, and studies code properties like perfectness, bounds, and duality within this framework.

Findings

Complete weight distribution of pomset block space determined.

Characterization of $I$-perfect pomset block codes provided.

Established bounds and duality theorems for MDS pomset block codes.

Abstract

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.