arXiv subjects

Atul Kumar Shriwastva

Publications and source records attributed to Atul Kumar Shriwastva.

5 recordsLinked to original sources

arXiv2025

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

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.

cs.IT↗

arXiv2023

Block Codes on Pomset Metric

Given a regular multiset $M$ on $[n]=\{1,2,\ldots,n\}$, a partial order $R$ on $M$, and a label map $π: [n] \rightarrow \mathbb{N}$ defined by $π(i) = k_i$ with $\sum_{i=1}^{n}π(i) = N$, we define a pomset block metric $d_{(Pm,π)}$ on the direct sum $ \mathbb{Z}_{m}^{k_1} \oplus \mathbb{Z}_{m}^{k_2} \oplus \ldots \oplus \mathbb{Z}_{m}^{k_n}$ of $\mathbb{Z}_{m}^{N}$ based on the pomset $\mathbb{P}=(M,R)$. The pomset block metric extends the classical pomset metric introduced by I. G. Sudha and R. S. Selvaraj and generalizes the poset block metric introduced by M. M. S. Alves et al over $\mathbb{Z}_m$. The space $ (\mathbb{Z}_{m}^N,~d_{(Pm,π)} ) $ is called the pomset block space and we determine the complete weight distribution of it. Further, $I$-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between MDS codes and $I$-perfect codes for any ideal $I$ is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

cs.IT↗

arXiv2022

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

In this paper, we establish the Singleton bound for pomset block codes ($(Pm,π)$-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 $(Pm,π)$-code is an MDS $(P,π)$-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.

cs.IT↗

arXiv2022

Linear isometries on Weighted Coordinates Poset Block Space

Given $[n]=\{1,2,\ldots,n\}$, a poset order $\preceq$ on $[n]$, a label map $π: [n] \rightarrow \mathbb{N}$ defined by $π(i)=k_i$ with $\sum_{i=1}^{n}π(i) = N$, and a weight function $w$ on $\mathbb{F}_{q}$, let $\mathbb{F}_{q}^N$ be the vector space of $N$-tuples over the field $\mathbb{F}_{q}$ equipped with $(P,w,π)$-metric where $ \mathbb{F}_q^N $ is the direct sum of spaces $ \mathbb{F}_{q}^{k_1}, \mathbb{F}_{q}^{k_2}, \ldots, \mathbb{F}_{q}^{k_n} $. In this paper, we determine the groups of linear isometries of $(P,w,π)$-metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset (block) metric spaces. In particular, we re-obtain the group of linear isometries of the $(P,w)$-mertic spaces and $(P,π)$-mertic spaces.

math.CO↗

arXiv2022

Weighted Coordinates Poset Block Codes

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

math.CO↗