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

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.