The Covering Problem in Rosenbloom-Tsfasman Spaces

We investigate the covering problem in RT spaces induced by the Rosenbloom-Tsfasman metric, extending the classical covering problem in Hamming spaces. Some connections between coverings in RT spaces and coverings in Hamming spaces are derived. Several lower and upper bounds are established for the smallest cardinality of a covering code in an RT space, generalizing results by Carnielli, Chen and Honkala, Brualdi et al., Yildiz et al. A new construction of MDS codes in RT spaces is obtained. Upper bounds are given on the basis of MDS codes, generalizing well-known results due to Stanton et al., Blokhuis and Lam, and Carnielli. Tables of lower and upper bounds are presented too.


Introduction
Rosenbloom and Tsfasman [22] introduced a new metric on linear spaces over finite fields, motivated by possible applications to interference in parallel channels of communication systems (see also [24]).Nowadays this metric is known as Rosenbloom-Tsfasman (RT) metric (or ρ metric).A concept similar to RT metric was implicity posed by Niederreiter [16], and later Brualdi et al. [4] generalized it by introducing a new family of metrics called poset metrics.
These seminal papers have shed new light on the subject.Since RT metric generalizes Hamming metric, central concepts on codes in Hamming spaces have been investigated in RT environmental: perfect code, MDS code, linear code, distribution, according to [4,11,17,13,20,22,24] for instance.In particular, Quistorff [17] presented the combinatorial foundations of the packing problem for RT spaces in a more general setting, extending the famous packing problem in Hamming spaces.
Both packing and covering problems play a central role in combinatorial coding theory due to many reasons.One remarkable feature is the wide use of links and tools from many fields of mathematics, computer science, information theory.Several applications have motivated the research, for instance: compression with distortion, decoding of errors and erasures, cellular telecommunication.See [8] for an overview on covering codes.
Since packing problem in RT spaces has been studied [17,22,24], it seems interesting to investigate the covering problem in RT space.However the literature on this subject has remained rather poor: the only result deals with the computation of a particular class of RT spaces (when the poset is a chain), according to [4,27].
In this paper we explore this gap, by studying the covering problem for an arbitrary RT space.We focus on the minimal cardinality of covering codes in RT spaces, mainly upper bounds on such covering codes.Sharp bounds are given in Theorems 8 and 13, and Corollaries 19 and 22.This paper is organized as follows.We review some concepts on posets and poset metric in Section 2. In Section 3 we introduce covering in RT spaces and present preliminary results which are needed in our investigation, including a sharp bound in [4,27].A connection between covering codes in Hamming and RT spaces is established in Section 4. Inductive relations are discussed in Section 5, generalizing several known bounds.Such relationships combined with well-known results on covering codes yield some optimal classes.In Section 6 we show a new method to construct MDS codes in RT spaces.In Section 7 bounds arising from MDS codes are obtained, extending constructions due to Stanton et al. [25], Blokhuis and Lam [3], and Carnielli [5].We conclude this work with some tables of lower and upper bounds for small instances.

Poset
Our results are based on the perspective given by Brualdi et al. [4], where a codeword is viewed as a vector and the RT metric is associated to a suitable poset metric.To present this approach, we briefly describe a few concepts and properties on partially ordered set, henceforth abbreviated poset.We refer to [9] for an overview on poset.
Let P be a finite poset whose partial order relation is denoted by .A poset is a chain when any two elements are comparable; a poset is an anti-chain when no two distinct elements are comparable.
A subset I of P is an ideal of P when the following property holds: if b ∈ I and a b, then a ∈ I.The ideal generated by a subset A of P is the ideal of smaller cardinality which the electronic journal of combinatorics 22(3) (2015), #P3.30

RT metric
We review now the RT metric as a poset metric, according to [4].Assume that the n positions of the coordinates of a vector in Z n q are in standard bijection with the elements of the set [n].A vector z = (z 1 , . . ., z n ) ∈ Z n q can be represented briefly by z = (z 1 . . .z n ).As usual, the support of a vector z is denoted by supp(z) = {i : z i = 0}.
Given x, y ∈ Z ms q and the RT poset [m × s], the RT distance between x and y is The set Z ms q endowed with the RT distance is a Rosenbloom-Tsfasman space, or simply, RT space.
The additive group Z q can be replaced by an additive group G with |G| = q in the definition of RT distance above.The RT distance can also be defined over an arbitrary set Q with |Q| = q without a structure of an additive group, see for instance [11,13,22].
Example 3. Consider the RT poset [2 × 3] presented in Example 2. Given x = (010100) and y = (100110) in Z 6  2 , the RT distance between x and y is In the RT space Z ms q , the RT ball centered at x of radius R is the set with cardinality according to [4].Since any RT ball of radius ms is the set Z ms q , Eq (1) yields 1 + ms i=1 min{m,i} j=1 q i−j (q − 1) j = q ms .
Thus one can determinate V RT q (m, s, R) by making the difference between the number of vectors in Z ms q and the number of vectors in the complementary set of a ball B RT (x, R), that is, the electronic journal of combinatorics 22(3) (2015), #P3.30

Covering codes in RT space
Covering codes in Hamming spaces are extended naturally to an arbitrary RT space, as described below.
Definition 4. Given an RT poset [m × s], let C be a subset of Z ms q .The code C is an R-covering of the RT space Z ms q when satisfies the property: for every x ∈ Z ms q , there is The number K RT q (m, s, R) denotes the smallest cardinality of an R-covering of the RT space Z ms q .
It is worth mentioning that an RT metric associated to an anti-chain (s = 1) of length m is equivalent to the well-known Hamming distance.Therefore, the number K RT q (m, s, R) generalizes the classical number K q (m, R) = K RT q (m, 1, R). Remark 5. Trivial values are easily computed for both extremal radius.The set C = Z ms q is the only 0-covering of the RT space Z ms q , thus K RT q (m, s, 0) = q ms .Given an arbitrary vector c ∈ Z ms q , the code C = {c} is an ms-covering of the RT space Z ms q .Therefore K RT q (m, s, ms) = 1.We begin our results with a general upper bound.
Proposition 6. (Trivial upper bound) For every m, s, q 2 and R such that 0 R ms, K RT q (m, s, R) q ms−R .
Proof.Proposition 1 implies that there is an ideal I of the RT poset [m × s] with R elements.The bound follows from the fact that the code is an R-covering of the RT space Z ms q .A closer look on the proof above reveals that every subset I with size R ensures an R-covering C of the Hamming space Z m q .In contrast, the restriction of I being an ideal is essential in RT spaces.We go back to Example 2 to illustrate this fact.The set I = {2, 3, 5, 6} (which is not an ideal of the RT poset [2 × 3]) induces the subspace C = {000000, 000100, 100000, 100100}, which is not a 4-covering of the RT space Z 6  2 , because d RT (c, 111111) = 6 for all c ∈ C. A simple but important lower bound on covering codes in RT spaces is stated below, extending the classical ball covering code.
Let C a subset of Z ms q .The code C is perfect (with respect to the RT poset [m × s]) provided that there exists an integer R such that the balls of radius R with centers at the codewords of C are pairwise disjoint and their union is the space Z ms q .Let C be an R-covering of the RT space Z ms q .The code C is perfect if and only if the equality holds in Proposition 7.
The codes mentioned in Remark 5 are trivial perfect codes.Particularly interesting, a classification of linear perfect codes with respect to a chain is established in [4, Theorem 2.1], whose proof yields implicitly the next result (see also [27,Theorem 2.3]).Characterizations of such codes in more general setting are presented in [1, Proposition 3.1] and [19,Theorem 7].
Theorem 8. [4,27] For any positive integer s, Proof.An alternative proof is presented here.An RT ball of radius R in the RT space Z s q has cardinality q R .Indeed, a look on the chain [1 × s] reveals that Ω 1 (i) = 1 and Ω j (i) = 0 for every 1 < i s.Eq. ( 1) produces Hence the ball covering bound yields K RT q (1, s, R) q s−R .On the other hand, a simple application of the trivial upper bound concludes the argument.
Both Propositions 6 and 7 are sharp at least for R = 0, R = ms and for a chain (when m = 1).Nevertheless, it is expected that these bounds can be far from the exact value for most instances, like in the classical covering codes.A careful analysis on the coordinates can be a tool to find improvements. Let q , the projection of x with respect to I is the vector More generally, for a non-empty subset A of Z n q , the projection of A with respect to I is the set π I (A) = {π I (a) : a ∈ A}.The set C is a 5-covering of the RT space Z 8 2 .Indeed, let I be the set formed by all maximal elements of the RT poset [4 × 2] (so I = {2, 4, 6, 8}).Given an arbitrary vector x ∈ Z 8  2 , we analyze a few cases: Case 1: If π I (x) ∈ {(0000), (0001), (0010), (0100), (1000)}, then a simple inspection reveals that x is covered by c 1 .
A closer look shows us that x is covered by c 2 .
Example 10.The optimal bound K RT 2 (2, 2, 2) = 3 holds.For the upper bound, choose C = {(0000), (0101), (1111)}.A similar argument used in Example 9 yields that C is a 2-covering of the RT space Z 4  2 , by using the ideals I = {1, 2} and J = {3, 4} of the RT poset [2 × 2].On the other hand, suppose by a contradiction that there is a 2-covering This statement is a contradiction with the fact that C is a perfect 2-covering.

A Relationship between coverings in Hamming and RT spaces
A connection between the Hamming and RT metrics is established in this section, and produces a systematic way for finding new lower bounds on K RT q (m, s, R) from known values of the classical covering codes.Proposition 11.For every q 2, K q (ms, R) K RT q (m, s, R) K RT q (1, ms, R).
Proof.Let C be an R-covering of the RT space Hence C is also an R-covering of the Hamming space Z ms q , and this proves that K q (ms, R) K RT q (m, s, R).
The upper bound on K RT q (m, s, R) is straightforward from Proposition 6 and Theorem 8.
Theorem 13.For every m (t − 1)q + 1, K RT q (m, s, ms − t) = q holds.In particular, K RT q (m, s, ms − 1) = q.Proof.The repetition code C = {(0, . . ., 0), (1, . . ., 1), . . ., (q − 1, . . ., q − 1)} is an (ms − t)-covering of the RT space Z ms q .Given an arbitrary vector x ∈ Z ms q , choose (t−1)q +1 coordinates that are maximal elements of the RT poset [m×s].The pigeonhole principle states that there is a symbol, say y, that appears at least t times in these coordinates.Take c = (y, . . ., y) ∈ C, and notice that For the lower bound, suppose by contradiction that C is an (ms − t)-covering of Z ms q whose cardinality is less than q.For each i ∈ [m × s], choose x i ∈ Z q such that c i = x i for all c ∈ C. The vector x = (x 1 , . . ., x ms ) satisfies for any c ∈ C, that is, the vector x is not covered by C. Hence K RT q (m, s, ms − t) q.Theorem 13 generalizes the results: • K q (2, 1) = q, due to Kalbfleisch and Stanton [14]; • K q (m, 1, m − t) = q for m (t − 1)q + 1, by Carnielli [5, Theorem 9], Chen and Honkala [7,Theorem 6].
The numbers K RT q (m, s, ms − 1) = K q (ms, ms − 1) = q reveal that Proposition 11 is optimal at least for a class of parameters.

Inductive Relations
Inductive relations between parameters play a central role in the literature on covering codes.In this section we focus on the behavior of inductive relations in RT spaces.All the results in the present section deal with upper bounds for covering codes in RT spaces.Sharp bounds are given in Corollaries 19 and 22.
The result below extends the very useful relation [8] for instance.
Proof.The RT poset [(m 1 + m 2 ) × s] can be viewed as a disjoint union of the RT poset [m 1 × s] and the RT poset [m 2 × s].This simple remark is the key of the proof.For i = 1, 2, let C i be an optimal R i -covering of the RT space Z m i s q .The set . For a vector (x, y) ∈ Z and the result follows.
Corollary 16.If n m and R ns, then Proof.Apply Theorem 15 with m 1 = m − n, m 2 = n, R 1 = 0, R 2 = R and use the trivial number K RT q (m − n, s, 0) = q (m−n)s .In a similar spirit, known results on covering codes can be adapted to RT spaces.Proposition 17.For n m and ns R, Proof.Let C be an (R − ns)-covering of the RT space Z (m−n)s q . Consider the ideal I = {1, . . ., ns} of the RT poset [m × s].Take the set Note that |C| = |C |.It remains to prove that C is an R-covering of the RT space Z ms q .Given an arbitrary x ∈ Z ms q , there is c ∈ C such that d RT (π The result above is imperceptible in Hamming spaces, because Proposition 21 applied to the case r = 0 is collapsed into the innocuous bound K RT q (m, s, R) K RT q (m, s, R).An exact class is derived from the previous proposition, more specifically: Proof.We firstly show that V RT 2 (2, s, 2s − 2) = 2 2s−1 .Indeed, Eq. ( 2) implies Suppose by a contradiction that there is a (2s − 2)-covering C of the RT space Z 2s 2 with |C| = 2.The ball covering bound implies , the covering C is perfect.However, given distinct x, y ∈ Z 2s 2 , note that B RT (x, 2s−2)∩B RT (y, 2s−2) = ∅.Indeed, write x = (x 1 , . . ., x s , x s+1 , . . ., x 2s ) and y = (y 1 , . . ., y s , y s+1 , . . ., y 2s ).Take z = (x 1 , . . ., x s , y s+1 , . . ., y 2s ).A simple inspection shows that d RT (x, z) s 2s − 2 and d RT (y, z) s 2s − 2, which implies z ∈ B RT (x, 2s − 2) ∩ B RT (y, 2s − 2).This contradicts the fact that C is a perfect (2s − 2)covering.
On the other hand, note that K RT 2 (2, 2, 2) = 3 from Example 10.By applying Proposition 21 when r = s − 2, we have Therefore the optimal value is proved.
Covering codes in Hamming spaces can be very useful to improve certain upper bounds on covering codes in RT spaces, as described in the next result.
Theorem 23.Let m, s, q 1 and p, k, n, r integers such that 0 < p s, 0 < k m and m = nr.Then the following inequality holds: where a = K RT q (1, s, p) = q s−p (see Theorem 8).
Proof.Let H 1 be an optimal p-covering of the RT space Z s q .Then |H 1 | = K RT q (1, s, p) = a, and a bijection identify the sets H 1 and Z a .Consider H 2 an optimal k-covering of the RT space Z m a , where the symbols are viewed as elements in H 1 .The set H 2 yields a (ks + (m − k)p)-covering of the RT space Z ms q .Indeed, given x ∈ Z ms q , write x = (y 0 , . . ., y m−1 ), where y i = (x is+1 , . . ., x (i+1)s ) ∈ Z m q for i = 0, 1, . . ., m − 1.For each y i ∈ Z s q , there exists Then there exists a vector w = (w 0 , . . ., w m−1 ) ∈ H 2 such that d RT (z, w) k (in respect to RT poset [n × r]).Thus d RT (z, w) ks (in respect to RT poset [m × s]).We claim that the vector w ∈ H 2 covers x in the RT space Z ms q .In the coordinates where w and z differs, Remark 24.Proposition 11 provides K a (nr, k) K RT a (n, r, k).Since m = nr in Theorem 23, a closer look reveals that the best choice to reduce the upper bound is n = m and r = 1.Therefore Theorem 23 can be rewritten as K RT q (m, s, ks + (m − k)p) K a (m, k).Appropriate parameters m, s, q, k and p sometimes provide more than one bound on K RT q (m, s, R).As an illustration, take the parameters m = 4, s = 4, q = 2, k = 1 and p = 2 in Theorem 23.Then K RT 2 (4, 4, 10) K 4 (4, 1) = 24.On the other hand, take k = 2 and p = 1.Theorem 23 states that K RT 2 (4, 4, 10) K 8 (4, 2) 23, which is better than 24.
On the other hand, the sharp bound . This bound is better than 12, according to Proposition 17.

A new class of MDS codes in RT spaces
The famous Singleton bound was extended to an RT linear space over a finite field in [22].The papers [11,13,17] consider this bound in a slightly more general setting, as follows.Let C be a code in the RT space Z ms q with cardinality q k , length ms, and minimum distance d = d RT (C).The parameters of C with respect to RT metric are denoted by [m, s, k, d] q .The Singleton bound states that A code C meeting the Singleton bound (Eq.4) is an MDS (maximum distance separable) code.A research problem arises naturally: the existence of such MDS codes.In this direction, Quistorff [17] presented a classification for the binary alphabet, and two classes of MDS codes in RT spaces are built in [22] (constructions using hyperderivative of a polynomial over a finite field are showed by Skriganov [24]).
[22] Given a prime power q, for any s and a such that s q and 0 a qs, there exist MDS linear codes with parameters [q, s, a + 1, qs − a] q and [q + 1, s, a + 1, (q + 1)s − a] q .
In this section we aim to establish a new class of MDS codes in RT spaces.Under some conditions we construct an MDS code with parameters [t, n, 2, tn − 1] q .
the electronic journal of combinatorics 22(3) (2015), #P3.30Theorem 27.Suppose that there is an MDS code C in the Hamming space Z n q with d H (C) = n − 1.For each t ∈ {2, . . ., n} there is an MDS code C t in the RT space Z tn q (RT poset [t × n]) with d RT (C t ) = tn − 1.
Given a positive integer q, N (q) denotes the maximum cardinality of a set of mutually orthogonal Latin squares of order q.An MDS code (in a Hamming space) with cardinality q 2 , length n, and minimum distance n−1 is equivalent to a set of n−2 mutually orthogonal Latin squares of order q (n − 2 N (q)), according to [23,Theorem 3].The following result is an immediate consequence of Theorem 27.
Corollary 28.Given positive integers q, t and n such that 3 n N (q) + 2 and 2 t n, there exists an MDS code with parameters [t, n, 2, tn − 1] q .
Since N (q) = q − 1 for a prime power q, the class of MDS codes with parameters [q + 1, 1, 2, q] q is also derived from Theorem 26.While Theorem 26 produces always linear codes, Theorem 27 generates nonlinear codes over an arbitrary alphabet.For instance, because N (12) 5 there is an MDS code of type [t, 7, 2, 7t − 1] 12 for any 2 t 7 by the previous corollary.
Theorem 27 states that C 3 is an MDS code in the RT space Z 9 2 with minimum distance 8 (with respect to the RT poset [3 × 3]).In contrast, C 3 is not an MDS code in the Hamming space Z 9 2 .
7 Upper bounds from MDS codes MDS codes have been a tool to construct covering codes in Hamming spaces.Such codes have been applied to covering codes in [3,5,6,18,25] as well as to closely related concepts [10].We refer to [8] for additional contributions on this topic.
In this section we explore several upper bounds for covering codes in RT spaces arising from MDS codes, extending a well-known result in the literature.
Theorem 30.Suppose that there is an MDS code in the RT space Z ms q with minimum distance d + 1.For every r 2, Proof.Throughout this proof, the set Z qr is regarded as the set Z qr = Z q × Z r by the bijection x+ry → (x, y).This strategy allows us to analyze information on the coordinates x and y separately.
Let H be an optimal d-covering of the RT space Z ms r , and let C be an MDS code with d RT (C) = d + 1 in the RT space Z ms q .The set G = {((c 1 , h 1 ), . . ., (c ms , h ms )) ∈ Z ms qr : (c 1 , . . ., c ms ) ∈ C, (h 1 , . . ., h ms ) ∈ H} is a d-covering of the RT space Z ms qr .Indeed, given an arbitrary z = ((x 1 , y 1 ), . . ., (x ms , y ms )) ∈ Z ms qr , take x = (x 1 , . . ., x ms ) and y = (y 1 , . . ., y ms ).Clearly, x ∈ Z ms q and y ∈ Z ms r .Since H is a d-covering of the RT space Z ms r , for y ∈ Z ms r there is h = (h 1 , . . ., h ms ) ∈ H such that d RT (y, h) d.Consider the ideal J = supp(y − h) .By Proposition 1, there is an ideal I ∈ I  d).An MDS code with minimum distance d+1 is also known as a d+1-Latin code, because its connection with Latin squares.Therefore Theorem 30 generalizes [5,Theorem 6].See also [8,Theorem 3.7.10].
The impact of Theorem 30 is discussed now.Classical MDS codes in Hamming spaces (see [23]) can be applied in our investigation.As mentioned in this section, MDS codes may give good bounds for covering codes in RT spaces.Some classes of codes with distance properties close to MDS codes have been studied, see near MDS code in [2] for instance.It would be interesting to investigate how these classes can be applied to covering codes in RT spaces.

Tables
We finally present tables of lower and upper bounds on K RT 2 (m, s, R) for "small" values of m, s and R. We do not take account the case s = 1 because this class corresponds to the classical numbers K q (m, R).The case m = 1 is omitted too, since its numbers are completely determined in Theorem 8. Updated tables by Kéri [15] are very useful for constructing some columns of our tables.
A few conventions are adopted.In the tables, the unmarked lower and upper bounds are derived from Proposition 7 or Proposition 6, respectively.When the bound is sharp, a capital letter at the right side explains the reason.When an upper bound is improved, we use a lower case letter at the right side of the upper bound, according to the keys in Table 2.
Singleton bound implies d RT (C t ) tn − 1.It remains to prove that d RT (C t ) tn−1.For this purpose, we describe properties related to codewords in C t as follows.Given a = (a 1 , . . ., a n ) ∈ C and b = (b 1 , . . ., b n ) ∈ C, note that a and b coincide at most in one coordinate, since d H (C) = n − 1.Therefore, for every i ∈ {1, . . ., n − 1} the vectors (a σ i (1) , . . ., a σ i (n) ) and (b σ i (1) , . . ., b σ i (n) d (RT ) of the RT poset [m × s] such that J ⊆ I.Because I ∈ I d (RT ) and C is an MDS code with d RT (C) = d + 1, there is a codeword c = (c 1 , . . ., c ms ) in C such that x and c coincide in all coordinates of I c .Thus supp(x − c) ⊆ I. Choose now g = ((c 1 , h 1 ), . . ., (c ms , h ms )) ∈ G.By construction, z and g coincide in all coordinates of I c .Hence d RT (z, g) = | supp(z − g) | |I| = d, and this completes the argument.Therefore K RT qr (m, s, d) |G| = |C||H| = q ms−d K RT r (m, s,

Table 1 below
displays all the parameters Ω j