Overview. Codes in the rank metric have seen a rapid increase of interest in the last decades, due to their application to network coding. However, researchers have been fascinated by these codes not only for this application, but also for their intrinsic mathematical structure. Indeed, rank-metric codes have been studied in connection with many mathematical areas, such as finite semifields, linear sets in finite geometry, tensors,
Codes in the rank metric have seen a rapid increase of interest in the last decades, due to their application to network coding. However, researchers have been fascinated by these codes not only for this application, but also for their intrinsic mathematical structure. Indeed, rank-metric codes have been studied in connection with many mathematical areas, such as finite semifields, linear sets in finite geometry, tensors,-analogues in combinatorics.
Rank-metric codes can be seen either as vector subspaces of the space of
Rank-metric codes can be seen either as vector subspaces of the space ofmatrices over a (possibly finite) field, e.g. , or as subspaces of vectors of length over a degree extension field, e.g. . In both cases, the most important parameters of a rank-metric code are given by , its dimension and its minimum rank distance, that is the minimum rank of a nonzero element of the code. These parameters are related by a very elegant inequality, which is known as the Singleton-like bound. Codes attaining this bound with equality are called maximum rank distance (MRD) codes, and they are considered to be optimal, due to their largest possible error-correction capability.
Constructions of MRD codes are known for every parameters if we consider -linear rank-metric codes. However, with the further requirement to be -linear, this is not anymore true. Indeed, constructions of -linear MRD codes are known when for any possible dimension and finite field size and they were already proposed in the seminal papers by Delsarte  and Gabidulin . When , the only known constructions concern -dimensional MRD codes with , when is even [3, 5], and the direct sum of copies of suitable MRD codes among those just described [18, Proposition 22].
As already well-known for classical coding theory involving the Hamming metric, also rank-metric codes have a geometric equivalent representation. This was initially described in several works [33, 5, 37] for some special cases, and was definitely established independently by Sheekey  and Randrianarisoa . Namely, -linear codes of length and -dimension can be equivalently represented in a geometric way as -subspaces of of -dimension . They were named as -systems and provide a sort of dual representation of rank-metric codes, which was shown to be useful in characterizing families of codes, as it happens for codes in the Hamming metric. Already Randrianarisoa in  exploited this geometric perspective to give a complete classification of -linear one-weight codes in the rank metric.
Recently, in  a new class of rank-metric codes has been introduced and investigated, the family of minimal rank-metric codes. The peculiarity of these codes is that the set of supports – which in the rank-metric setting is given by -subspaces of – form an antichain with respect to the inclusion and they are the -analogues of minimal linear codes in the Hamming metric, which have been shown to have interesting combinatorial and geometric properties and applications to secret sharing schemes, as proposed by Massey [24, 25]. Using the geometric viewpoint, minimal rank-metric codes have been shown in  to correspond to linear cutting blocking sets, which are special -systems with the following intersection property: for every , it holds . Furthermore, always in , constructions of minimal rank-metric codes were proposed using scattered subspaces in a -dimensional -space.
Our contribution. Motivated by the study of minimal rank-metric codes, we make further progresses on linear cutting blocking sets and their properties. We give a new characterization of them in terms of their evasiveness properties. In particular, in Theorem 3.3 we show that a -system of of -dimension is a linear cutting blocking set if and only if it is -evasive. This means that every -dimensional -subspace of intersects in a space of -dimension at most . In a coding theoretic language, this translates in a very elegant and concise characterization described in Theorem 3.4 which can be read as follows:
Theorem 3.4. An -linear (nondegenerate) rank-metric code in is minimal if and only if its second generalized rank weight is strictly greater than the field extension degree .
This result does not seem to have an analogue in the Hamming metric, and thus it is genuinely new.
Since every -system containing a linear cutting blocking set is itself a linear cutting blocking set, it is then natural to look only for the small ones; or, in other words, for the existence of short minimal rank-metric codes. In this direction, we provide the first answers for small parameters. We completely settle the case for every prime power and then we construct a family of linear cutting blocking sets of -dimension in , when is an odd power of
is an odd power of. Concretely, the latter is proved in Theorem 4.5, where we show that the -system,
is a linear cutting blocking set of .
As a byproduct, we also show that the codes associated to give rise to a new family of -dimensional MRD codes. This immediately follows from the geometric result given in Theorem 4.1, where is proved to be scattered. We then compare these codes with the known constructions of -linear MRD codes, realizing that not only they are new, but they are structurally new. Indeed, the codes we construct cannot be obtained as direct sum of smaller MRD codes, in contrast to all the previously known constructions. We show that this also implies that these new codes have better parameters, being their second generalized rank weight strictly larger. This is in strong contrast with what happens for the Hamming metric, in which MDS codes have all the same generalized (Hamming) weights. Thus, the new construction provides the first very concrete evidence that MRD codes can have substantially different parameters.
Outline. The paper is structured as follows. Section 2 introduces the main objects and notions that we need in the paper, giving a brief recap on rank-metric codes, -systems and their evasiveness and cutting properties. In Section 3 we explore the link between linear cutting blocking sets and evasive -systems, showing that minimal rank-metric codes can be characterized in terms of their second generalized rank weight. Section 4 contains the study of the special -system in (1), which is shown to be scattered and -evasive with the aid of some technical results. After discussing the properties of the generalized rank weights of the associated MRD codes, we conclude in Section 5, listing some open problems and new research directions.
In this section we recall the basic notions on -systems, evasive subspaces, linear cutting blocking sets and their relations with linear rank-metric codes. We first introduce the setting. Let be a prime power and let be positive integers. We denote by the finite field with element and by the extension field of degree . Furthermore, for any positive integer , is a vector space of dimension over . This will be always identified with . Also, any -dimensional -vector subspace of will be called point of .
2.1 Rank-metric codes
Rank-metric codes were originally introduced by Delsarte in the late ’s in , for a pure mathematical interest and with no applications in mind. They were reintroduced a few years later by Gabidulin in , and afterwards several applications of codes in the rank-metric were proposed, such as crisscross error correction , cryptography , distributed storage , and network coding . Mathematically speaking, one can either define them as set of matrices over a (finite) field, or as set of vectors defined over an extension field. In this work we will only consider the vector representation, and in particular we will focus on rank-metric codes which are linear over the extension field.
On the vector space we fix the metric induced by the -rank. More precisely, the rank weight is defined, for , as
If and we fix an -basis of , then there exists a matrix such that . The rank support of defined as
where denotes the -span of the rows of , and it is well-defined since it does not depend on the choice of .
The rank weight induces a metric, which is given by the rank distance, defined for as .
An (rank-metric) code is a -dimensional -subspace of endowed with the rank distance.
Before introducing the metric properties and invariants of a rank-metric code, we introduce the notion of minimality of codewords and of codes; see also .
Let be an code. A nonzero codeword is minimal if for every ,
Furthermore, if every nonzero codeword of is minimal, then is said to be a minimal rank-metric code.
Important invariants of rank-metric codes are given by their generalized rank weights. They have been first introduced and studied by Kurihara et al. in [17, 16], by Oggier and Sboui in  and by Ducoat and Kyureghyan in . They are the analogue of generalized weights in Hamming metric and are of great interest due to their combinatorial properties (see [30, 15, 12]) and their applications to network coding (see [22, 23]).
Let and , and denote by the image of under the componentwise map , that is
We say that is Galois closed if for every . Observe that this is equivalent to require that for a generator of . For instance, we can just consider to be the -Frobenius automorphism. We denote by the set of Galois closed -subspaces of .
Let be an code, and let . The -th generalized rank weight of is the integer
For brevity, when we want to underline the generalized rank weights of we will call it an code, where .
Observe that the first generalize weight coincides with the minimum rank distance of , that is
When only the minimum rank distance is relevant, we will write that is an code.
The minimum rank distance of a code measures its error correction capability, and hence it is a fundamental parameter and it is crucial to have it as large as possible. However, there are some constraints on the parameters that one has to take into account. The most important one is given by the well-known Singleton bound.
Theorem 2.4 (Singleton Bound ).
Let be an code. Then
A code is said to be maximum rank distance (MRD) if Bound (2) is met with equality.
The first construction of MRD codes was provided already by Delsarte  and Gabidulin , when . These codes are known today as Delsarte-Gabidulin codes. When , another construction was also provided more recently by Sheekey in . Additional constructions can be obtained when and via the direct sum of copies of an MRD code, e.g. a Delsarte-Gabidulin code. These constructions have been recently extended using geometric arguments in [3, 5], for odd, and , giving MRD codes. They are obtained as direct sum of MRD codes (like Delsarte-Gabidulin codes) and MRD codes. In particular, these are the only constructions of MRD codes with .
Also, the generalized rank weights are important for applications and give a measure on the security performance and the error correction capability of secure network coding; see . As for the minimum rank distance, one can derive bounds on the parameters of a code involving the generalized rank weights. In this case the bounds are more complicated.
Proposition 2.5 (Bounds ).
Let be an code. Then for each we have
Apart from the bounds, the most important properties of the generalized rank weights are given by the strict monotonicity and the Wei-type duality. The latter involves the notion of dual code. If is an code, we define to be its orthogonal complement with respect to the standard inner product on . In other words, is the code given by
and it is called the dual code of .
Let be an code and let be its dual code. Then
. (Wei-type duality)
We conclude by recalling the notions of (non)degeneracy and of equivalence of rank-metric codes. There are several equivalent ways to define nondegenerate rank-metric codes; see e.g. [1, Proposition 3.2]. Here we give the following.
An code is said to be nondegenerate if .
Also concerning equivalence of codes there are a few ways to introduce this notion. Here we only consider equivalence of codes given by linear isometries of the ambient space .
Two codes are said to be (linearly) equivalent if there exists such that ,
In this section we recall some notions and results on -systems. They were introduced by Sheekey in  and by Randrianarisoa in , as the natural geometric objects describing codes. In particular, we will focus on evasive subspaces and linear cutting blocking sets.
We first introduce the notion of weight with respect to an -subspace.
Let be an -subspace of . For an -subspace of , we define the weight of in the quantity .
We now recall the definition of -system, which was given in  for the first time.
An system is an -subspace of , with and such that . For each , the parameter is defined as
When the parameters are not relevant/known we will write that is an system. Furthermore, when none of the parameters is relevant, we will generically refer to as a -system.
Two systems are (linearly) equivalent if there exists such that .
The notation and the language used for studying -systems are inherited from the theory of rank-metric codes. This is due to their strong interconnection that was first observed in , and in , and then further developed in [1, 21].
Let denote the set of equivalence classes of systems, and let denote the set of equivalence classes of nondegenerate codes. One can define the maps
Theorem 2.11 (see ).
The maps and are well-defined and they are one the inverse of each other. Hence, they define a one-to-one correspondence between equivalence classes of codes and equivalence classes of systems.
In light of Theorem 2.11, from now on, for any system , we say that a nondegenerate code is associated to if . Similarly, an system will be said to be associated to a nondegenerate code if .
A special family of -systems is given by the so-called evasive subspaces, which generalize scattered subspaces.
Let be positive integers such that . An system is said to be an -evasive subspace (or simply -evasive) if for each -subspace of with . When , an -evasive subspace is called -scattered. Furthermore, when , a -scattered subspace will be simply called scattered.
From [6, Theorem 2.3], if is an -scattered of , then . When the equality is reached, then is said be maximum.
Evasive subspaces are a special family of evasive sets, which were introduced first by Pudlák and Rödl . These objects were then analyzed by Guruswami [13, 14], Dvir and Lovett  in connection with list decodability of codes with optimal rate and constant list-size. A mathematical theory of evasive subspace was recently developed in .
The following result highlights the relations between evasive subspaces and the parameters of the associated rank-metric codes.
Theorem 2.13 (see [21, Theorem 3.3]).
Let be an code, and let . Then, the following are equivalent.
is an -evasive subspace.
In particular, if and only if is -evasive but not -evasive.
We conclude this section by recalling another family of -systems which was recently introduced in .
An system is said to be -cutting if for every -subspace of of codimension we have . When , we simply say that is cutting (or a linear cutting blocking set).
The study of these objects was due to their connection to minimal rank-metric codes, as one can see from the following result.
Theorem 2.15 (see [1, Corollary 5.7]).
Let be an code, and let be any of the associated systems. Then, is a minimal rank-metric code if and only if is a linear cutting blocking set.
The following bound on the parameters of linear cutting blocking sets was derived in .
Proposition 2.16 (see [1, Corollary 5.10]).
Let be a cutting system, with . Then .
Moreover, always in  it was observed that linear cutting blocking sets are related with scattered subspaces when . In this case, scattered subspaces were used to construct linear cutting blocking sets, as the following result shows.
Proposition 2.17 (see [1, Theorem 6.3]).
If is a scattered system with , then is cutting.
In the next section we will further investigate the properties and the parameters of linear cutting blocking sets, and their connection with evasive subspaces. In particular, we will answer the following two natural questions: Can we generalize the Proposition 2.17 to larger values of ? Does the converse of Proposition 2.17 hold? The answers are both contained in a more general result given in Theorem 3.3.
3 Linear cutting blocking sets and evasive subspaces
In this section we derive new results on evasive subspaces and linear cutting blocking sets. In particular, our aim is to prove the main result of Theorem 3.3 which shows how these two objects are related.
3.1 Evasive subspaces
We first start with evasive subspaces, showing in which cases an -evasive -system can also be -evasive, for some special parameters .
The next result is an improvement on [2, Proposition 2.6].
Let be an -evasive system and let be such that . Then is -evasive.
By [2, Proposition 2.6], the system is -evasive and -evasive. Let be an -dimensional -subspace of such that . Let us consider the projection , where is a -dimensional -subspace with . In this way, is an -subspace of of -dimension . By the assumption on the parameters, is not scattered, hence there exists a point with . This implies that the -dimensional -subspace spanned by and has weight at least in , a contradiction. ∎
Also, we have the following result.
If is an -evasive system with , then is -evasive.
By way of contradiction suppose that there exists a -dimensional -subspace with . Then each point of has weight at least in and can be partitioned in -subspaces of dimension . Since is -evasive, the maximum number of vectors of contained in is
which is less than , a contradicion. ∎
3.2 Linear cutting blocking sets
We start by generalizing the result in Proposition 2.17 to the case of arbitrary . Furthermore, it is natural to ask whether the converse of Proposition 2.17 is true, or at least under which condition it holds. Here we give a complete answer to this question.
Let be an system. Then, is -evasive if and only if it is cutting.
() Assume by contradiction that is not cutting. Then, there exists an -hyperplane of such that , where . In particular, we have , and . However, this contradicts the hypothesis that is -evasive.
() By way of contradiction, suppose that there exists a -dimensional -subspace, say , of such that
Since is cutting, each of the hyperplanes through has vectors in U. It follows that
a contradiction. ∎
From Theorem 3.3 we can actually characterize minimal rank-metric codes in terms of their second generalized rank weight. Even though it directly follows from Theorem 3.3 and Theorem 2.13, the following result will be named as a theorem, due to its importance and conciseness.
Let be a nondegenerate code. Then, is minimal if and only if .
We know that there are some cases in which the lower bound in Proposition 2.16 is reached. Indeed, there exists a scattered -subspace of of dimension 7  and hence a linear cutting blocking set of of minimal dimension. However, in general, the lower bound is not met for every set of parameters.
If then there are no cutting systems.
If we instead restrict to the case that is equal to , we have that linear cutting blocking sets of dimension are equivalent to maximum -scattered -systems. More precisely, we have the following.
Let be a system and let be a code associated to . The following are equivalent.
is maximum -scattered.
We conclude by providing two inductive constructions of linear cutting blocking sets.
Let be a linear cutting blocking set in a hyperplane of of dimension and let be a point not in . Then is a cutting system.
Easy computations show that is a union of lines through . Let be a hyperplane of different from . Since is cutting, and there exists at least a line trough intersecting in a point defined by a vector of . The assertion follows. ∎
Let , , be a cutting system of . If is -evasive for each , with and , then is a cutting system.
Let be a hyperplane of . Then . Also, and
Since is a hyperplane of and , we get
and hence the assertion. ∎
3.3 Dual of a linear cutting blocking set
Let be a nondegenerate bilinear form on and define
where denotes the trace function of over . Then is a nondegenerate bilinear form on , when is regarded as a -dimensional vector space over . Let and be the orthogonal complement maps defined by and on the lattices of the -subspaces and -subspaces of , respectively. Recall that if is an -subspace of and is an -subspace of then is an -subspace of , , and . It easy to see that for each -subspace of . For a more detailed explanation, we refer to [36, Chapter 7]).
With the notation above, is called the dual of (with respect to ). Up to -equivalence, the dual of an -subspace of does not depend on the choice of the nondegenerate bilinear forms and on . For more details see . If is an -dimensional -subspace of and is a -dimensional -subspace of , then
Let be a scattered system. Then the following are equivalent:
is a -evasive system,
is a cutting system.
Let be a scattered system. Then the following are equivalent:
is a cutting system,
is a -evasive system.
3.4 The case
We conclude this section with a focus on a special case, namely . In particular, in this case we will determine the shortest length that a minimal rank-metric code of dimension can have with respect to the field extension .
Formally, denote by the smallest dimension of a linear cutting blocking set in , or, equivalently, the length of the shortest minimal rank-metric code of dimension , that is
From Proposition 3.7 we can deduce the recursive inequality given by
On the other hand, Proposition 2.16 can be rewritten as
and we have seen that this is not always an equality; see Corollary 3.5.
In this section we determine the exact value for every prime power . By Corollary 3.5, we can already deduce that . We will see that actually and this will be a consequence of the following more general bound on evasive -systems.
Let be such that . Let be an -evasive system. Then, for every , we have