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

Atul Kumar Shriwastva, R. S. Selvaraj

TL;DR

This work unifies and extends weight-distribution analysis for codes under the weighted coordinates poset block metric $(P,w,\pi)$. By introducing $I$-balls and the notions of $I$-perfect and $t$-perfect codes, it connects perfectness to MDS properties and derives duality theorems, particularly focusing on equal-block-length scenarios. The authors provide explicit weight-distribution formulas for the general $(P,w,\pi)$-space and specialized settings, including $(P,w)$-space, $(P,\pi)$-space, $\pi$-space, $P$-space, hierarchical posets, and NRT-weighted block spaces, with corresponding results on $I$-perfectness, r-perfectness, and duality. The results yield concrete counting formulas (via partitions, $D_r^k$, and $PRT$ sets) that subsume classic metrics and recover known distributions in special cases, and they establish duality relations for MDS and perfect codes across chain posets. The work thus advances the design and analysis of codes with complex block-structured metrics, with implications for MDS constructions, perfect codes, and weight-distribution studies in generalized poset-block frameworks.

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 ($(P,w,π)$-metric), also known as the $(P,w,π)$-space, thereby obtaining it for $(P,w)$-space, $(P,π)$-space, $π$-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 $(P,w,π)$-space. We investigate the relationship between the $I$-perfect codes and $t$-perfect codes in $(P,w,π)$-space. Given an ideal $I$, we investigate how the maximum distance separability (MDS) is related with $I$-perfect codes and $t$-perfect codes in $(P,w,π)$-space. Duality theorem is derived for an MDS $(P,w,π)$-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.

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

TL;DR

This work unifies and extends weight-distribution analysis for codes under the weighted coordinates poset block metric . By introducing -balls and the notions of -perfect and -perfect codes, it connects perfectness to MDS properties and derives duality theorems, particularly focusing on equal-block-length scenarios. The authors provide explicit weight-distribution formulas for the general -space and specialized settings, including -space, -space, -space, -space, hierarchical posets, and NRT-weighted block spaces, with corresponding results on -perfectness, r-perfectness, and duality. The results yield concrete counting formulas (via partitions, , and sets) that subsume classic metrics and recover known distributions in special cases, and they establish duality relations for MDS and perfect codes across chain posets. The work thus advances the design and analysis of codes with complex block-structured metrics, with implications for MDS constructions, perfect codes, and weight-distribution studies in generalized poset-block frameworks.

Abstract

In this paper, we determine the complete weight distribution of the space endowed by the weighted coordinates poset block metric (-metric), also known as the -space, thereby obtaining it for -space, -space, -space, and -space as special cases. Further, when is a chain, the resulting space is called as Niederreiter-Rosenbloom-Tsfasman (NRT) weighted block space and when 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 -ball for an ideal in and study the characteristics of it in -space. We investigate the relationship between the -perfect codes and -perfect codes in -space. Given an ideal , we investigate how the maximum distance separability (MDS) is related with -perfect codes and -perfect codes in -space. Duality theorem is derived for an MDS -code when all the blocks are of same length. Finally, the distribution of codewords among -balls is analyzed in the case of chain poset, when all the blocks are of same length.
Paper Structure (19 sections, 53 theorems, 55 equations)

This paper contains 19 sections, 53 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Coordinates Poset Block Metrics
  4. Weight Distribution of $(P,w,\pi)$-space
  5. $(P,w)$-space
  6. $(P,\pi)$-space
  7. $\pi$-space
  8. $P$-space
  9. Weighted Coordinates Hierarchical Poset Block Spaces
  10. Hierarchical poset block space
  11. Hierarchical poset space
  12. $I$-Perfect block codes and $t$-perfect block codes
  13. MDS and $I$-perfect $(P,w,\pi)$-codes with equal block length
  14. Duality theorem
  15. Codes on NRT weighted block spaces
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $P = ([n], \preceq)$ be a poset. Let $0 \leq s \leq t \leq n$. Then for each $I \in \mathcal{I}^t$ there exists $J \in \mathcal{I}^s$ such that $J \subseteq I$. Moreover, for each $J \in \mathcal{I}^s$ there exists $I \in \mathcal{I}^t$ such that $J \subseteq I$.

Theorems & Definitions (91)

  • Proposition 2.1
  • Theorem 2.1: aks
  • Definition 3.1: $(P,w,\pi)$-weight as
  • Theorem 3.2: wj,as
  • Remark 3.3: as
  • Definition 3.4
  • Theorem 3.5: Singleton Bound for $(P,w,\pi)$-block Code as
  • Corollary 3.6: Singleton Bound for $(P,w)$-code aks
  • Proposition 4.1
  • proof
  • ...and 81 more