Weight Distribution of the Weighted Coordinates Poset Block Space and Singleton Bound

TL;DR

This paper determines the complete weight distribution of the weighted coordinates poset block space, explores properties of perfect codes, and establishes duality theorems, advancing coding theory in structured metric spaces.

Contribution

It provides the first complete weight distribution for the $(P,w, ext{pi})$-space and analyzes the relationships between perfect codes, MDS codes, and duality in these spaces.

Findings

Complete weight distribution for $(P,w, ext{pi})$-space derived.

Relationship between $I$-perfect, $t$-perfect, and MDS codes established.

Duality theorem for MDS $(P,w, ext{pi})$-codes proved.

Abstract

In this paper, we determine the complete weight distribution of the space $ \mathbb{F}_q^N $ endowed by the weighted coordinates poset block metric ($(P,w,\pi)$-metric), also known as the $(P,w,\pi)$-space, thereby obtaining it for $(P,w)$-space, $(P,\pi)$-space, $\pi$-space, and $P$-space as special cases. Further, when $P$ is a chain, the resulting space is called as Niederreiter-Rosenbloom-Tsfasman (NRT) weighted block space and when $P$ is hierarchical, the resulting space is called as weighted coordinates hierarchical poset block space. The complete weight distribution of both the spaces are deduced from the main result. Moreover, we define an $I$-ball for an ideal $I$ in $P$ and study the characteristics of it in $(P,w,\pi)$-space. We investigate the relationship between the $I$-perfect codes and $t$-perfect codes in $(P,w,\pi)$-space. Given an ideal $I$, we investigate how the maximum distance separability (MDS) is related with $I$-perfect codes and $t$-perfect codes in $(P,w,\pi)$-space. Duality theorem is derived for an MDS $(P,w,\pi)$-code when all the blocks are of same length. Finally, the distribution of codewords among $r$-balls is analyzed in the case of chain poset, when all the blocks are of same length.