# Symmetry Group of Ordered Hamming Block Space

## TL;DR

This paper characterizes the symmetry group of the ordered Hamming block space, a metric space combining poset and block structures, unifying and extending known symmetry groups of related metrics.

## Contribution

It provides a complete description of the symmetry group of the ordered Hamming block space, generalizing previous results for Niederreiter-Rosenbloom-Tsfasman and error-block metrics.

## Key findings

- Reobtains symmetry group for Niederreiter-Rosenbloom-Tsfasman space
- Derives symmetry group for error-block metric space
- Unifies symmetry analysis for multiple metric spaces

## Abstract

Let $P = (\{1,2,\ldots,n,\leq)$ be a poset that is an union of disjoint chains of the same length and $V=\mathbb{F}_q^N$ be the space of $N$-tuples over the finite field $\mathbb{F}_q$. Let $V_i = \mathbb{F}_q^{k_i}$, $1 \leq i \leq n$, be a family of finite-dimensional linear spaces such that $k_1+k_2+\ldots +k_n = N$ and let $V = V_1 \oplus V_2 \oplus \ldots \oplus V_n$ endow with the poset block metric $d_{(P,\pi)}$ induced by the poset $P$ and the partition $\pi=(k_1,k_2,\ldots,k_n)$, encompassing both Niederreiter-Rosenbloom-Tsfasman metric and error-block metric. In this paper, we give a complete description of group of symmetries of the metric space $(V,d_{(P,\pi)})$, called the ordered Hammming block space. In particular, we reobtain the group of symmetries of the Niederreiter-Rosenbloom-Tsfasman space and obtain the group of symmetries of the error-block metric space.