Linear isometries on Weighted Coordinates Poset Block Space

Abstract

Given $[n]=\{1,2,\dots ,n\}$, a poset order $⪯$ on $[n]$, a label map $\pi :[n]\to \mathbb{N}$ defined by $\pi (i)=k_i$ with ${\sum }_{i=1}^n\pi (i)=N$, 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 $(P,w,\pi )$-metric where ${\mathbb{F}}_q^N$ is the direct sum of spaces ${\mathbb{F}}_q^{k_1},{\mathbb{F}}_q^{k_2},\dots ,{\mathbb{F}}_q^{k_n}$. In this paper, we determine the groups of linear isometries of $(P,w,\pi )$-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 $(P,\pi )$-mertic spaces.

13 Pages