MDS and $I$-Perfect Codes in Pomset block Metric

TL;DR

This paper develops bounds and characterizes optimal codes in pomset block metrics over rings, establishing relationships between MDS and perfect codes, and analyzing their duality and weight distribution.

Contribution

It introduces the Singleton bound for pomset block codes, characterizes MDS and perfect codes, and explores their duality and weight distribution in this new metric setting.

Findings

Established the Singleton bound for pomset block codes.

Characterized the conditions for MDS codes in pomset metrics.

Derived the weight distribution of MDS pomset block codes.

Abstract

In this paper, we establish the Singleton bound for pomset block codes ($(Pm,\pi)$-codes) of length $N$ over the ring $\mathbb{Z}_m$. We give a necessary condition for a code to be MDS in the pomset (block) metric and prove that every MDS $(Pm,\pi)$-code is an MDS $(P,\pi)$-code. Then we proceed on to find $I$-perfect and $r$-perfect codes. Further, given an ideal with partial and full counts, we look into how MDS and $I$-perfect codes relate to one another. For chain pomset, we obtain the duality theorem for pomset block codes of length $N$ over $\mathbb{Z}_m$; and, the weight distribution of MDS pomset block codes is then determined.