ABSTRACT
Let be a poset that is an union of disjoint chains of the same length and be the space of N-tuples over the finite field . Let , with , be a family of finite-dimensional linear spaces such that and let endow with the poset block metric induced by the poset P and the partition , 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 , 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 um conjunto parcialmente ordenado dado por uma união disjunta de cadeias de mesmo comprimento e o espaço vetorial das N-uplas sobre o corpo finito . Seja um produto direto de V, em blocos de subespaços com , munido com a métrica de blocos ordenados induzida pela ordem P e pela partição . Neste trabalho descrevemos o grupo de isometrias do espaço métrico .
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 elements in , the space of N-tuples over the finite field , with the largest minimum distance possible. There are many possible metrics that can be defined in , but the most common ones are the Hamming and Lee metrics.
In 1987 Harald Niederreiter generalized the classical problem of coding theory (see 88 H. Niederreiter. A combinatorial problem for vector spaces over finite fields. Discrete Mathematics, 96 (1991), 221-228.): given positive integers s and , to f ind sets C of vectors , for and , with the largest minimum , where the minimum is extended over all integers with for and for which the subset is linearly dependent in . The classical problem corresponds to the special case where and for all .
Brualdi, Graves and Lawrence (see 22 R. Brualdi, J.S. Graves & M. Lawrence. Codes with a poset metric. Discrete Mathematics, 147 (2008), 57-72.) 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 ( 44 K. Feng, L. Xu & F.J. Hickernell. Linear error-block codes. Finite Fields and Their Applications, (12) (2006), 638-652., 2006) introduced the block metric, by partitioning the set of coordinate positions of 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 11 M.M.S. Alves, L. Panek & M. Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2 (2008), 95-111.) 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.55 M. Firer, M.M. Alves, J.A. Pinheiro & L. Panek. “Poset codes: partial orders, metrics and coding theory”. Springer (2018)..
A particular instance of poset block codes and spaces, with one-dimensional blocks, are the spaces introduced by Niederreiter in 1991 (see 88 H. Niederreiter. A combinatorial problem for vector spaces over finite fields. Discrete Mathematics, 96 (1991), 221-228.) and Rosenbloom and Tsfasman in 1997 (see 1212 M.Y. Rosenbloom & M.A. Tsfasman. Codes for the m-metric. Probl. Inf. Transm., 33 (1997), 45-52.), 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 1212 M.Y. Rosenbloom & M.A. Tsfasman. Codes for the m-metric. Probl. Inf. Transm., 33 (1997), 45-52.) and Park e Barg (see 1111 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.).
In 77 K. Lee. The automorphism group of a linear space with the Rosenbloom-Tsfasman metric. Eur. J. Combin., (24) (2003), 607-612., 33 S. Cho & D. Kim. Automorphism group of crown-weight space. Eur. J. Combin., 1(27) (2006), 90-100. and 1010 L. Panek, M. Firer, H. Kim & J. Hyun. Groups of linear isometries on poset structures. Discrete Mathematics, 308 (2008), 4116-4123. the groups of linear isometries of poset metrics were determined for the Rosenbloom-Tsfasman space, crown space and arbitrary poset-space respectively. In 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771. 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 66 J. Hyun. A subgroup of the full poset-isometry group. SIAM Journal of Discrete Mathematics, 2(24) (2010), 589-599. 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 11 M.M.S. Alves, L. Panek & M. Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2 (2008), 95-111..
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 be a finite set with n elements and let ≤ be a partial order on [n]. We call the pair a poset and say that k is smaller than j if and . An ideal in is a subset that contains every element that is smaller than some of its elements, i.e., if and , then . Given a subset , 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 we have that either or . In this case, n is said to be the length of the chain and the set can be labeled in such a way that . For the simplicity of the notation, in this situation we will always assume that the order P is defined as .
Let q be a power of a prime, be the finite field of q elements and the N-dimensional vector space of N-tuples over . Let be a partition of N, that is,
with an integer. For each integer k i , let be the k i -dimensional vector space over the finite field and define
called the π-direct product decomposition of V. A vector can be uniquely decomposed as
with for each . We will call this the π-direct product decomposition of v. Given a poset , we define the poset block weight (or simply the (P, π)-weight) of a vector to be
where 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 , for all . The (P, π)-weight induces a metric d (P,π) on V , that we call the poset block metric (or simply (P, π)-metric):
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 that preserves distance, that is,
for all . 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 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771. the group of isometries of a product of Niederreiter-Rosenbloom-Tsfasman spaces is characterized. In 66 J. Hyun. A subgroup of the full poset-isometry group. SIAM Journal of Discrete Mathematics, 2(24) (2010), 589-599. 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 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771.. 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 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771.. In this work, we prove some results on isometries, also anologous to those of 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771., 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 be the linear order , let be a partition of N and let
where , for , be the π-direct product decomposition of the vector space 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, will be the linear order .
We note that, given and in the total ordered block space V,
For each , let
be a map that is a bijection with respect to the first block space V i , that is, given , the map defined by
is a bijection. Let S q,π,i be the set of such maps F i . Given , with 1 ≤ i ≤ n, we define a map by
Theorem 1.Letbe the linear orderand letbe the π-direct product decomposition ofendowed 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.
Proof. Given and , let . Since each is a bijection in relation to the first block space V i , it follows that
and
for any . It follows that
and hence is distance preserving. Since V is a finite metric space, it follows that is also a bijection.
Now let T be an isometry of V . Let us write
We prove first that , that is, T j does not depend on the first coordinates. In other words, we want to prove that
regardless of the values of the first coordinates. Since
and since T is an isometry, it follows that
and so,
for any and . Thus,
and the first statement is proved. Now, we need to prove that each is a bijection, what is equivalent to prove those maps are injective. If is not injective, then there are in V i such that
Considering i minimal with this property, it follows that
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 be the π-direct product decomposition of with . Since V has q N elements we can identify the group S q,π,1 of functions such that is a permutation of , with operation
with and , with the direct product1 1 If H 1 ,..., H l are groups, then their direct product, denoted by H1 × ... × Hl, is the group with elements h1, ..., hl, hi ∈ Hi for each 1 ≤ i ≤ l, and with operation h1, ..., hlh1', ..., hl' = h1h1', ..., hlhl'. . With this notations, it follows the following result.
Theorem 2.Letbe the linear orderand letbe the π-direct product decomposition ofendowed with the poset block metric induced by the poset P and the partition π. If, then the group of isometries Isom(V, d(P,π) ) has a semi-direct product2 2 Let G be a group with identity 1G 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, if N ≅ N1, Q ≅ Q1, N1 ∩ Q1 = 1G, N1Q1 = G and N1 is a normal subgroup of G. structure given by
Proof. Let be the isometry group of
where for each is the linear order and . Let
and
where each is the projection map given by . We claim that Isom(V, d (P,π) ) is a semi-direct product of H by K. Clearly, , because each isometry of (V, d (P,π) ) is a composition with and . Let . Since and, since L is also in K, it follows that . Hence, 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 , it suffices to check that for each . Let and . Then and for some and . If , then
Since F 1 is a bijection with respect to the first block space V 1, it follows that TLT . This shows that H is a normal subgroup of Isom(V, d (P,π) ) and that
In order to simplify notation, we will denote the elements of by(πX ), where
The group acts on by
and acts by
Both groups act as groups of isometries and both act faithfully. Therefore these actions establish isomorphisms of these groups with subgroups . Using the aforementioned isomorphisms involving H and K, it follows that
which concludes the proof.
Corollary 3. Let be the linear order and let be the π-direct product decomposition of endowed with the poset block metric induced by the poset P and the partition π. If
and
then 3 3 Given groups Q and N and a homomorphism θ : Q → AutN, then N × Q equipped with the operation a, xb, y : = aθxb, xy is a semi-direct product of N by Q (see 13, Theorem 7.22), denoted by N ⋊ θQ. If G = N ⋊ Q and θxa = xax-1, for all x ∈ Q and a ∈ N, then G ≅ N ⋊ θQ (see 13, Theorem 7.23).
with given by
for all .
Corollary 4. Let be the linear order and let be the π-direct product decomposition of endowed with the poset block metric induced by the poset P and the partition . Then
Now, if the partition , it follows the following result (see 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771., Corollary 3.1):
Corollary 5.Letbe the linear orderand letbe the vector space endowed with the poset metric induced by the poset P. Then the group of isometries Isom(V, dP ) is a semi-direct product
In particular,
4 ISOMETRIES OF NRT BLOCK SPACE
In this section, we consider an order , that is, the union of m disjoint chains P 1 , P 2 , ..., P m of order n. We identify the elements of with the set of ordered pairs of integers (i, j), with , where , where is just the usual order on ℕ. We denote . Each P i is a chain and those are the connected components of .
Let be a partition of and for each let . Let
where
and , for all . The space V with the poset metric induced by the order is called the (m, n, π)-NRT block space. Note that if , then induces just the error-block metric on V, and in particular, if , then induces just the Hamming metric on . Hence the induced metric from the poset can be viewed as a generalization of the error-block metric.
Let as in (4.1), called the canonical decomposition of V. Given the canonical decompositions , we have that
where , 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 with the metric determined by the chain . Let be the group of isometries , of . The direct product acts on V in the following manner: given ,
Lemma 1.Let (V, d (P,π) ) be the (m, n, π)-NRT block space overand letbe the group of isometries of. Given, with, the mapdefined by
is an isometry of (V, d (P,π) ).
Proof. Given , consider the canonical decompositions . Then,
which concludes the proof.
Let S m be the permutation group of {1, 2, ..., m}. We will call a permutation admissible if implies that . Cleary, the set Sπ of all admissible permutations is a subgroup of S m .
Let us consider the canonical decomposition 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 , we define
Lemma 2.Let (V, d (P,π) ) be the (m, n, π)-NRT block space V and let. Then T σ is an isometry of (V, d (P,π) ).
Proof. Given , we consider their canonical decompositions . Then,
which concludes the proof.
The Lemmas 1 and 2 assure that the groups 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 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
And
for every . Since is normal in G (m;n;π) and G (m,n,π) is generated by and S π , it follows that
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
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 .
Lemma 4.Let (V, d (P,π) ) be the (m, n, π)-NRT block space and letbe the canonical decomposition of V. If
and is an isometry such that , then for each index there is another index such that
And
for all.
Proof. In the following we denote the subspace . We begin by showing that for each index there is another index such that . Since
it follows that is a vector of (P, π)-weight 1. Thus for some index , but also
If . Hence . Now apply the same reasoning to T −1. If and therefore . So that . Therefore . Since T is bijective, it follows that . For induction on k, suppose that for each s there exists an index l such that
and for all and for all . We note that Usn = Us. Without loss of generality, let us consider . Let P l be the chain that begins at (l, 1) such that and suppose that , it follows that
We will use this to show that . First suppose that . In this case, and therefore, if , then
a contradiction. Hence , and suppose now there is another summand . Then and therefore . By the induction hypothesis, it follows that T −1(u l ) is a vector in with . Hence
again a contradiction. Hence, . From the induction hypothesis and from the fact that T is a weight-preserving bijection, it follows that
where implies . Therefore, is a bijection, it follows that . Hence .
We recall that we defined an action of the group S π of the admissible permutations of S m on the canonical decomposition of V by
and that we defined an action of on V by
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, with σ in S π and g in.
Proof. Let T be an isometry of V , with . By the Lemma 4, for each there is a σ (i) such that . Since T is a bijection, it follows that the map is an admissible permutation of the set {1,..., m}. We define by
Thus, , we have that is an isometry of U i . Defining it follows that , and hence, .
Theorem 6.Let (V, d (P,π) ) be the (m, n, π)-NRT block space. The group of isometries of V is isomorphic to
Proof. Let G (m,n,π) be the group of isometries of V generated by the action of and S π . Let T be an isometry of V and let . The translation is clearly an isometry of V and is an isometry that fixes the origin. Hence, by the previous lemma, it follows that . Consider the canonical decomposition of v on the chain spaces, . Since the restriction for each i, is the translation by v i , it follows that is an isometry of U i . Thus, and hence, that 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 .
If (P is an antichain) and , where
with positive integers such that , it follows that (S π only permutes those blocks with same dimensions). Therefore it follows the following result.
Corollary 7. If P is an antichain, then
When , the (P, π)-weight is the usual Hamming weight on . 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 . The isometry group of is isomorphic to .
If , then every permutation in S m is admissible. Hence, it follows the following result (see 99 L. Panek, M. Firer & M.M.S. Alves. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Mathematics, 309 (2009), 763-771., Theorem 4.1):
Corollary 9. Let be the vector space endowed with the poset metric d P induced by the poset which is union of chains P 1 , ..., P m of length n. Then
where . In particular,
5 AUTOMORPHISMS
The group of automorphisms of (V, d (P,π) ) is easily deduced from the Lemma 5 and Theorem 6. Let 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 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 of linear isometries of . Therefore . 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
Corollary 2. Let be a partition of N. If
with , then
Proof. Note initially that there is a bijection from Aut (U i ) and the family of all ordered bases of U i . Let be an ordered basis of U i . If is an ordered basis of U i . If is an ordered basis of U i , then there exists a unique automorphism T with . Since the number of ordered basis of U i is equal to
follows that . Since
for each i, from Theorem 1
Since , it follows the result.
Restricting to the Hamming case again, it follows that
and , and therefore, it follows the following corollary:
Corollary 3.The automorphism group of.
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
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1M.M.S. Alves, L. Panek & M. Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2 (2008), 95-111.
-
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-
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-
4K. Feng, L. Xu & F.J. Hickernell. Linear error-block codes. Finite Fields and Their Applications, (12) (2006), 638-652.
-
5M. Firer, M.M. Alves, J.A. Pinheiro & L. Panek. “Poset codes: partial orders, metrics and coding theory”. Springer (2018).
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7K. Lee. The automorphism group of a linear space with the Rosenbloom-Tsfasman metric. Eur. J. Combin., (24) (2003), 607-612.
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10L. Panek, M. Firer, H. Kim & J. Hyun. Groups of linear isometries on poset structures. Discrete Mathematics, 308 (2008), 4116-4123.
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11W. Park & A. Barg. The ordered Hamming metric and ordered symmetric channels. In “IEEE Internacional Symposium on Information Theory Proceedings”. IEEE (2011), pp. 2283-2287.
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12M.Y. Rosenbloom & M.A. Tsfasman. Codes for the m-metric. Probl. Inf. Transm., 33 (1997), 45-52.
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13J.J. Rotman. “An introduction to the theory of groups”. Springer (1995).
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1
If H 1 ,..., H l are groups, then their direct product, denoted by , is the group with elements for each , and with operation .
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2
Let G be a group with identity 1G 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 1313 J.J. Rotman. “An introduction to the theory of groups”. Springer (1995)., p. 167), denoted by is a normal subgroup of G.
-
3
Given groups Q and N and a homomorphism , then equipped with the operation is a semi-direct product of N by Q (see 1313 J.J. Rotman. “An introduction to the theory of groups”. Springer (1995)., Theorem 7.22), denoted by . If and , for all and , then (see 1313 J.J. Rotman. “An introduction to the theory of groups”. Springer (1995)., Theorem 7.23).
Publication Dates
-
Publication in this collection
03 Aug 2020 -
Date of issue
May-Aug 2020
History
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Received
15 Dec 2017 -
Accepted
18 Feb 2020