Group of Isometries of Niederreiter-Rosenbloom-Tsfasman Block Space

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Group of Isometries of

Niederreiter-Rosenbloom-Tsfasman Block Space

LUCIANO PANEK^1* and NAYENE MICHELE PAI ˜AO PANEK²

Received on December 15, 2017 / Accepted on February 18, 2020

ABSTRACT. LetP= ({1,2, . . . ,n},≤)be a poset that is an union of disjoint chains of the same length andV=F^Nq be the space ofN-tuples over the finite fieldFq. LetVi=F^kqⁱ, with 1≤i≤n, be a family of finite-dimensional linear spaces such thatk1+k2+. . .+kn=Nand letV=V1×V2×. . .×Vnendow with the poset block metricd_(P,π)induced by the posetPand the partitionπ= (k₁,k₂, . . . ,k_n), 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(V,d_(P,π)), 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.

1 INTRODUCTION

One of the main classical problem of the coding theory is to find sets withq^kelements inF^N_q, the space ofN-tuples over the finite fieldFq, with the largest minimum distance possible. There are many possible metrics that can be defined inF^Nq, 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 integerssandm₁, . . . ,m_s, to find setsCof vectorsc_{i j}∈F^Nq, for 1≤i≤sand 1≤j≤m_i, with the largest minimum sum∑^s_i=1di, where the minimum is extended over all integersd1, . . . ,ds

with 0≤d_i≤m_ifor 1≤i≤sand∑^s_i=1d_i≥1 for which the subset{c_i,_j: 1≤i≤sand 1≤j≤d_i} is linearly dependent inF^Nq. The classical problem corresponds to the special case wheres>m andm_i=1 for all 1≤i≤s.

*Corresponding author: Luciano Panek – E-mail: luciano.panek@unioeste.br

1Centro de Engenharias e Ciˆencias Exatas, UNIOESTE, Av. Tarqu´ınio Joslin dos Santos, 1300, 85870-650, Foz do Iguac¸u, PR, Brazil – E-mail: luciano.panek@unioeste.br https://orcid.org/0000-0002-9425-6351

2Centro de Engenharias e Ciˆencias Exatas, UNIOESTE, Av. Tarqu´ınio Joslin dos Santos, 1300, CEP 85870-650, Foz do Iguac¸u, PR, Brazil – E-mail: nayene.panek@unioeste.br https://orcid.org/0000-0001-6439-7172

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Brualdi, Graves and Lawrence (see [2]) also provided in 1995 a wider situation for the Niederre- iter’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 ofF^Nq 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 Fireret 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 theNiederreiter-Rosenbloom-Tsfasman block space(orNRT block space, for short).

2 POSET BLOCK METRIC SPACE

Let[n]:={1,2, . . . ,n}be a finite set withnelements and let≤be a partial order on[n]. We call the pairP:= ([n],≤)aposetand say thatkissmaller than jifk≤ jandk6= j. Anidealin ([n],≤)is a subsetI⊆[n]that contains every element that is smaller than some of its elements, i.e., if j∈Iandk≤j, thenk∈I. Given a subsetX⊆[n], we denote byhXithe smallest ideal containingX, called theideal generated by X. An order on the finite set [n] is called alinear orderor achainif any two elements are comparable, that is, giveni,j∈[n]we have that either i≤jor j≤i. In this case,nis said to bethe lengthof the chain and the set can be labeled in such a way thati1<i2< . . . <i_n. For the simplicity of the notation, in this situation we will always assume that the orderPis defined as 1<2< . . . <n.

Letqbe a power of a prime,Fqbe the finite field ofqelements andV :=F^Nq theN-dimensional vector space ofN-tuples overFq. Letπ= (k₁,k₂, . . . ,k_n)be a partition ofN, that is,

N=k1+k2+. . .+k_n,

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withk_i>0 an integer. For each integerk_i, letV_i:=F^kqⁱ be thek_i-dimensional vector space over the finite fieldFqand define

V=V1×V₂×. . .×V_n,

called theπ-direct product decompositionofV. A vectorv∈V can be uniquely decomposed as v= (v₁,v2, . . . ,vn),

withv_i∈V_ifor each 1≤i≤n. We will call this theπ-direct product decompositionofv. Given a posetP= ([n],≤), we define theposet block weightω_(P,π)(v)(or simply the(P,π)-weight) of a vectorv= (v₁,v₂, . . . ,v_n)to be

ω_(P_,π)(v):=| hsupp(v)i |

wheresupp(v):={i∈[n]:vi6=0}is theπ-supportof the vectorvand|X|is the cardinality of the setX. The block structure is said to betrivialwhenk_i=1, for all 1≤i≤n. The(P,π)-weight induces a metricd_(P_,π)onV, that we call theposet block metric(or simply(P,π)-metric):

d_(P_,π)(u,v):=ω_(P_,π)(u−v).

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

Anisometryof(V,d_(P,π))is a bijectionT :V→V that preserves distance, that is, d_(P,π)(T(u),T(v)) =d_(P_,π)(u,v),

for allu,v∈V. The setIsom(V,d_(P,π))of all isometries of(V,d_(P,π))is a group with the nat- ural operation of composition of functions, and we call it theisometry groupof(V,d_(P,π)). An automorphismis a linear isometry.

In [9] the group of isometries of a product of Niederreiter-Rosenbloom-Tsfasman spaces is char- acterized. 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).

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3 ISOMETRIES OF LINEAR ORDERED BLOCK SPACE

LetP= ([n],≤)be the linear order 1<2< . . . <n, letπ= (k₁,k₂, . . . ,k_n)be a partition ofN and let

V =V1×V₂×. . .×V_n,

whereV_i =F^kqⁱ, for i=1,2, . . . ,n, be the π-direct product decomposition of the vector space V=F^Nq endow with the poset block metricd_(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= ([n],≤)will be the linear order 1<2< . . . <n.

We note that, givenu= (u₁, . . . ,u_n)andv= (v₁, . . . ,v_n)in the total ordered block spaceV, d_(P_,π)(u,v) =max{i:u_i6=v_i}.

For eachi∈ {1,2, . . . ,n}, let

F_i:V_i×V_i+1×. . .×V_n→V_i

be a map that is a bijection with respect to the first block spaceV_i, that is, given(v_i+1, . . . ,v_n)∈ Vi+1×. . .×V_n, the mapFe_v_i+1,...,v_n:V_i→V_idefined by

Fe_v_i+1_,...,v_n(v_i) =F_i(v_i,v_i+1, . . . ,v_n)

is a bijection. LetSq,π,ibe the set of such mapsF_i. GivenF_i∈Sq,π,i, with 1≤i≤n, we define a mapT_(F

1,F₂,...,Fn):V→Vby

T_(F₁_,F₂_,...,F_n₎(v₁, . . . ,v_n):= (F₁(v₁, . . . ,v_n),F₂(v₂, . . . ,v_n), . . . ,F_n(v_n)).

Theorem 1.Let P= ([n],≤)be the linear order1<2< . . . <n and let V=V₁×V₂×. . .×V_n be theπ-direct product decomposition of V =F^Nq endowed with the poset block metric induced by the poset P and the partitionπ. Then, the group Isom(V,d_(P_,π))of isometries of(V,d_(P,π))is the set of all maps T_(F₁_,F₂_,...,F_n₎:V→V .

Proof. Givenu= (u₁, . . . ,u_n)andv= (v₁, . . . ,v_n)∈V, letl=d_(P_,π)(u,v) =max{i:u_i6=v_i}.

Since eachFi:Vi×V_i+1×. . .×V_n→Vi is a bijection in relation to the first block spaceVi, it follows that

F_l(u_l,u_l+1, . . . ,u_n)6=F_l(v_l,v_l+1, . . . ,v_n) and

F_t(u_t,u_t+1, . . . ,u_n) =F_t(v_t,v_t+1, . . . ,v_n) for anyt>l. It follows that

d_(P_,π) T_(F₁_,...,F_n₎(u),T_(F₁_,...,F_n₎(v)

=max{i:F_i(u_i, . . . ,u_n)6=F_i(v_i, . . . ,v_n)}=l

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and hence T_(F

1,F₂,...,Fn) is distance preserving. SinceV is a finite metric space, it follows that

T_(F₁_,F₂_,...,F_n₎is also a bijection.

Now letT be an isometry ofV. Let us write

T(v₁,v₂, . . . ,v_n) = (T₁(v₁,v₂, . . . ,v_n), . . . ,T_n(v₁,v₂, . . . ,v_n)).

We prove first thatT_j(v₁,v2, . . . ,v_n) =F_j v_j,vj+1, . . . ,v_n

, that is,T_jdoes not depend on the first j−1 coordinates. In other words, we want to prove that

T_j v₁, . . . ,vj−1,v_j, . . . ,v_n

=T_j u₁, . . . ,uj−1,v_j, . . . ,v_n regardless of the values of the first j−1 coordinates. Since

d_(P_,π) u₁, . . . ,uj−1,v_j, . . . ,v_n

, v₁, . . . ,vj−1,v_j, . . . ,v_n

=max

i {i:v_i6=u_i}

≤ j−1 and sinceT is an isometry, it follows that

d_(P,π) T v₁, . . . ,vj−1,v_j, . . . ,v_n

,T u₁, . . . ,uj−1,v_j, . . . ,v_n

=

=d_(P_,π) u1, . . . ,uj−1,v_j, . . . ,v_n

, v1, . . . ,vj−1,v_j, . . . ,v_n

≤ j−1, and so,

T_j v₁, . . . ,vj−1,v_j, . . . ,v_n

=T_j u₁, . . . ,uj−1,v_j, . . . ,v_n ,

for any(v₁, . . . ,vj−1),(u₁, . . . ,uj−1)∈V1×. . .×V_j−1and(v_j, . . . ,vn)∈Vj×. . .×V_n. Thus, T(v₁,v₂, . . . ,v_n) = (F₁(v₁,v₂, . . . ,v_n),F₂(v₂, . . . ,v_n), . . . ,F_n(v_n))

and the first statement is proved. Now, we need to prove that eachFe_v_i+1_,...,v_n is a bijection, what is equivalent to prove those maps are injective. IfFe_v_i+1,...,v_n is not injective, then there arev_i6=u_i inVisuch that

Fe_v_i+1_,...,v_n(v_i) =Fe_v_i+1_,...,v_n(u_i). Consideringiminimal with this property, it follows that

i = d_(P,π)((v₁, . . . ,v_i, . . . ,v_n),(v₁, . . . ,u_i, . . . ,v_n))

= d_(P,π)(T(v₁, . . . ,v_i, . . . ,v_n),T(v₁, . . . ,u_i, . . . ,v_n))<i

contradicting the assumption thatT is an isometry of(V,d_(P,π)).

Let S_m be the symmetric group of permutations of a set withm elements andV =V₁×V₂×

. . .×V_nbe theπ-direct product decomposition ofV =F^Nq withπ= (k₁,k2, . . . ,kn). SinceV has

q^N elements we can identify the groupS_q,π,1of functionsF:V₁×V₂×. . .×V_n→V₁such that

Fe_v₂,...,v_n is a permutation ofV₁=F^kq¹, with operation

(F·G)(v):=F(G(v₁,v₂, . . . ,v_n),v2, . . . ,v_n),

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withF,G∈Sq,π,1 andv= (v₁,v2, . . . ,vn)∈V, with the direct product¹ S_qk1

q^N−k¹

. With this notations, it follows the following result.

Theorem 2.Let P= ([n],≤)be the linear order1<2< . . . <n and let V=V₁×V₂×. . .×V_nbe theπ-direct product decomposition of V=F^Nq endowed with the poset block metric induced by the poset P and the partitionπ. Ifπ= (k₁,k₂, . . . ,k_n), then the group of isometries Isom(V,d_(P_,π)) has a semi-direct product²structure given by

(S_qk1)^q^N−k¹o

. . .

(Sq^kn−1)^q^N−k¹^−k²^−...−^kn−1o(S_qkn)^q^N−k¹^−k²^−...−^kn−1⁻^kn . . .

.

Proof. LetG₍

P,bbπ)be the isometry groupIsom(bV,d₍

P,bbπ))of Vb=Vb₁×Vb₂×. . .×Vbn−1,

whereVbi=Vi+1for eachi=1,2, . . . ,n−1,Pb= ([n−1],≤)is the linear order 1<2< . . . <n−1 andπb= (k₂,k₃, . . . ,k_n). Let

H={T ∈Isom(V,d_(P_,π)):T = (F₁,Pr_V₂, . . . ,Pr_V_n)withF₁∈S_q,π,1} and

K={T ∈Isom(V,d_(P_,π)):T= (Pr_V₁,F₂, . . . ,F_n)withF_i∈S_q,π,i},

where each Pr_V_i:V_i×V_i+1×. . .×V_n→V_iis the projection map given by Pr_V_i(v_i,v_i+1, . . . ,v_n) =v_i. We claim thatIsom(V,d_(P,π))is a semi-direct product ofHbyK. Clearly,Isom(V,d_(P_,π)) =HK, because each isometry of (V,d_(P_,π)) is a composition T₁◦T₂ with T₁∈H and T₂∈K. Let L∈H∩K. Since L∈H, L(x₁,x₂, . . . ,x_n+1) = (x⁰₁,x₂, . . . ,x_n+1) and, since L is also in K, it follows that x⁰₁=x1. Hence, L =idV 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(V,d_(P,π)) =HK, it suffices to check thatT HT⁻¹⊂Hfor eachT∈K. LetL∈HandT ∈K. ThenL(x₁, . . . ,x_n) = (F₁(x₁, . . . ,x_n),x₂, . . . ,x_n)andT(x₁, . . . ,x_n) = (x₁,Te(x₂, . . . ,x_n))for some F₁∈S_q,π,1 andTe∈ Isom(bV,d₍

P,bbπ)). If(x₁, . . . ,x_n)∈V, then T◦L◦T⁻¹

(x₁, . . . ,x_n) = (T◦L)

x₁,Te⁻¹(x₂, . . . ,x_n)

=T F₁

x₁,Te⁻¹(x₂, . . . ,x_n)

,Te⁻¹(x₂, . . . ,x_n)

= F₁

x₁,Te⁻¹(x₂, . . . ,x_n)

,Te◦Te⁻¹(x₂, . . . ,x_n)

= F₁

x₁,Te⁻¹(x₂, . . . ,x_n)

,x₂, . . . ,x_n .

1IfH₁, . . . ,Hl are groups, then theirdirect product, denoted byH₁×. . .×H_l, is the group with elements(h1, . . . ,hl), hi∈Hifor each 1≤i≤l, and with operation(h1, . . . ,h_l)(h⁰₁, . . . ,h⁰_l) = (h1h⁰₁, . . . ,h_lh⁰_l).

2LetGbe a group with identity 1Gand letN₁andQ₁ be subgroups ofG. We recall that the groupGis asemi-direct productofNbyQ(see [13], p. 167), denoted byG=NoQ, ifN∼=N₁,Q∼=Q₁,N₁∩Q₁={1_G},N₁Q₁=GandN₁is a normal subgroup ofG.

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SinceF₁is a bijection with respect to the first block spaceV₁, it follows thatT LT⁻¹∈H. This shows thatHis a normal subgroup ofIsom(V,d_(P_,π))and that

Isom(V,d_(P_,π)) =HoK.

In order to simplify notation, we will denote the elements of S_qk1

q^N−k¹

by(π_X), where (π_X):= (π_X)

X∈F^N−kq ¹

.

The groupG_(b_P,b_π)acts onV =V₁×Vbby

T(x₁, . . . ,x_n) = (x₁,T(x₂, . . . ,x_n)) and(S_qk1)^q^N−k¹ acts by

(π_X) (x₁, . . . ,xn) = π_(x₂_,...,x_n₎(x₁),x2, . . . ,xn .

Both groups act as groups of isometries and both act faithfully. Therefore these actions establish isomorphisms of these groups with subgroupsHandK:H∼= (S_qk1)^q^N−k¹ andK∼=G₍

P,bπ)b. Using the aforementioned isomorphisms involvingHandK, it follows that

Isom(V,d_(P_,π)) = (S_qk1)^q^N−k¹oG_(b_P,

π)b,

which concludes the proof.

Corollary 3.Let P= ([n],≤)be the linear order1<2< . . . <n and let V=V₁×V₂×. . .×V_n be theπ-direct product decomposition of V=F^Nq endowed with the poset block metric induced by the poset P and the partitionπ. If

H={T∈Isom(V,d_(P,π)):T = (F₁,Pr_V₂, . . . ,Pr_V_n)with F₁∈S_q,π,1} and

K={T ∈Isom(V,d_(P_,π)):T= (Pr_V₁,F₂, . . . ,F_n)with F_i∈S_q,π,i}, then³

Isom(V,d_(P,π))∼=HoθK withθ:K→Aut(H)given by

θ_T(L)(x₁,x₂, . . . ,x_n) = F1

x1,Te⁻¹(x₂, . . . ,x_n)

,x2, . . . ,x_n

for all L= (F₁,Pr_V₂, . . . ,Pr_V_n)∈H, T = (Pr_V₁,F₂, . . . ,F_n)∈K and (x₁, . . . ,x_n)∈V withTe:=

(F₂, . . . ,F_n).

3Given groupsQandNand a homomorphismθ:Q→Aut(N), thenN×Qequipped with the operation(a,x)(b,y):=

(aθx(b),xy)is a semi-direct product ofNbyQ(see [13], Theorem 7.22), denoted byNoθQ. IfG=NoQandθx(a) = xax⁻¹, for allx∈Qanda∈N, thenG∼=NoθQ(see [13], Theorem 7.23).

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Corollary 4.Let P= ([n],≤)be the linear order1<2< . . . <n and let V=V₁×V₂×. . .×V_n be theπ-direct product decomposition of V =F^Nq endowed with the poset block metric induced by the poset P and the partitionπ= (k₁,k2, . . . ,kn). Then

|Isom(V,d_(P_,π))|= (q^k¹!)^q^N−k¹·(q^k²!)^q^N−k¹^−k²·. . .·(q^kⁿ⁻¹!)^q·(q^kⁿ!).

Now, if the partitionπ= (1,1, . . . ,1), it follows the following result (see [9], Corollary 3.1):

Corollary 5.Let P= ([n],≤)be the linear order1<2< . . . <n and let V =Fⁿq 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

(S_q)^qⁿ⁻¹o(. . .((S_q)^qoSq). . .).

In particular,

|Isom(V,dP)|= (q!)

qn−1

q−1 .

4 ISOMETRIES OF NRT BLOCK SPACE

In this section, we consider an order P= ([m·n],≤), that is, the union of m disjoint chains

P₁,P₂, . . . ,P_m of ordern. We identify the elements of[m·n] with the set of ordered pairs of

integers(i,j), with 1≤i≤m, 1≤j≤n, where(i,j)≤(k,l)iff i=kand j≤_Nl, where≤_Nis just the usual order onN. We denoteP_i={(i,j): 1≤j≤n}. EachP_iis a chain and those are the connected components of([m·n],≤).

Letπ= (k₁₁, . . . ,k_1n, . . . ,k_m1, . . . ,k_mn)be a partition ofN=mnand for each 1≤i≤mletπ_i=

(k_i1, . . . ,k_in). Let

V=U₁×U₂×. . .×U_m, (4.1)

where

U_i:=V_i1×V_i2×. . .×V_in

andV_{i j}=F^kq^{i j}, for all 1≤i≤m,1≤j≤n. The spaceVwith the poset metric induced by the order P= ([m·n],≤)is called the(m,n,π)-NRT block space. Note that ifn=1, thenP= ([m·1],≤) induces just the error-block metric onV, and in particular, ifπ= (1,1, . . . ,1), thenP= ([m·1],≤) induces just the Hamming metric onF^mq. Hence the induced metric from the posetP= ([m·n],≤) can be viewed as a generalization of the error-block metric.

LetV =U₁×U₂×. . .×U_m as in (4.1), called the canonical decomposition of V. Given the

canonical decompositionsu= (u₁, . . . ,u_m)andv= (v₁, . . . ,v_m)withu_i,v_i∈U_i, we have that d_(P_,π)(u,v) =

m i=1

∑

d_(P

i,π_i)(u_i,v_i),

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where d_(P_i_,π_i₎, the restriction ofd_(P_,π) toU_i, is a linear poset block metric. We note that the restriction ofd_(P,π)to eachU_iturns it into a poset space defined by a linear order, that is, eachU_i is isometric to(U_i,d_([n],π

i))with the metricd_([n],π_i₎determined by the chain 1<2< . . . <n. Let G_i,π_i_,nbe the group of isometries of(U_i,d_([n],π_i₎). The direct product∏^mi=1G_i,π_i_,nacts onV in the following manner: givenT= (T₁, . . . ,T_m)∈∏^m_i=1Gi,π_i,nandv= (v₁, . . . ,v_m)∈V,

T(v):= (T₁(v₁), . . . ,T_m(v_m)).

Lemma 1.Let(V,d_(P_,π))be the(m,n,π)-NRT block space overFqand let G_i,π_i_,nbe the group of isometries of U_i,d_([n],π_i₎

. Given T_i∈Gi,π_i,n, with1≤i≤m, the map T= (T₁, . . . ,T_m)defined by

T(v):= (T₁(v₁), . . . ,T_m(v_m)) is an isometry of(V,d_(P_,π)).

Proof. Given u,v ∈V, consider the canonical decompositions u = (u₁, . . . ,u_m) and v = (v₁, . . . ,v_m)withu_i,v_i∈U_i. Then,

d_(P_,π)(T(u),T(v)) =

m

∑

i=1

d_(P_i_,π_i₎(T_i(u_i),T_i(v_i))

=

m

∑

i=1

d_(P_i_,π_i₎(u_i,v_i)

=d_(P,π)(u,v),

LetS_mbe the permutation group of{1,2, . . . ,m}. We will call a permutationσ∈S_madmissibleif σ(i) =jimplies thatk_il=k_jl, for all 1≤l≤n. Cleary, the setS_πof all admissible permutations is a subgroup ofS_m.

Let us consider the canonical decomposition v= (v₁, . . . ,vm) of a vector v in the (m,n,π)- NRT block space V. The group S_π acts onV as a group of isometries: given σ ∈S_π and v= (v₁, . . . ,v_m)∈V, we define

T_σ(v) = (v_σ(1),v_σ(2), . . . ,v_σ(m)).

Lemma 2. Let(V,d_(P_,π)) be the(m,n,π)-NRT block space V and letσ ∈S_π. Then T_σ is an isometry of(V,d_(P_,π)).

Proof. Given u,v∈V, we consider their canonical decompositions u= (u₁, . . . ,u_m) andv= (v₁, . . . ,v_m)withu_i,v_i∈U_i. Then,

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d_(P_,π)(T_σ(u),T_σ(v)) =

m

∑

i=1

d_(P_i_,π_i₎ u_σ(i),v_σ(i)

=

m i=1

∑

d_(P_i_,π_i₎(u_i,v_i)

=d_(P,π)(u,v),

The Lemmas 1 and 2 assure that the groups∏^m_i=1G_i,π_i_,n andS_π are both isometry groups of the(m,n,π)-NRT block spaceV, and so is the groupG_(m,n,π) generated by both of them. We identify∏^m_i=1G_i,π_i_,nandS_πwith their images inG_(m,n,π)and make an abuse of notation, denoting the images inG_(m,n,π)by the same symbols. With this notation, analogous calculations as those of Theorem 2 show that

m

∏

i=1

Gi,π_i,n

!

∩Sπ={id_V} and

σ◦

m

∏

i=1

G_i,π_i_,n

!

◦σ⁻¹=

m

∏

i=1

G_i,π_i_,n,

for everyσ∈S_π. Since∏^mi=1G_i,π_i_,nis normal inG_(m,n,π)andG_(m,n,π)is generated by∏^mi=1G_i,π_i_,n andSπ, it follows that

G_(m,n,π)=

m

∏

i=1

Gi,π_i,n

!

·Sπ, 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

m

∏

i=1

Gi,π_i,n

! oSπ.

We need two more lemmas in order to prove that every isometry of the(m,n,π)-NRT block space Vis inG_(m,n,π), i.e., thatG_(m,n,π)is the group of isometries ofV. We will identify the block space U_iofV with the subspace ofVof vectorsv= (v₁, . . . ,v_m)such thatv_j=0 forj6=i.

Lemma 4.Let(V,d_(P,π))be the(m,n,π)-NRT block space and let V=U1×U₂× · · · ×U_mbe the canonical decomposition of V . If

π= (k₁₁, . . . ,k_1n, . . . ,k_m1, . . . ,k_mn)

and T:V→V is an isometry such that T(0) =0, then for each index1≤i≤m there is another index1≤ j≤m such that

T(U_i) =U_j

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and

k_il=dim(V_il) =dim(V_jl) =k_jl, for all1≤l≤n.

Proof. In the following we denote the subspaceVi1×V_i2×. . .×V_ikbyU_ik. We begin by showing that for each index 1≤i≤mthere is another index 1≤ j≤m such that T(U_i1) =U_j1 and k_i1=k_j1. Letv_i∈U_i1, withv_i6=0. Since

d_(P_,π)(T(v_i),0) =d_(P_,π)(v_i,0) =1,

it follows thatv_j=T(v_i)is a vector of(P,π)-weight 1. ThusT(v_i)∈U_j1for some index 1≤j≤ m. Ifv⁰_i∈Ui1, withv⁰_i6=v_iandv⁰_i6=0, thenT(v⁰_i) =v_kfor somev_k∈U_k1withv_k6=0, but also

d_(P_,π)(T(v_i),T(v⁰_i)) =d_(P_,π)(v_i,v⁰_i) =1.

Ifk6=j, thend_(P,π)(T(v_i),T(v⁰_i)) =d_(P,π)(v_j,v_k) =2. Hencek=jandT(U_i1)⊆U_j1. Now apply the same reasoning toT⁻¹. Ifv_i∈U_i1, withv_i6=0, andT(v_i) =v_jwithv_j∈U_j1, thenT⁻¹(v_j)∈ U_i1 and thereforeT⁻¹(U_j1)⊆U_i1. So thatU_j1⊆T(U_i1). ThereforeT(U_i1) =U_j1. Since T is bijective, it follows thatki1=kj1. For induction onk, suppose that for each sthere exists an indexlsuch that

T(U_sk) =U_lk

andks j=k_{l j} for all 1≤ j≤kand for all 1≤k≤n. We note thatUsn=Us. Without loss of generality, let us considers=1,P₁={(1,1), . . . ,(1,n)}. LetP_lbe the chain that begins at(l,1) such thatT(U₁₁) =U_l1and suppose thatU_1(k−1)is taken byT ontoU_l(k−1)withk_1j=k_{l j}for all 1≤ j≤k−1. Letv= (v₁₁, . . . ,v1k),v1i∈V1iwithv1k6=0, and letT(v) = (u₁, . . . ,um),ui∈Ui. SinceT(0) =0, it follows that

ω_(P,π)(v) =ω_(P_,π)(T(v)) =ω_(P_,π)(u₁) +. . .+ω_(P,π)(u_m).

We will use this to show that T(v) =u_l. First suppose thatu_l =0. In this case,ω_(P_,π)(v) =

∑j6=lω_(P_,π)(u_j)and therefore, ifu₁₁∈U₁₁, withu₁₁6=0 andT(u₁₁) =u_l1, then k=d_(P_,π)(u₁₁,v) =d_(P,π)(T(u₁₁),T(v)) =

∑

j6=l

ω_(P,π)(u_j) +ω_(P_,π)(u_l1) =k+1,

a contradiction. Hence u_l 6=0. Letu_l= (u_l1, . . . ,u_lt),u_li∈V_li, and suppose now there is another summandu_i6=0. Thenk=

∑

j

ω_(P_,π)(u_j)>ω_(P,π)(u_l)and thereforet<k. By the induction hypothesis, it follows thatT⁻¹(u_l)is a vector inV1(k−1)withω_(P_,π)(T⁻¹(u_l))<k.Hence

k=d_(P_,π)(T⁻¹(u_l),v) =d_(P_,π)(u_l,T(v)) =

∑

j6=l

ω_(P_,π)(u_j)<k,

again a contradiction. Hence,T(v)∈U_lk. From the induction hypothesis and from the fact thatT is a weight-preserving bijection, it follows that

T(v₁₁, . . . ,v_1k) = (u_l1, . . . ,u_lk),

wherev_1k6=0 impliesu_lk6=0. Therefore,T(U_1k) =U_lk. Sincek_1j=k_{l j}for all 1≤j≤k−1 and T is a bijection, it follows thatk_1k=k_lk. HenceT(U₁) =U_lwithk1j=k_{l j}for all 1≤j≤n.

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We recall that we defined an action of the groupS_πof the admissible permutations ofS_mon the canonical decompositionU₁×U₂× · · · ×U_mofV by

Tσ(v):= (v_σ(1),v_σ(2), . . . ,v_σ(m)) and that we defined an action of∏^m_i=1Gi,π_i,nonVby

(g₁,g₂, . . . ,g_m)(v₁,v₂, . . . ,v_m) = (g₁(v₁), . . . ,g_m(v_m)).

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 Tσ◦g, withσ in Sπand g in∏^m_i=1Gi,π_i,n.

Proof. LetT be an isometry ofV, withT(0) =0. By the Lemma 4, for each 1≤i≤mthere is aσ(i)such thatT(U_i) =U_σ(i)withk_il=k_σ(i)lfor all 1≤l≤n. SinceT is a bijection, it follows that the mapi7→σ(i)is an admissible permutation of the set{1, . . . ,m}. We defineTσ:V→V by

T_σ(v):= (v_σ(1),v_σ(2), . . . ,v_σ(m)).

Thus,T =T_σ⁻¹(T_σT) =T_σ−1(T_σT), whereσ ∈Sπ. Letg=TσT. Sinceg(U_i) = (T_σT)(U_i) = T_σ(U_σ(i)) =U_σ−1σ(i)=U_i, we have thatg|_U_i is an isometry ofU_i. Definingg_i:=g|_U_i it follows

thatg= (g₁, . . . ,g_m), and hence,g∈∏^m_i=1Gi,π_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

m

∏

i=1

Gi,π_i,n

! oSπ.

Proof. LetG_(m,n,π)be the group of isometries ofV generated by the action of∏^m_i=1G_i,π_i_,nand S_π. LetT be an isometry ofV and letv=T(0). The translationS−v(u):=u−vis clearly an isometry ofV and(S_−v◦T) (0) =S−v(v) =0 is an isometry that fixes the origin. Hence, by the previous lemma, it follows thatS−v◦T∈G_(m,n,π). Consider the canonical decomposition ofvon the chain spaces,v= (v₁, . . . ,v_m),v_i∈U_i. Since the restrictionS_v_i ofS_v= (S_v₁, . . . ,S_v_m)toU_i, withSv_i(u_i) =ui+vifor eachi, is the translation byvi, it follows that is an isometry ofUi. Thus, S_v∈∏^m_i=1G_i,π_i_,n⊂G_(m,n,π)and hence, thatT=S_v◦(S−v◦T)is inG_(m,n,π). ThusG_(m,n,π)is the isometry group ofV. By Proposition 3, it follows thatG_(m,n,π)is isomorphic to(∏^m_i=1Gi,π_i,n)oSπ.

Ifn=1 (Pis an antichain) andπ= (k₁,k₂, . . . ,k_m), where

k1=. . .=km₁ =l1, . . . ,k_m₁+...+m_l−1+1=. . .=km₁+...+m_l =lr

withl₁> . . . >l_r andm₁, . . . ,m_l positive integers such thatm₁+. . .+m_l=m, it follows that

G_i,(k_i_),1=S_q_ki, for 1≤i≤m, andS_π=S_m₁×. . .×S_m_l (S_πonly permutes those blocks with same dimensions). Therefore it follows the following result.

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Corollary 7.If P is an antichain, then

Isom(V,d_(P_,π)) =

m

∏

i=1

S_q_ki

! o

l

∏

i=1

S_m_i

! .

Whenn=1 andπ= (1,1, . . . ,1), the(P,π)-weight is the usual Hamming weight onF^mq. In this case, eachG_i,(1),1in Corollary 7 is equal toS_qand every permutation inS_m is also admissible.

Thus, we reobtain the isometry groups of Hamming space:

Corollary 8. Let d_H be the Hamming metric over F^mq. The isometry group of (F^mq,d_H) is isomorphic to S^m_qoSm.

Ifπ= (1,1, . . . ,1), then every permutation inSmis admissible. Hence, it follows the following result (see [9], Theorem 4.1):

Corollary 9.Let V=F^mn_q be the vector space endowed with the poset metric d_Pinduced by the poset P= ([mn],≤)which is union of chains P₁, . . . ,P_mof length n. Then

Isom(V,d_P) = (G_n)^moS_m, where G_n:= (S_q)^qⁿ⁻¹o(. . .((S_q)^qoS_q). . .). In particular,

|Isom(V,d_P)|= (q!)^m·

qn−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.

LetT =T_σ◦gbe a isometry. SinceT_σ is linear, it follows that the linearity ofT is a matter of whethergis linear or not. Now, ifg= (g₁,g2, . . . ,g_m)is linear, then each componentg_imust also be linear. Since eachg_iis an isometry, it follows thatg_iis in the groupAut U_i,d_([n],π_i₎

of linear isometries of(U_i,d_([n],π_i₎). Thereforeg∈∏^m_i=1Aut U_i,d_([n],π_i₎

. 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

m

∏

i=1

Aut U_i,d_([n],π_i₎

! oS_π.

Corollary 2.Let n=1andπ= (k₁,k₂, . . . ,k_m)be a partition of N. If k₁=. . .=k_m₁=l₁, . . . ,k_m₁_+...+m_l−1₊₁=. . .=k_n=l_r

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with l₁>l₂> . . . >l_r, then Aut F^Nq,d_(P,π)

=

m

∏

i=1

q^kⁱ−1 q^kⁱ−q . . .

q^kⁱ−q^kⁱ⁻¹

!

·

l

∏

j=1

m_j!

! .

Proof. Note initially that there is a bijection fromAut(U_i)and the family of all ordered bases of U_i. Let e₁,e₂, . . . ,e_k_i

be an ordered basis ofU_i. IfT∈Aut(V_i), then T(e₁),T(e₂), . . . ,T e_k_i is an ordered basis ofU_i. If v₁,v₂, . . . ,v_k_i

is an ordered basis ofU_i, then there exists a unique automorphismT withT(e_j) =vjfor all j∈ {1,2, . . . ,ki}. Since the number of ordered basis of U_iis equal to

q^kⁱ−1 q^kⁱ−q . . .

q^kⁱ−q^kⁱ⁻¹ follows that|Aut(U_i)|= q^kⁱ−1

q^kⁱ−q

. . . q^kⁱ−q^kⁱ⁻¹ . Since Aut U_i,d_([n],π_i₎

=Aut(U_i) for eachi, from Theorem 1

Aut F^Nq,d_(P,π) =

m

∏

i=1

|Aut(U_i)|

!

· |S_π|.

Since|S_π|=∏^l_j=1mj!, it follows the result.

Restricting to the Hamming case again, it follows that Aut U_i,d_([n],π_i₎

=Aut(U_i) =Aut(Fq)'F^∗q

andS_π=S_m, and therefore, it follows the following corollary:

Corollary 3.The automorphism group of F^mq,d_H is F^∗q

m

oS_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.

RESUMO. SejaP= ({1,2, . . . ,n},≤)um conjunto parcialmente ordenado dado por uma união disjunta de cadeias de mesmo comprimento eV=F^Nq o espaço vetorial dasN-uplas sobre o corpo finitoFq. SejaV =V1×V2×. . .×Vn um produto direto deV, em blocos de subespaçosVi=F^kqⁱ comk1+k2+. . .+kn=N, munido com a métrica de blocos or- denadosd_(P,π) induzida pela ordemPe pela partiçãoπ= (k1,k2, . . . ,kn). Neste trabalho descrevemos o grupo de isometrias do espaço métrico(V,d_(P,π)).

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

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