Weighted Coordinates Poset Block Codes

TL;DR

This paper introduces a generalized weighted coordinates poset block metric for coding theory, extending previous metrics and deriving key properties like weight distribution and bounds.

Contribution

It defines the weighted coordinates poset block metric, generalizes existing metrics, and determines the complete weight distribution and bounds for codes under this new metric.

Findings

Derived the complete weight distribution of the $(P,w, ext{pi})$-space.

Established the Singleton bound for $(P,w, ext{pi})$-codes.

Re-obtained the Singleton bound for $(P, ext{pi})$-metric and $P$-metric.

Abstract

Given $[n]=\{1,2,\ldots,n\}$, a partial order $\preceq$ on $[n]$, a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, the direct sum $ \mathbb{F}_{q}^{k_1} \oplus \mathbb{F}_{q}^{k_2}\oplus \ldots \oplus \mathbb{F}_{q}^{k_n} $ of $ \mathbb{F}_q^N $, and a weight function $w$ on $ \mathbb{F}_q $, we define a poset block metric $d_{(P,w,\pi)}$ on $\mathbb{F}_{q}^{N}$ based on the poset $P=([n],\preceq)$. The metric $d_{(P,w,\pi)}$ is said to be weighted coordinates poset block metric ($(P,w,\pi)$-metric). It extends the weighted coordinates poset metric ($(P,w)$-metric) introduced by L. Panek and J. A. Pinheiro and generalizes the poset block metric ($(P,\pi)$-metric) introduced by M. M. S. Alves et al. We determine the complete weight distribution of a $(P,w,\pi)$-space, thereby obtaining it for $(P,w)$-space, $(P,\pi)$-space, $\pi$-space, and $P$-space as special cases. We obtain the Singleton bound for $(P,w,\pi)$-codes and for $(P,w)$-codes as well. In particular, we re-obtain the Singleton bound for any code with respect to $(P,\pi)$-metric and $P$-metric. Moreover, packing radius and Singleton bound for NRT block codes are found.