Group of Isometries of Niederreiter-Rosenbloom-Tsfasman Block Space

ABSTRACT

Let $P=\left(\left\{1,2,...,n\right\}\le \right)$ 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 $V_i={\mathbb{F}}_q^{k_i}$ , with $1\le i\le n$, be a family of finite-dimensional linear spaces such that $k_1+k_2+...+k_n=N$ and let $V=V_1\times V_2\times ...\times V_n$ endow with the poset block metric $d_{\left(P,\pi \right)}$ induced by the poset P and the partition $\pi =\left(k_1,k_2,...,k_n\right)$, encompassing both Niederreiter-Rosenbloom-Tsfasman metric and error-block metric. In this paper, we give a complete description of group of isometries of the metric space $\left(V,d_{\left(P,\pi \right)}\right)$, also called the Niederreiter-Rosenbloom-Tsfasman block space. In particular, we reobtain the group of isometries of the Niederreiter-Rosenbloom-Tsfasman space and obtain the group of isometries of the error-block metric space.

Keywords:
error-block metric; poset metric; Niederreiter-Rosenbloom-Tsfasman metric; ordered Hamming metric; isometries; automorphisms

RESUMO

Seja $P=\left(\left\{1,2,...,n\right\},\le \right)$ um conjunto parcialmente ordenado dado por uma união disjunta de cadeias de mesmo comprimento e $V={\mathbb{F}}_q^N$ o espaço vetorial das N-uplas sobre o corpo finito ${\mathbb{F}}_q$. Seja $V=V_1\times V_2\times ...\times V_n$ um produto direto de V, em blocos de subespaços $V_i={\mathbb{F}}_q^{k_i}$ com $k_1+k_2+...+k_n=N$, munido com a métrica de blocos ordenados $d_{\left(P,\pi \right)}$ induzida pela ordem P e pela partição $\pi =\left(k_1,k_2,...,k_n\right)$. Neste trabalho descrevemos o grupo de isometrias do espaço métrico $\left(V,d_{\left(P,\pi \right)}\right)$.

Palavras-chave:
métrica de bloco; métrica de ordem; métrica de Niederreiter-Rosenbloom-Tsfasman; isometrias; automorfismos

1 INTRODUCTION

One of the main classical problem of the coding theory is to find sets with $q^k$ elements in ${\mathbb{F}}_q^N$, the space of N-tuples over the finite field ${\mathbb{F}}_q$, with the largest minimum distance possible. There are many possible metrics that can be defined in ${\mathbb{F}}_q^N$, but the most common ones are the Hamming and Lee metrics.

In 1987 Harald Niederreiter generalized the classical problem of coding theory (see ⁸): given positive integers s and $m_1,...,m_s$, to f ind sets C of vectors $c{}_{ij}\in {\mathbb{F}}_q^N$, for $1\le i\le s$ and $1\le j\le m_i$, with the largest minimum $sum\sum _{i=1}^s{d}_i$, where the minimum is extended over all integers $d_1,...,d_s$ with $0\le d_i\le m_i$ for $1\le i\le s$ and $\sum _{i=1}^s{d}_i\ge 1$ for which the subset $\left\{c_{i,j}:1\le i\le sand1\le j\le d_i\right\}$ is linearly dependent in ${\mathbb{F}}_q^N$. The classical problem corresponds to the special case where $s>m$ and $m_i=1$ for all $1\le i\le s$.

Brualdi, Graves and Lawrence (see ²) also provided in 1995 a wider situation for the Niederreiter’s problem: using partially ordered sets (posets) and defining the concept of poset codes, they started to study codes with a poset metric. Later Feng, Xu and Hickernell ( ⁴, 2006) introduced the block metric, by partitioning the set of coordinate positions of ${\mathbb{F}}_q^N$ into families of blocks. Both kinds of metrics are generalizations of the Hamming metric, in the sense that the latter is attained when considering the trivial order (in the poset case) or one-dimensional blocks (in the block metric case). In 2008, Alves, Panek and Firer (see ¹) combined the poset and block structure, obtaining a further generalization, the poset block metrics. As a unified reading we cite the book of Firer et al.⁵.

A particular instance of poset block codes and spaces, with one-dimensional blocks, are the spaces introduced by Niederreiter in 1991 (see ⁸) and Rosenbloom and Tsfasman in 1997 (see ¹²), where the posets taken into consideration have a finite number of disjoint chains of equal size. This spaces are of special interest since there are several rather disparate applications, as noted by Rosenbloom and Tsfasman (see ¹²) and Park e Barg (see ¹¹).

In ⁷, ³ and ¹⁰ the groups of linear isometries of poset metrics were determined for the Rosenbloom-Tsfasman space, crown space and arbitrary poset-space respectively. In ⁹ we describe the full isometry group (which includes non-linear isometries) of a poset metric that is a product of Rosenbloom-Tsfasman spaces and in ⁶ the author studied the full isometry group to any poset metric. The full description of the group of linear isometries of a poset block space were determined by Alves, Panek and Firer in ¹.

In this work, we describe the group of isometries (not necessarily linear ones) of the poset block space whose underlying poset is a finite union of disjoint chains of same length. We call this space the Niederreiter-Rosenbloom-Tsfasman block space (or NRT block space, for short).

2 POSET BLOCK METRIC SPACE

Let $\left[n\right]:=\left\{1,2,...,n\right\}$ be a finite set with n elements and let ≤ be a partial order on [n]. We call the pair $P:=\left(\left[n\right],\le \right)$ a poset and say that k is smaller than j if $k\le j$ and $k\ne j$. An ideal in $\left(\left[n\right],\le \right)$ is a subset $I\subseteq \left[n\right]$ that contains every element that is smaller than some of its elements, i.e., if $j\in I$ and $k\le j$, then $k\in I$. Given a subset $X\subseteq \left[n\right]$, we denote by 〈X〉 the smallest ideal containing X, called the ideal generated by X . An order on the finite set [n] is called a linear order or a chain if any two elements are comparable, that is, given $i,j\in \left[n\right]$ we have that either $i\le j$ or $j\le i$. In this case, n is said to be the length of the chain and the set can be labeled in such a way that $i_1<i_2<...<i_n$. For the simplicity of the notation, in this situation we will always assume that the order P is defined as $1<2<...<n$.

Let q be a power of a prime, ${\mathbb{F}}_q$ be the finite field of q elements and $V:={\mathbb{F}}_q^N$ the N-dimensional vector space of N-tuples over ${\mathbb{F}}_q$. Let $\pi =\left(k_1,k_2,...,k_n\right)$ be a partition of N, that is,

$N=k_1+k_2+...+k_n,$

with $k_i>0$ an integer. For each integer k _i , let $V_i:{\mathbb{F}}_q^{k_i}$ be the k _i -dimensional vector space over the finite field ${\mathbb{F}}_q$ and define

$V=V_1\times V_2\times ...\times V_n,$

called the π-direct product decomposition of V. A vector $v\in V$ can be uniquely decomposed as

$v=\left(v_1,v_2,...,v_n\right),$

with $v_i\in V_i$ for each $1\le i\le n$. We will call this the π-direct product decomposition of v. Given a poset $P=\left(\left[n\right],\le \right)$, we define the poset block weight${\omega }_{\left(P,\pi \right)}\left(v\right)$ (or simply the (P, π)-weight) of a vector $v=\left(v_1,v_2,...,v_n\right)$ to be

where $supp\left(v\right):=\left\{i\in \left[n\right]:v_i\ne 0\right\}$ is the π-support of the vector v and |X| is the cardinality of the set X . The block structure is said to be trivial when $k_i=1$, for all $1\le i\le n$. The (P, π)-weight induces a metric d _(P,π) on V , that we call the poset block metric (or simply (P, π)-metric):

$d_{\left(P,\pi \right)}\left(u,v\right):={\omega }_{\left(P,\pi \right)}\left(u-v\right).$

The pair (V, d _(P,π) ) is a metric space and where no ambiguity may rise, we say it is a poset block space, or simply a (P, π)-space.

An isometry of (V, d _(P,π) ) is a bijection $T:V\to V$ that preserves distance, that is,

$d_{\left(P,\pi \right)}\left(T\left(u\right),T\left(v\right)\right)=d_{\left(P,\pi \right)}\left(u,v\right),$

for all $u,v\in V$. The set Isom(V, d _(P,π) ) of all isometries of (V, d _(P,π) ) is a group with the natural operation of composition of functions, and we call it the isometry group of (V, d _(P,π) ). An automorphism is a linear isometry.

In ⁹ the group of isometries of a product of Niederreiter-Rosenbloom-Tsfasman spaces is characterized. In ⁶ is studied a subgroup of the full isometry group for any given poset. In this work, we will describe the full isometry group of an important class of poset block spaces, namely, those induced by posets that are an union of disjoint chains of the same length. This class includes the block metric spaces over chains and the Niederreiter-Rosembloom-Tsfasman spaces with trivial block structures.

We remark that the initial idea is the same as in ⁹. The main differences are that we follow a more coordinate free approach an that the dimensions of the blocks pose a new restraint. We first study the isometry group of NRT block space induced by one simple chain (Theorem 1), analogous to those of ⁹. In this work, we prove some results on isometries, also anologous to those of ⁹, plus a result on preservation of block dimensions (Lemma 4), and conclude that Isom(V, d(P,π)) is the semi-direct product of the direct product of the isometry groups induced by each chain and the automorphism group of the permutations of chains that preserves the block dimensions (Theorem 6).

3 ISOMETRIES OF LINEAR ORDERED BLOCK SPACE

Let $P=\left(\left[n\right],\le \right)$ be the linear order $1<2<...<n$, let $\pi =\left(k_1,k_2,...,k_n\right)$ be a partition of N and let

$V=V_1\times V_2\times ...\times V_n,$

where $V_i={\mathbb{F}}_q^{k_i}$, for $i=1,2,...,n$, be the π-direct product decomposition of the vector space $V={\mathbb{F}}_q^N$ endow with the poset block metric d _{(p, π)} . In this section we will describe the full isometry group of the poset block space (V, d _(P,π) ). This description will be used in the next section to describe the isometry group of the NRT block space. In this section, $P=\left(\left[n\right],\le \right)$ will be the linear order $1<2<...<n$.

We note that, given $u=\left(u_1,...,u_n\right)$ and $v=\left(v_1,...,v_n\right)$ in the total ordered block space V,

$d_{\left(P,\pi \right)}\left(u,v\right)=max\left\{i:u_i\ne v_i\right\}.$

For each $i\in \left\{1,2,...,n\right\}$, let

$F_i:V_i\times V{i+1}_{}\times ...\times V_n\to V_i$

be a map that is a bijection with respect to the first block space V _i , that is, given $\left(v_{i+1},...,v_n\right)\in V_{i+1}\times ...\times V_n$, the map ${\tilde{F}}_{v_{i+1},...,v_n}:V_i\to V_i$ defined by

${\tilde{F}}_{v_{i+1},...,v_n}\left(v_i\right)=F_i\left(v_i,v_{i+1},...,v_n\right)$

is a bijection. Let S _q,π,i be the set of such maps F _i . Given $F_i\in S_{q,\pi ,i}$, with 1 ≤ i ≤ n, we define a map $T_{\left(F_1,F_2,...,F_n\right)}:V\to V$ by

$T_{\left(F_1,F_2,...,F_n\right)}\left(v_1,...,v_n\right):=\left(F_1\left(v_1,...,v_n\right),...,F_n\left(v_n\right)\right)$

Theorem 1.Let$P=\left(\left[n\right],\le \right)$be the linear order$1<2<...<n$and let$V=V_1\times V_2\times ...\times V_n$be the π-direct product decomposition of$V={\mathbb{F}}_q^N$endowed with the poset block metric induced by the poset P and the partition π. Then, the group Isom(V, d_(P,π) ) of isometries of (V, d _(P,π) ) is the set of all maps$T_{\left(F_1,F_2,...,F_n\right)}:V\to V$.

Proof. Given $u=\left(u_1,...,u_n\right)$ and $v=\left(v_1,...,v_n\right)\in V$, let $l=d_{\left(P,\pi \right)}\left(u,v\right)=max\left\{i:u_i\ne v_i\right\}$. Since each $F_i:V_i\times V_{i+1}\times ...V_n\to V_i$ is a bijection in relation to the first block space V _i , it follows that

$F_l\left(u_l,u_{l+1},...,u_n\right)\ne F_l\left(v_l,v_{l+1},...,v_n\right)$

and

$F_t\left(u_t,u_{t+1},...,u_n\right)=F_t\left(v_t,v_{t+1},...,v_n\right)$

for any $t>l$. It follows that

$d_{\left(P,\pi \right)}\left(T_{\left(F_1,...,F_n\right)}\left(u\right),T_{\left(F_1,...,F_n\right)}\left(v\right)\right)=max\left\{i:F_i\left(u_i,...,u_n\right)\ne F_i\left(v_i,...,v_n\right)\right\}=l$

and hence $T_{\left(F_1,F_2,...,F_n\right)}$ is distance preserving. Since V is a finite metric space, it follows that $T_{\left(F_1,F_2,...,F_n\right)}$ is also a bijection.

Now let T be an isometry of V . Let us write

$T\left(v_1,v_2,...,v_n\right)=\left(T_1\left(v_1,v_2,...,v_n\right),...,T_n\left(v_1,v_2,...,v_n\right)\right).$

We prove first that $T_j\left(v_1,v_2,...,v_n\right)=F_j\left(v_j,v_{j+1},...,v_n\right)$, that is, T _j does not depend on the first $j-1$ coordinates. In other words, we want to prove that

$T_j\left(v_1,...,v_{j-1},v_j,...,v_n\right)=T_j\left(u_1,...,u_{j-1},v_j,...,v_n\right)$

regardless of the values of the first $j-1$ coordinates. Since

$d_{\left(P,\pi \right)}\left(\left(u_1,...,u_{j-1},v_j,...,v_n\right),\left(v_1,...,v_{j-1},v_j,...,v_n\right)\right)=\underset{\underset{\le j-1}{i}}{max}\left\{i:v_i\ne u_i\right\}$

and since T is an isometry, it follows that

$\begin{array}{c}{d}_{\left(P,\pi \right)}\left(T\left(v_1,...,v_{j-1},v_j,...,v_n\right),T\left(u_1,...,u_{j-1},v_j,...,v_n\right)\right)=\\ =d_{\left(P,\pi \right)}\left(\left(u_1,...,u_{j-1},v_j,...,v_n\right),\left(v_1,...,v_{j-1},v_j,...,v_n\right)\right)\le j-1,\end{array}$

and so,

$T_j\left(v_1,...,v_{j-1},v_j,...,v_n\right)=T_j\left(u_1,...,u_{j-1},v_j,...,v_n\right),$

for any $\left(v_1,...,v_{j-1}\right),\left(u_1,...,u_{j-1}\right)\in V_1\times ...\times V_{j-1}$ and $\left(v_j,...,v_n\right)\in V_j\times ...\times V_n$. Thus,

$T\left(v_1,v_2,...,v_n\right)=\left(F_1\left(v_1,v_2,...,v_n\right),F_2\left(v_2,...,v_n\right),...,F_n\left(v_n\right)\right)$

and the first statement is proved. Now, we need to prove that each $\tilde{F}{}_{v_{i+1},...,v_n}$ is a bijection, what is equivalent to prove those maps are injective. If $\tilde{F}{}_{v_{i+1},...,v_n}$ is not injective, then there are $v_i\ne u_i$ in V _i such that

$\tilde{F}{}_{v_{i+1},...,v_n}\left(v_i\right)={\tilde{F}}_{v_{i+1},...,v_n}\left(u_i\right).$

Considering i minimal with this property, it follows that

$\begin{array}{c}i=d_{\left(P,\pi \right)}\left(\left(v_1,...,v_i,...,v_n\right),\left(v_1,...,u_i,...,v_n\right)\right)\\ =d_{\left(P,\pi \right)}\left(T\left(v_1,...,v_i,...,v_n\right),T\left(v_1,...,u_i,...,v_n\right)\right)<1\end{array}$

contradicting the assumption that T is an isometry of (V, d _(P,π) ).

Let S _m be the symmetric group of permutations of a set with m elements and $V=V_1\times V_2\times ...\times V_n$ be the π-direct product decomposition of $V={\mathbb{F}}_q^N$ with $\pi =\left(k_1,k_2,...,k_n\right)$. Since V has q ^N elements we can identify the group S _q,π,1 of functions $F:V_1\times V_2\times ...\times V_n\to V_1$ such that ${\tilde{F}}_{v_2,...,v_n}$ is a permutation of $V_1={\mathbb{F}}_q^{k_1}$, with operation

$\left(F·G\right)\left(v\right):=F\left(G\left(v_1,v_2,...,v_n\right),v_2,...,v_n\right),$

with $F,G\in S_{q,\pi ,1}$ and $v=\left(v_1,v_2,...,v_n\right)\in V$, with the direct product¹ ${\left(S_{q{k}_1}\right)}^{q^{N-k_1}}$. With this notations, it follows the following result.

Theorem 2.Let$P=\left(\left[n\right],\le \right)$be the linear order$1<2<...<n$and let$V=V_1\times V_2\times ...\times V_n$be the π-direct product decomposition of$V={\mathbb{F}}_q^N$endowed with the poset block metric induced by the poset P and the partition π. If$\pi =\left(k_1,k_2,...,k_n\right)$, then the group of isometries Isom(V, d_(P,π) ) has a semi-direct product²structure given by

${\left(S_{q{k}_1}\right)}^{q^{N-k_1}}⋊\left(...\left({\left(S_{q{k}_{n-1}}\right)}^{q^{N-k_1-k_2-...-k_{n-1}}}⋊{\left(S_{q{k}_n}\right)}^{q^{N-k_1-k_2-...-k_{n-1}-k_n}}\right)...\right).$

Proof. Let $G_{\left(\hat{P},\hat{\pi }\right)}$ be the isometry group $Isom\left(\hat{V},d_{\left(\hat{P},\hat{\pi }\right)}\right)$ of

$\hat{V}={\hat{V}}_1\times {\hat{V}}_2\times ...\times {\hat{V}}_{n-1},$

where ${\hat{V}}_i=V_{i+1}$ for each $i=1,2,...,n-1,\hat{P}=\left(\left[n-1\right],\le \right)$ is the linear order $1<2<...<n-1$ and $\hat{\pi }=\left(k_2,k_3,...,k_n\right)$. Let

$H=\left\{T\in Isom\left(V,d_{\left(P,\pi \right)}\right):T=\left(F_1,Pr{v}_2,...,Pr{v}_n\right)with{F}_1\in S_{q,\pi ,l}\right\}$

and

$K=\left\{T\in Isom\left(V,d_{\left(P,\pi \right)}\right):T=\left(Pr{v}_1,F_2,...,F_n\right)with{F}_i\in S_{q,\pi ,i}\right\},$

where each $P{r}_{V_i}:V_i\times V_{i+1}\times ...\times V_n\to V_i$ is the projection map given by $P{r}_{V_i}\left(v_i,v_{i+1},...,v_n\right)=v_i$. We claim that Isom(V, d _(P,π) ) is a semi-direct product of H by K. Clearly, $Isom\left(V,d_{\left(P,\pi \right)}\right)=HK$, because each isometry of (V, d _(P,π) ) is a composition $T_1\circ T_2$ with $T_1\in H$ and $T_2\in K$. Let $L\in H\cap K$. Since $L\in H,L\left(x_1,x_2,...,x_{n+1}\right)=\left(x_1^{\text{'}},x_2,...,x_{n+1}\right)$ and, since L is also in K, it follows that $x_1^{\text{'}}=x_1$. Hence, $L=i{d}_v$ and the groups H and K intersect trivially. Now, we prove that H is a normal subgroup of Isom(V, d _(P,π) ). In fact, since $Isom\left(V,d_{\left(P,\pi \right)}\right)=HK$, it suffices to check that $TH{T}^{-1}\subset H$ for each $T\in K$. Let $L\in H$ and $T\in K$. Then $L\left(x_1,..,x_n\right)=\left(F_1\left(x_1,...,x_n\right),x_2,...,x_n\right)$ and $T\left(x_1,...,x_n\right)=\left(x_1,\tilde{T}\left(x_2,...,x_n\right)\right)$ for some $F_1\in S_{q,\pi ,1}$ and $\tilde{T}\in Isom\left(\hat{V},d_{\left(\hat{P},\hat{\pi }\right)}\right)$. If $\left(x_1,...,x_n\right)\in V$, then

$\begin{array}{c}\left(T\circ L\circ T^{-1}\right)\left(x_1,...,x_n\right)=\left(T\circ L\right)\left(x_1,{\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right)\\ =T\left(F_1\left(x_1,{\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right),{\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right)\\ =\left(F_1\left(x_1,{\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right),\tilde{T}\circ {\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right)\\ =\left(F_1\left(x_1,{\tilde{T}}^{-1}\left(x_2,...,x_n\right)\right),x_2,...,x_n\right).\end{array}$

Since F ₁ is a bijection with respect to the first block space V ₁, it follows that TLT $TL{T}^{-1}\in H$. This shows that H is a normal subgroup of Isom(V, d _(P,π) ) and that

$Isom\left(V,d_{\left(P,\pi \right)}\right)=H⋊K.$

In order to simplify notation, we will denote the elements of ${\left(S_{q{k}_1}\right)}^{Q^{N-k_1}}$ by(π_X ), where

$\left({\pi }_x\right):=\left({\pi }_x\right)X\in {\mathbb{F}}_q^{N-k_1}.$

The group $G_{\left(\hat{P},\hat{\pi }\right)}$ acts on $V=V_1\times \hat{V}$ by

$T\left(x_1,...,x_n\right)=\left(x_1,T\left(x_2,...,x_n\right)\right)$

and ${\left(S_{q{k}_1}\right)}^{Q^{N-k_1}}$ acts by

$\left({\pi }_X\right)\left(x_1,...,x_n\right)=\left({\pi }_{\left(x_2,...,x_n\right)}\left(x_1\right),x_2,...,x_n\right).$

Both groups act as groups of isometries and both act faithfully. Therefore these actions establish isomorphisms of these groups with subgroups $HandK:H\cong {\left(S_{q{k}_1}\right)}^{q^{N-k_1}}andK\cong G_{\left(\hat{P},\hat{\pi }\right)}$. Using the aforementioned isomorphisms involving H and K, it follows that

$Isom\left(V,d_{\left(P,\pi \right)}\right)={\left(S_{q{k}_1}\right)}^{q^{N-k_1}}⋊G_{\left(\hat{P},\hat{\pi }\right)},$

which concludes the proof.

Corollary 3. Let $P=\left(\left[n\right],\le \right)$ be the linear order $1<2<...<n$ and let $V=V_1\times V_2\times ...\times V_n$ be the π-direct product decomposition of $V={\mathbb{F}}_q^N$ endowed with the poset block metric induced by the poset P and the partition π. If

$H=\left\{T\in Isom\left(V,d_{\left(P,\pi \right)}\right):T=\left(F_1,P{r}_{V_2},...,P{r}_{V_n}\right)with{F}_1\in S_{q,\pi ,l}\right\}$

and

$K=\left\{T\in Isom\left(V,d_{\left(P,\pi \right)}\right):T=\left(P{r}_{V_1},F_2,...,F_n\right)with{F}_i\in S_{q,\pi ,i}\right\},$

then ³

$Isom\left(V,d_{\left(P,\pi \right)}\right)\cong H⋊{}_{\theta }K$

with $\theta :K\to Aut\left(H\right)$ given by

${\theta }_T\left(L\right)\left(x_1,x_2,...,x_n\right)=\left(F_1\left(x_1,\tilde{T}{}^{-1}\left(x_2,...,x_n\right)\right),x_2,...,x_n\right)$

for all $L=\left(F_1,P{r}_{V_2},...,P{r}_{V_n}\right)\in H,T=\left(P{r}_{V_1},F_2,...,F_n\right)\in Kand\left(x_1,...,x_n\right)\in Vwith\tilde{T}:=\left(F_2,...,F_n\right)$ .

Corollary 4. Let $P=\left(\left[n\right],\le \right)$ be the linear order $1<2<...<n$ and let $V=V_1\times V_2\times ...\times V_n$ be the π-direct product decomposition of $V={\mathbb{F}}_q^n$ endowed with the poset block metric induced by the poset P and the partition $\pi =\left(k_1,k_2,...,k_n\right)$ . Then

$\left|Isom\left(V,d_{\left(P,\pi \right)}\right)\right|={\left(q^{k_1}!\right)}^{q^{N-k_1}}·{\left(q^{k_2}!\right)}^{q^{N-k_1-k_2}}·...·{\left(q^{k_{n-1}}!\right)}^q·\left(q^{k_n}!\right).$

Now, if the partition $\pi =\left(l,l,...,l\right)$, it follows the following result (see ⁹, Corollary 3.1):

Corollary 5.Let$P=\left(\left[n\right],\le \right)$be the linear order$1<2<...<n$and let$V={\mathbb{F}}_q^n$be the vector space endowed with the poset metric induced by the poset P. Then the group of isometries Isom(V, d_P ) is a semi-direct product

${\left(S_q\right)}^{q^{n-1}}⋊\left(...\left({\left(S_q\right)}^q⋊S_q\right)...\right).$

In particular,

$\left|Isom\left(V,d_P\right)\right|={\left(q!\right)}^{\frac{q^n-1}{q-1}}.$

4 ISOMETRIES OF NRT BLOCK SPACE

In this section, we consider an order $P=\left(\left[m·n\right],\le \right)$, that is, the union of m disjoint chains P ₁ , P ₂ , ..., P _m of order n. We identify the elements of $\left[m·n\right]$ with the set of ordered pairs of integers (i, j), with $1\le i\le m,1\le j\le n$, where $\left(i,j\right)\le \left(k,l\right)iffi=kandj\le {}_{\mathrm{\mathbb{N}}}l$, where $\le {}_{\mathrm{\mathbb{N}}}$ is just the usual order on ℕ. We denote $P_i=\left\{\left(i,j\right):1\le j\le n\right\}$. Each P _i is a chain and those are the connected components of $\left(\left[m·n\right],\le \right)$.

Let $\pi =\left(k_{1l},...,k_{1n},...,k_{ml},...,k_{mn}\right)$ be a partition of $N=mn$ and for each $1\le i\le m$ let ${\pi }_i=\left(k_{il},...,k_{in}\right)$. Let

$V=U_1\times U_2\times ...\times U_m,$ (4.1)

where

$U_i:=V_{i1}\times V_{i2}\times ...\times V_{in}$

and $V_{ij}={\mathbb{F}}_q^{k_{ij}}$, for all $1\le i\le m,1\le j\le n$. The space V with the poset metric induced by the order $P=\left(\left[m·n\right],\le \right)$ is called the (m, n, π)-NRT block space. Note that if $n=1$, then $P=\left(\left[m·l\right],\le \right)$ induces just the error-block metric on V, and in particular, if $\pi =\left(1,1,...,1\right)$, then $P=\left(\left[m·l\right],\le \right)$ induces just the Hamming metric on ${\mathbb{F}}_q^m$. Hence the induced metric from the poset $P=\left(\left[m·n\right],\le \right)$ can be viewed as a generalization of the error-block metric.

Let $V=U_1\times U_2\times ...\times U_m,$ as in (4.1), called the canonical decomposition of V. Given the canonical decompositions $u=\left(u_1,...,u_m\right)andv=\left(v_1,...,v_m\right)with{u}_i,v_i\in U_i$, we have that

$d_{\left(P,\pi \right)}\left(u,v\right)=\sum _{i=1}^m{d}_{\left(P_i,{\pi }_i\right)}\left(u_i,v_i\right),$

where $d_{\left(P_i,{\pi }_i\right)}$, the restriction of d _(P,π) to U _i , is a linear poset block metric. We note that the restriction of d _(P,π) to each U _i turns it into a poset space defined by a linear order, that is, each U _i is isometric to $\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$ with the metric $d_{\left(\left[n\right],{\pi }_i\right)}$ determined by the chain $1<2<...<n$. Let $G_{i,{\pi }_i,n}$ be the group of isometries , of $\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$. The direct product ${\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}$ acts on V in the following manner: given $T=\left(T_1,...,T_m\right)\in {\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}andv=\left(v_1,...,v_m\right)\in V$,

$T\left(v\right):=\left(T_1\left(v_1\right),...,T_m\left(v_m\right)\right).$

Lemma 1.Let (V, d _(P,π) ) be the (m, n, π)-NRT block space over${\mathbb{F}}_q$and let$G_{i,{\pi }_i,n}$be the group of isometries of$\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$. Given$T_i\in G_{i,{\pi }_i,n}$, with$1\le i\le m$, the map$T=\left(T_1,...,T_m\right)$defined by

$T\left(v\right):=\left(T_1\left(v_1\right),...,T_m\left(v_m\right)\right)$

is an isometry of (V, d _(P,π) ).

Proof. Given $u,v\in V$, consider the canonical decompositions $u=\left(u_1,...,u_m\right)andv=\left(v_1,...,v_m\right)with{u}_i,v_i\in U_i$. Then,

$\begin{array}{c}{d}_{\left(P,\pi \right)}\left(T\left(u\right),T\left(v\right)\right)=\sum _{i=1}^m{d}_{\left(P_i,{\pi }_i\right)}\left(T_i\left(u_i\right),T_i\left(v_i\right)\right)\\ =\sum _{i=1}^m{d}_{\left(P_i,{\pi }_i\right)}\left(u_i,v_i\right)\\ =d_{\left(P,\pi \right)}\left(u,v\right),\end{array}$

which concludes the proof.

Let S _m be the permutation group of {1, 2, ..., m}. We will call a permutation $\sigma \in S_{\pi }$admissible if $\sigma \left(i\right)=j$ implies that $k_{il}=k_{jl},forall1\le l\le n$. Cleary, the set Sπ of all admissible permutations is a subgroup of S _m .

Let us consider the canonical decomposition $v=\left(v_1,...,v_m\right)$ of a vector v in the (m, n, π)NRT block space V. The group S _π acts on V as a group of isometries: given $\sigma \in S_{\pi }andv=\left(v_1,...,v_m\right)\in V$, we define

$T_{\sigma }\left(v\right)=\left(v_{\sigma \left(1\right)},v_{\sigma \left(2\right)},...,v_{\sigma \left(m\right)}\right).$

Lemma 2.Let (V, d _(P,π) ) be the (m, n, π)-NRT block space V and let$\sigma \in S_{\pi }$. Then T _σ is an isometry of (V, d _(P,π) ).

Proof. Given $u,v\in V$, we consider their canonical decompositions $u=\left(u_1,...,u_m\right)andv=\left(v_1,...,v_m\right)with{u}_i,v_i\in U_i$. Then,

$\begin{array}{c}{d}_{\left(P,\pi \right)}\left(T_{\sigma }\left(u\right),T_{\sigma }\left(v\right)\right)=\sum _{i=1}^m{d}_{\left(P_i,{\pi }_i\right)}\left(u_{\sigma \left(i\right)},v_{\sigma \left(i\right)}\right)\\ =\sum _{i=1}^m{d}_{\left(P_i,{\pi }_i\right)}\left(u_i,v_i\right)\\ =d_{\left(P,\pi \right)}\left(u,v\right),\end{array}$

which concludes the proof.

The Lemmas 1 and 2 assure that the groups ${\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}$ and S _π are both isometry groups of the (m, n, π)-NRT block space V, and so is the group G _{(m, n, π)} generated by both of them. We identify ${\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}$ and Sπ with their images in G _{(m, n, π)} and make an abuse of notation, denoting the images in G _{(m, n, π)} by the same symbols. With this notation, analogous calculations as those of Theorem 2 show that

$\left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)\cap S_{\pi }=\left\{i{d}_v\right\}$

And

$\sigma \circ \left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)\circ {\sigma }^{-1}=\prod _{i=1}^m{G}_{i,{\pi }_i,n},$

for every $\sigma \in S_{\pi }$. Since ${\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}$ is normal in G _(m;n;π) and G _(m,n,π) is generated by ${\Pi }_{i=1}^m{G}_{i,{\pi }_i,n}$ and S _π , it follows that

$G_{\left(m,n,\pi \right)}=\left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)·S_{\pi },$

and therefore, it follows the following proposition:

Proposition 3. The group G _(m,n,π) has the structure of a semi-direct product given by

$\left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)⋊S_{\pi }.$

We need two more lemmas in order to prove that every isometry of the (m, n, π)-NRT block space V is in G _(m,n,π) , i.e., that G _(m,n,π) is the group of isometries of V . We will identify the block space U _i of V with the subspace of V of vectors $v=\left(v_1,...,v_m\right)suchthat{v}_j=0forj\ne i$.

Lemma 4.Let (V, d _(P,π) ) be the (m, n, π)-NRT block space and let$V=U_1\times U_2\times ...\times U_m$be the canonical decomposition of V. If

$\pi =\left(k_{1l},...,k_{1n},...,k_{m1},...,k_{mn}\right)$

and $T:V\to V$ is an isometry such that $T\left(0\right)=0$ , then for each index $1\le i\le m$ there is another index $1\le j\le m$ such that

$T\left(U_i\right)=U_j$

And

$k_{il}=dim\left(V_{il}\right)=dim\left(V_{jl}\right)=k_{jl},$

for all$1\le l\le n$.

Proof. In the following we denote the subspace $V_{i1}\times V_{i2}\times ...\times V_{ik}by{U}_{ik}$. We begin by showing that for each index $1\le i\le m$ there is another index $1\le j\le m$ such that $T\left(U_{i1}\right)=U_{j1}and{k}_{i1}=k_{j1}.Let{v}_i\in U_{i1},with{v}_i\ne 0$. Since

$d_{\left(P,\pi \right)}\left(T\left(v_i\right),0\right)=d_{\left(P,\pi \right)}\left(v_i,0\right)=1,$

it follows that $v_j=T\left(v_i\right)$ is a vector of (P, π)-weight 1. Thus $T\left(v_i\right)\in U_{j1}$ for some index $1\le j\le m.If{v}_i^{\text{'}}\in U_{il},with{v}_i^{\text{'}}\ne v_{i}and{v}_i^{\text{'}}\ne 0,thenT\left(v_i^{\text{'}}\right)=v_{k}forsome{v}_k\in U_{k1}with{v}_k\ne 0$, but also

$d_{\left(P,\pi \right)}\left(T\left(v_i\right),T\left(v_i^{\text{'}}\right)\right)=d_{\left(P,\pi \right)}\left(v_i,v_i^{\text{'}}\right)=1.$

If $k\ne j,then{d}_{\left(P,\pi \right)}\left(T\left(v_i\right),T\left(v_i^{\text{'}}\right)\right)=d_{\left(P,\pi \right)}\left(v_j,v_k\right)=2$. Hence $k=jandT\left(U_{i1}\right)\subseteq U_{j1}$. Now apply the same reasoning to T ⁻¹. If $v_i\in U_{i1},with{v}_i\ne 0,andT\left(v_i\right)=v_{j}with{v}_j\in U_{j1},then{T}^{-1}\left(v_j\right)\in U_{i1}$ and therefore $T^{-1}\left(U_{j1}\right)\subseteq U_{i1}$. So that $U_{j1}\subseteq T^{-1}\left(U_{i1}\right)$. Therefore $T\left(U_{i1}\right)=U_{j1}$. Since T is bijective, it follows that $k_{i1}=k_{j1}$. For induction on k, suppose that for each s there exists an index l such that

$T\left(U_{sk}\right)=U_{lk}$

and $k_{sj}=k_{lj}$ for all $1\le j\le k$ and for all $1\le k\le n$. We note that Usn = Us. Without loss of generality, let us consider $s=1,P1=\left\{\left(1,1\right),...,\left(1,n\right)\right\}$. Let P _l be the chain that begins at (l, 1) such that $T\left(U_{11}\right)=U_{l1}$ and suppose that ${U_{1\left(k-1\right)}}_{}istakenbyTonto{U}_{l\left(k-1\right)}with{k}_{1j}=k_{lj}forall1\le j\le k-1.Letv=\left(v_{11},...,v_{lk}\right),v_{1i}\in V_{1i}with{v}_{1k}\ne 0,andletT\left(v\right)=\left(u_1,...,u_m\right),u_i\in U_i.SinceT\left(0\right)=0$, it follows that

${\omega }_{\left(P,\pi \right)}\left(v\right)={\omega }_{\left(P,\pi \right)}\left(T\left(v\right)\right)={\omega }_{\left(P,\pi \right)}\left(u_1\right)+...+{\omega }_{\left(P,\pi \right)}\left(u_m\right).$

We will use this to show that $T\left(v\right)=u_l$. First suppose that $u_l=0$. In this case, ${\omega }_{\left(P,\pi \right)}\left(v\right)=\sum _{j\ne l}{\omega }_{\left(P,\pi \right)}\left(u_j\right)$ and therefore, if $u_{11}\in U_{11},with{u}_{11}\ne 0andT\left(u_{11}\right)=u_{l1}$, then

$k=d_{\left(P,\pi \right)}\left(u_{11},v\right)=d_{\left(P,\pi \right)}\left(T\left(u_{11}\right),T\left(v\right)\right)=\sum _{j\ne l}{\omega }_{\left(P,\pi \right)}\left(u_j\right)+{\omega }_{\left(P,\pi \right)}\left(u_{l1}\right)=k+1,$

a contradiction. Hence $u_l\ne 0.Let{u}_l=\left(u_{l1},...,u_{lt}\right),u_{li}\in v_{li}$, and suppose now there is another summand $u_i\ne 0$. Then $k=\sum _j{\omega }_{\left(P,\pi \right)}\left(u_j\right)>{\omega }_{\left(P,\pi \right)}\left(u_l\right)$ and therefore $t<k$. By the induction hypothesis, it follows that T ⁻¹(u _l ) is a vector in $V_{1\left(k-1\right)}$ with ${\omega }_{\left(P,\pi \right)}\left(T^{-1}\left(u_l\right)\right)<k$. Hence

$k=d_{\left(P,\pi \right)}\left(T^{-1}\left(u_l\right),v\right)=d_{\left(P,\pi \right)}\left(u_l,T\left(v\right)\right)=\sum _{j\ne l}{\omega }_{\left(P,\pi \right)}\left(u_j\right)<k,$

again a contradiction. Hence, $T\left(v\right)\in U_{lk}$. From the induction hypothesis and from the fact that T is a weight-preserving bijection, it follows that

$T\left(v_{11},...,v_{1k}\right)=\left(u_{l1},...,u_{lk}\right),$

where $v_{lk}\ne 0$ implies $u_{lk}\ne 0$. Therefore, $T\left(U_{1k}\right)=U_{lk}.Since{k}_{1j}=k_{lj}forall1\le j\le k-1andT$ is a bijection, it follows that $k_{1k}=k_{lk}$. Hence $T\left(U_1\right)=U_{l}with{k}_{1j}=k_{lj}forall1\le j\le n$.

We recall that we defined an action of the group S _π of the admissible permutations of S _m on the canonical decomposition $U_1\times U_2\times ...\times U_m$ of V by

$T_{\sigma }\left(v\right):=\left(v_{\sigma \left(1\right)},v_{\sigma \left(2\right)},...,v_{\sigma \left(m\right)}\right)$

and that we defined an action of $\prod _{i=1}^m{G}_{i,{\pi }_i,n}$ on V by

$\left(g_1,g_2,...,g_m\right)\left(v_1,v_2,...,v_m\right)=\left(g_1\left(v_1\right),g_m\left(v_m\right)\right).$

Lemma 5.Let (V, d _(P,π) ) be the (m, n, π)-NRT block space. Each isometry of V that preserves the origin is a product$T_{\sigma }\circ g$, with σ in S _π and g in$\prod _{i=1}^m{G}_{i,{\pi }_i,n}$.

Proof. Let T be an isometry of V , with $T\left(0\right)=0$. By the Lemma 4, for each $1\le i\le m$ there is a σ (i) such that $T\left(U_i\right)=U_{\sigma \left(i\right)}with{k}_{il}=k_{\sigma \left(i\right)l}forall1\le l\le n$. Since T is a bijection, it follows that the map $i\mapsto \sigma \left(i\right)$ is an admissible permutation of the set {1,..., m}. We define $T_{\sigma }:V\to V$ by

$T_{\sigma }\left(v\right):=\left(v_{\sigma \left(1\right)},v_{\sigma \left(2\right)},...,v_{\sigma \left(m\right)}\right).$

Thus, $T=T_{\sigma }^{-1}\left(T_{\sigma }T\right)=T_{\sigma -1}\left(T_{\sigma }T\right),where\sigma \in S_{\pi }.Letg=T_{\sigma }T.Sinceg\left(U_i\right)=\left(T_{\sigma }T\right)\left(U_i\right)=T_{\sigma }\left(U_{\sigma \left(i\right)}\right)=U_{\sigma -1\sigma \left(i\right)}=U_i$, we have that $g|U_i$ is an isometry of U _i . Defining $g_i:=g|U_i$ it follows that $g=\left(g_1,...,g_m\right)$, and hence, $g\in \prod _{i=1}^m{G}_{i,\pi {}_i,n}$.

Theorem 6.Let (V, d _(P,π) ) be the (m, n, π)-NRT block space. The group of isometries of V is isomorphic to

$\left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)⋊S_{\pi }.$

Proof. Let G _(m,n,π) be the group of isometries of V generated by the action of $\prod _{i=1}^m{G}_{i,{\pi }_i,n}$ and S _π . Let T be an isometry of V and let $v=T\left(0\right)$. The translation $S_{-v}\left(u\right):=u-v$ is clearly an isometry of V and $\left(S_{-v}\circ T\right)\left(0\right)=S_{-v\left(v\right)}=0$ is an isometry that fixes the origin. Hence, by the previous lemma, it follows that $S_{-v}\circ T\in G_{\left(m,n,\pi \right)}$. Consider the canonical decomposition of v on the chain spaces, $v=\left(v_1,...,v_m\right),v_i\in U_i$. Since the restriction $S_{v_i}of{S}_v=\left(S_{v_1},...,S_{v_m}\right)to{U}_i,with{S}_{v_i}\left(u_i\right)=u_i+v_i$ for each i, is the translation by v _i , it follows that is an isometry of U _i . Thus, $S_v\in \prod _{i=1}^m{G}_{i,{\pi }_i,n}\subset G_{\left(m,n,\pi \right)}$ and hence, that $T=S_v\circ \left(S_{-v}\circ T\right)$ is in G _{(m, n, p).} Thus G _{(m, n, p)} is the isometry group of V . By Proposition 3, it follows that G _(m,n,π) is isomorphic to $\left(\prod _{i=1}^m{G}_{i,{\pi }_i,n}\right)⋊S_{\pi }$.

If $n=1$ (P is an antichain) and $\pi =\left(k_1,k_2,...,k_m\right)$, where

$k_1=...=k_{m_1}=l_1,...,k_{m_{1+...+m_{l-1}}+1}=...=k_{m_1+...+m_l}=l_r$

with $l_1>...>l_{r}and{m}_1,...,m_l$ positive integers such that $m_1+...+m_l=m$, it follows that $G_{i,\left(k_i\right),1}=S_{q{k}_i},for1\le i\le m,and{S}_{\pi }=S_{m1}\times ...\times S_{ml}$ (S _π only permutes those blocks with same dimensions). Therefore it follows the following result.

Corollary 7. If P is an antichain, then

$Isom\left(V,d_{\left(P,\pi \right)}\right)=\left(\prod _{i=1}^m{S}_{q{k}_i}\right)⋊\left(\prod _{i=1}^l{S}_{m_i}\right),$

When $n=1and\pi =\left(1,1,...,1\right)$, the (P, π)-weight is the usual Hamming weight on ${\mathbb{F}}_q^m$. In this case, each G _i,(1) ,1 in Corollary 7 is equal to S _q and every permutation in S _m is also admissible. Thus, we reobtain the isometry groups of Hamming space:

Corollary 8. Let d _H be the Hamming metric over ${\mathbb{F}}_q^m$ . The isometry group of $\left({\mathbb{F}}_q^m,d_H\right)$ is isomorphic to $S_q^m⋊S_m$ .

If $\pi =\left(1,1,...,1\right)$, then every permutation in S _m is admissible. Hence, it follows the following result (see ⁹, Theorem 4.1):

Corollary 9. Let $V={\mathbb{F}}_q^{mn}$ be the vector space endowed with the poset metric d _P induced by the poset $P=\left(\left[mn\right],\le \right)$ which is union of chains P ₁ , ..., P _m of length n. Then

$Isom\left(V,d_P\right)={\left(G_n\right)}^m⋊S_m,$

where $G_n:={\left(S_q\right)}^{q^{n-1}}⋊\left(...\left({\left(S_q\right)}^q⋊S_q\right)...\right)$ . In particular,

$\left|Isom\left(V,d_P\right)\right|={\left(q!\right)}^{m·\frac{q^n-1}{q-1}+m}·m!.$

5 AUTOMORPHISMS

The group of automorphisms of (V, d _(P,π) ) is easily deduced from the Lemma 5 and Theorem 6. Let $T=T_{\sigma }\circ g$ be a isometry. Since T _σ is linear, it follows that the linearity of T is a matter of whether g is linear or not. Now, if $g=\left(g_1,g_2,...,g_m\right)$ is linear, then each component g _i must also be linear. Since each g _i is an isometry, it follows that g _i is in the group $Aut\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$ of linear isometries of $\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$. Therefore $g\in \prod _{i=1}^{m}Aut\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)$. On the other hand, any element of this group is a linear isometry. Hence, it follows the following result:

Theorem 1.The automorphism group Aut (V, d _(P,π) ) of (V, d _(P,π) ) is isomorphic to

$\left(\prod _{i=1}^{m}Aut\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)\right)⋊S_{\pi }.$

Corollary 2. Let $n=1and\pi =\left(k_1,k_2,...,k_m\right)$ be a partition of N. If

$k_1=...=k_{m_1}=l_1,...,k_{m_1+...+m_{l-1}+1}=...=k_n=l_r$

with $l_1>l_2>...>l_r$ , then

$\left|Aut\left({\mathbb{F}}_q^N,d_{\left(P,\pi \right)}\right)\right|=\left(\prod _{i=1}^m\left(q^{k_i}-1\right)\left(q^{k_i}-q\right)...\left(q^{k_i}-q^{k_i-1}\right)\right)·\left(\prod _{j=1}^l{m}_j!\right).$

Proof. Note initially that there is a bijection from Aut (U _i ) and the family of all ordered bases of U _i . Let $\left(e_1,e_2,...,e_{k_i}\right)$ be an ordered basis of U _i . If $T\in Aut\left(V_i\right),then\left(T\left(e_1\right),T\left(e_2\right),...,T\left(e_{k_i}\right)\right)$ is an ordered basis of U _i . If $\left(v_1,v_2,...,v_{k_i}\right)$ is an ordered basis of U _i , then there exists a unique automorphism T with $T\left(e_j\right)=v_{j}forallj\in \left\{1,2,...,k_i\right\}$. Since the number of ordered basis of U _i is equal to

$\left(q^{k_i-1}\right)\left(q^{k_i}-q\right)...\left(q^{k_i}-q^{k_i-1}\right)$

follows that $\left|Aut\left(U_i\right)\right|=\left(q^{k_i-1}\right)\left(q^{k_i}-q\right)...\left(q^{k_i}-q^{k_i-1}\right)$. Since

$Aut\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)=Aut\left(U_i\right)$

for each i, from Theorem 1

$\left|Aut\left({\mathbb{F}}_q^N,d_{\left(P,\pi \right)}\right)\right|=\left(\prod _{i=1}^m\left|Aut\left(U_i\right)\right|\right)·\left|S_{\pi }\right|.$

Since $\left|S_{\pi }\right|=\prod _{j=1}^l{m}_j!$, it follows the result.

Restricting to the Hamming case again, it follows that

$Aut\left(U_i,d_{\left(\left[n\right],{\pi }_i\right)}\right)=Aut\left(U_i\right)=Aut\left({\mathbb{F}}_q\right)\simeq {\mathbb{F}}_q^{*}$

and $S_{\pi }=S_m$, and therefore, it follows the following corollary:

Corollary 3.The automorphism group of$\left({\mathbb{F}}_q^m,d_H\right)is{\left({\mathbb{F}}_q^{*}\right)}^m⋊S_m$.

ACKNOWLEDGEMENT

The authors thank the reviewers for carefully reading the manuscript and for all the suggestions and corrections that improved the presentation of the work.

REFERENCES

• ¹
  M.M.S. Alves, L. Panek & M. Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2 (2008), 95-111.
• ²
  R. Brualdi, J.S. Graves & M. Lawrence. Codes with a poset metric. Discrete Mathematics, 147 (2008), 57-72.
• ³
  S. Cho & D. Kim. Automorphism group of crown-weight space. Eur. J. Combin., 1(27) (2006), 90-100.
• ⁴
  K. Feng, L. Xu & F.J. Hickernell. Linear error-block codes. Finite Fields and Their Applications, (12) (2006), 638-652.
• ⁵
  M. Firer, M.M. Alves, J.A. Pinheiro & L. Panek. “Poset codes: partial orders, metrics and coding theory”. Springer (2018).
• ⁶
  J. Hyun. A subgroup of the full poset-isometry group. SIAM Journal of Discrete Mathematics, 2(24) (2010), 589-599.
• ⁷
  K. Lee. The automorphism group of a linear space with the Rosenbloom-Tsfasman metric. Eur. J. Combin., (24) (2003), 607-612.
• ⁸
  H. Niederreiter. A combinatorial problem for vector spaces over finite fields. Discrete Mathematics, 96 (1991), 221-228.
• ⁹
  L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771.
• ¹⁰
  L. Panek, M. Firer, H. Kim & J. Hyun. Groups of linear isometries on poset structures. Discrete Mathematics, 308 (2008), 4116-4123.
• ¹¹
  W. Park & A. Barg. The ordered Hamming metric and ordered symmetric channels. In “IEEE Internacional Symposium on Information Theory Proceedings”. IEEE (2011), pp. 2283-2287.
• ¹²
  M.Y. Rosenbloom & M.A. Tsfasman. Codes for the m-metric. Probl. Inf. Transm., 33 (1997), 45-52.
• ¹³
  J.J. Rotman. “An introduction to the theory of groups”. Springer (1995).

• 1
  If H ₁ ,..., H _l are groups, then their direct product, denoted by $H_1\times ...\times H_l$, is the group with elements $\left(h_1,...,h_l\right),h_i\in H_i$ for each $1\le i\le l$, and with operation $\left(h_1,...,h_l\right)\left(h_1^{\text{'}},...,h_l^{\text{'}}\right)=\left(h_1{h}_1^{\text{'}},...,h_l{h}_l^{\text{'}}\right)$.
• 2
  Let G be a group with identity 1_G and let N ₁ and Q ₁ be subgroups of G. We recall that the group G is a semi-direct product of N by Q (see ¹³, p. 167), denoted by $G=N⋊Q,ifN\cong N_1,Q\cong Q_1,N_1\cap Q_1=\left\{1_G\right\},N_1{Q}_1=Gand{N}_1$ is a normal subgroup of G.
• 3
  Given groups Q and N and a homomorphism $\theta :Q\to Aut\left(N\right)$, then $N\times Q$ equipped with the operation $\left(a,x\right)\left(b,y\right):=\left(a{\theta }_x\left(b\right),xy\right)$ is a semi-direct product of N by Q (see ¹³, Theorem 7.22), denoted by $N⋊{}_{\theta }Q$. If $G=N⋊Q$ and ${\theta }_x\left(a\right)=xa{x}^{-1}$, for all $x\in Q$ and $a\in N$, then $G\cong N⋊{}_{\theta }Q$ (see ¹³, Theorem 7.23).

Publication Dates

• Publication in this collection
  03 Aug 2020
• Date of issue
  May-Aug 2020

History

• Received
  15 Dec 2017
• Accepted
  18 Feb 2020