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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.11978 |
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Table of Contents:
- 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.