Constant-Rank Codes and Their Connection
to Constant-Dimension Codes
Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.
Network coding, random linear network coding, error control codes, subspace codes, constant-dimension codes, constant-weight codes, rank metric codes, subspace metric, injection metric.
While random linear network coding [1, 2, 3] has proved to be a powerful tool for disseminating information in networks, it is highly susceptible to errors caused by various sources, such as noise, malicious or malfunctioning nodes, or insufficient min-cut. If received packets are linearly combined at random to deduce the transmitted message, even a single error in one erroneous packet could render the entire transmission useless. Thus, error control for random linear network coding is critical and has received growing attention recently. Error control schemes proposed for random linear network coding assume two types of transmission models: some [4, 5, 6, 7, 8] depend on and take advantage of the underlying network topology or the particular linear network coding operations performed at various network nodes; others [9, 10] assume that the transmitter and receiver have no knowledge of such channel transfer characteristics. The contrast is similar to that between coherent and noncoherent communication systems.
Error control for noncoherent random linear network coding was first considered in 111A related work  considers security issues in noncoherent random linear network coding.. Motivated by the property that random linear network coding is vector-space preserving, an operator channel that captures the essence of the noncoherent transmission model was defined in . Similar to codes defined in complex Grassmannians for noncoherent multiple-antenna channels, codes defined in Grassmannians over a finite field [12, 13] play a significant role in error control for noncoherent random linear network coding. We refer to these codes as constant-dimension codes (CDCs) henceforth. These codes can use either the subspace metric  or the injection metric . The standard advocated approach to random linear network coding (see, e.g., ) involves transmission of packet headers used to record the particular linear combination of the components of the message present in each received packet. From coding theoretic perspective, the set of subspaces generated by the standard approach may be viewed as a suboptimal CDC with minimum injection distance in a Grassmannian, because the whole Grassmannian forms a CDC with minimum injection distance . Hence, studying random linear network coding from coding theoretic perspective results in better error control schemes.
General studies of subspace codes started only recently (see, for example, [15, 16]). On the other hand, there is a steady stream of works related to codes in Grassmannians. For example, Delsarte  proved that a Grassmannian endowed with the injection distance forms an association scheme, and derived its parameters. The nonexistence of perfect codes in Grassmannians was proved in [13, 17]. In , it was shown that Steiner structures yield diameter-perfect codes in Grassmannians; properties and constructions of these structures were studied in ; in , it was shown that Steiner structures result in optimal CDCs. Related work on certain intersecting families and on byte-correcting codes can be found in  and , respectively. An application of codes in Grassmannians to linear authentication schemes was considered in . In , a Singleton bound for CDCs and a family of codes that are nearly Singleton-bound-achieving are proposed, a recursive construction of CDCs which outperform the codes in  was given in , while a class of codes with even greater cardinality was given in . Despite the asymptotic optimality of the Singleton bound and the codes proposed in , neither is optimal in finite cases: upper bounds tighter than the Singleton bound exist and can be achieved in some special cases . Thus, two research problems about CDCs remain open: the maximal cardinality of a CDC with finite dimension and minimum distance is yet to be determined, and it is not clear how to construct an optimal code that achieves the maximal cardinality.
In this paper, we introduce a novel approach to solving the two aforementioned problems. Namely, we aim to solve these problems via constant-rank codes (CRCs), which are the counterparts in rank metric codes of constant Hamming weight codes. There are several reasons for our approach. First, it is difficult to solve the two problems above directly based on CDCs since projective spaces lack a natural group structure . Also, the rank metric is very similar to the Hamming metric in many aspects, and hence familiar results from the Hamming space can be readily adapted. Furthermore, existing results for rank metric codes in the literature are more extensive than those for CDCs. Finally, the rank metric has been shown relevant to error control for both noncoherent  and coherent  random linear network coding.
Based on our approach, this paper makes two main contributions. Our first main contribution is that we establish a connection between CRCs and CDCs. Via this connection, we show that optimal CDCs correspond to optimal CRCs over matrices with sufficiently many rows. This connection converts the aforementioned open research problems about CDCs into research problems about CRCs, thereby allowing us to take advantage of existing results on rank metric codes in general to tackle such problems. Despite previous works on rank metric codes, constant-rank codes per se unfortunately have received little attention in the literature. Our second main contribution is our investigation of CRCs. In particular, we derive upper and lower bounds on the maximum cardinality of a CRC, propose explicit constructions of optimal or asymptotically optimal CRCs, and establish asymptotic bounds on the maximum rate of CRCs. Our investigation of CRCs not only is important for our construction of CDCs, but also serves as a powerful tool to study CDCs and rank metric codes.
The rest of the paper is organized as follows. Section II reviews some necessary background. In Section III, we determine the connection between optimal CRCs and optimal CDCs. In Section IV, we study the maximum cardinality of CRCs, and present our results on the asymptotic behavior of the maximum rate of a CRC.
Ii-a Rank metric codes
Error correction codes with the rank metric [26, 27, 28] have been receiving steady attention in the literature due to their applications in storage systems , public-key cryptosystems , space-time coding , and network coding [9, 10]. Below we review some important properties of rank metric codes established in [26, 27, 28].
For all , it is easily verified that is a metric over , referred to as the rank metric henceforth. Please note that the rank metric for the vector representation of rank metric codes is defined differently . Since the connection between the matrix representation of rank metric codes and CDCs is more natural, we consider the matrix representation of rank metric codes henceforth. We denote the number of matrices of rank () in as , where , , and for . The term is often referred to as a Gaussian binomial , and satisfies
for all , where . decreases with and satisfies . We denote the volume (i.e., the number of points) of the intersection of two spheres in of radii and and with rank distance between their centers as . A closed-form formula for is determined in .
A rank metric code is a subset of , and its minimum rank distance, denoted as , is simply the minimum rank distance over all possible pairs of distinct codewords. It is shown in [26, 27, 28] that the minimum rank distance of a code of cardinality in satisfies In this paper, we refer to this bound as the Singleton bound for rank metric codes and codes that attain the equality as maximum rank distance (MRD) codes. We refer to the subclass of MRD codes introduced in  as generalized Gabidulin codes. These codes are based on the vector view of rank metric codes, described as follows. The columns of a matrix can be mapped into elements of the field according to a basis of over . Hence can be mapped into the vector , and the rank of is equal to the maximum number of linearly independent coordinates of . Generalized Gabidulin codes are linear MRD codes over for . For all , , the number of codewords of rank in an linear MRD code over is denoted by , and it is known that 
We will omit the dependence of the quantities defined above on , , and when there is no ambiguity in some proofs.
Ii-B Constant-dimension codes
We refer to the set of all subspaces of with dimension as the Grassmannian of dimension and denote it as , where ; we refer to as the projective space. For , their intersection is also a subspace in , and we denote the smallest subspace containing the union of and as . Both the subspace metric [9, (3)] and injection metric [14, Def. 1] are metrics over .
The Grassmannian endowed with either the subspace metric or the injection metric forms an association scheme [9, 12]. Since for all and the injection distance provides a more natural distance spectrum, i.e., for all , we consider only the injection metric for Grassmannians and CDCs henceforth. We denote the number of subspaces in at distance from a given subspace as .
Iii Connection between constant-dimension codes and constant-rank codes
In this section, we first establish some connections between the rank metric and the injection metric. We then define constant-rank codes and we show how optimal constant-rank codes can be used to construct optimal CDCs.
Let us denote the row space and the column space of over as and , respectively. Following the convention of coding theory, a generator matrix of a subspace is any full rank matrix whose row space is the subspace . The notations introduced above are naturally extended to codes as follows: for , and .
For , , and with rank , and if and only if there exist a generator matrix of and a generator matrix of such that .
The proof of Lemma 1 is straightforward and hence omitted. We remark that is referred to as a rank factorization . We now derive a relation between the rank distance between two matrices and the injection distances between their respective row and column spaces.
For all ,
By Lemma 1, we have and , where , , , are generator matrices of , , , and , respectively. Hence and . Sylvester’s law of nullity in [37, Corollary 6.1] or in [38, 0.4.5 (c)], states that for any matrices with columns and with rows. Therefore,
Since and , we obtain the claim.
A constant-rank code (CRC) of constant rank in is a nonempty subset of such that all elements have rank . Proposition 1 below shows how a CRC leads to two CDCs with their minimum injection distance related to the minimum rank distance of the CRC.
Let be a CRC of constant rank and minimum distance in . Then and have minimum distances at least .
Proposition 1 follows directly from Theorem 1 and hence its proof is omitted. When the minimum rank distance of a CRC is greater than its constant rank, Proposition 2 below shows how the CRC leads to two CDCs with the same cardinality, and the relations between their distances can be further strengthened.
If is a CRC of constant rank and minimum rank distance () in , then and have cardinality and their minimum injection distances satisfy .
We remark that the requirement of having a minimum distance greater than the constant rank is a strong condition on the CRC. Indeed, any codeword of a linear code has rank at least equal to the minimum distance of a code. Therefore, no set of codewords of a linear code (and, in particular, a linear MRD code) satisfies this condition. Therefore, while CRCs with minimum distance no more than their constant-rank will be directly constructed from linear MRD codes in Section IV-B, designing CRCs with minimum distance greater than their constant-rank will require translates of codes instead, which are not as easy to manipulate.
Let be a CDC in and be a CDC in such that . Then there exists a CRC with constant rank and cardinality satisfying and . Furthermore, its minimum distance satisfies .
Denote the generator matrices of the component subspaces of and as and , respectively and define the code formed by the codewords for . Then and by Lemma 1 and the lower bound on follows from Theorem 1. Let and be distinct codewords in such that . By Theorem 1, we obtain . Similarly, we also obtain .
The connections between general CRCs and CDCs derived above naturally imply relations between optimal CRCs and optimal CDCs. We denote the maximum cardinality of a CRC in with constant rank and minimum rank distance as . If is a CRC in with constant rank , then its transpose code forms a CRC in with the same constant rank, minimum distance, and cardinality. Therefore , and henceforth in this paper we assume without loss of generality. We further observe that is a non-decreasing function of and , and a non-increasing function of , and that is a non-decreasing function of and a non-increasing function of .
For all , , and any ,
Combining the bounds in (4), we obtain that the cardinalities of optimal CRCs over matrices with sufficiently many rows equal the cardinalities of CDCs with related distances. Furthermore, we show that optimal CDCs can be constructed from such optimal CRCs.
For all , , and , if either or , where . Furthermore, if is an optimal CRC in with constant rank and minimum distance for or , then is an optimal CDC in with minimum distance .
First, the case where directly follows from (4) for . Second, if and , by (3) we obtain . Also, by [32, Lemma 1], we obtain for all , and hence (3) yields . Thus, when , the lower bound in (4) simplifies to . Combining with the upper bound in (4), we obtain .
The second claim immediately follows from Proposition 2. Theorem 2 implies that to determine and to construct optimal CDCs, it is sufficient to determine and to construct optimal CRCs over matrices with sufficiently many rows. We observe that this implies that remains constant for all . When , remains constant for . When , , but remains constant for , and this is shown in Section IV-B.
In comparison to existing constructions of CDCs [9, 10, 24, 15, 20, 35], our construction based on CRCs has two advantages. First and foremost, by Theorem 2, our construction leads to optimal CDCs for all parameter values. In contrast, none of previously proposed constructions lead to optimal CDCs for all parameter values. For example, the construction based on liftings of rank metric codes [9, 10] leads to suboptimal CDCs (though sometimes they may be nearly optimal). This is because CDCs of dimension based on liftings of rank metric codes have the highest possible covering radius , which implies there exists a subspace that can be added to such CDCs without decreasing the minimum distance. The CDCs constructed in similar approaches  are not optimal for the same reason. The optimality for some constructions [15, 25] are not clear. The construction based on Steiner structures  and that based on computational techniques  lead to optimal CDCs, but are applicable to special cases only. The second advantage of our construction is an additional degree of freedom, which is the number of rows of the matrices. By Theorem 2, optimal CRCs lead to optimal CDCs provided that , and hence the parameter may vary anywhere above the lower bound . On the other hand, the constructions in the literature use fixed dimensions and do not introduce any new parameter. For instance, in order to obtain a CDC in by lifting a rank metric code, the original code must be in . This additional degree of freedom is significant for code design, as it may be easier to construct optimal CRCs with larger . Thus our construction is a very promising approach to solving the two open research problems mentioned in Section I.
Iv Constant-rank codes
Having proved that optimal CRCs over matrices with sufficiently many rows lead to optimal CDCs, in this section we investigate the properties of CRCs.
We now derive bounds on the maximum cardinality of CRCs. We first remark that the bounds on derived in Section III can be used in this section. Also, since and for , we shall assume henceforth222Since the minimum distance of a code is defined using pairs of distinct codewords, the minimum distance for a code of cardinality one is defined to be zero sometimes..
We first derive the counterparts of the Gilbert and the Hamming bounds for CRCs in terms of intersections of spheres with rank radii.
For all , , and ,
The proof of the lower bound is straightforward and hence omitted. Let be a CRC with constant rank and minimum distance in . For all and , if we denote the set of matrices in with rank and distance from as , then for all . Clearly for all , and hence , which yields the upper bound.
We now derive upper bounds on . We begin by proving the counterpart in rank metric codes of a well-known bound on constant-weight codes proved by Johnson in .
Proposition 6 (Johnson bound for rank metric codes)
For all , , .
Let be an optimal CRC in with constant rank and minimum distance . For all and all , we define if and otherwise. For any , the row space of is contained in subspaces in and hence ; for all , . Summing over all possible pairs, we obtain
Hence there exists such that . By Lemma 1, all the codewords with can be expressed as , where and is a generator matrix of . Therefore, the code forms a CRC in with constant rank , minimum distance , and cardinality , and hence .
The Singleton bound for rank metric codes yields upper bounds on . For any , let denote the maximum cardinality of a code in with minimum rank distance such that all codewords have ranks belonging to . Then , where . We now determine the counterpart of the Singleton bound for CRCs.
Proposition 7 (Singleton bound for CRCs)
For all , , where .
Let be an optimal CRC in with constant rank and minimum distance , and consider the code obtained by puncturing coordinates of the codewords in . Since , the codewords of all have ranks between and . Also, since , any two codewords have distinct puncturings, and we obtain and . Hence .
For all , , .
We now derive the counterpart in rank metric codes of the Bassalygo-Elias bound  and we also tighten the bound when . For a code (), for ; we refer to ’s as the rank distribution of .
Proposition 9 (Bassalygo-Elias bound for rank metric codes)
For , , , and any code with minimum rank distance and rank distribution ’s,
Furthermore, if , then
Although the RHS of (5) and (6) can be maximized over , it is difficult to do so since is not available for most rank metric codes with the exception of linear MRD codes. Thus, we derive a bound using the rank weight distribution of linear MRD codes.
For all , , .
Applying (5) to an MRD code over , we obtain . Summing for all , we obtain since .
For all , , for and otherwise.
Iv-B Constructions of CRCs
We first give a construction of asymptotically optimal CRCs when . We assume the matrices in are mapped into vectors in according to a fixed basis of over .
For all , , .
The codewords of rank in an linear MRD code over form a CRC in with constant rank and minimum distance . Thus, .
We now prove the lower bound on . First, for , . Second, suppose . By (2), can be expressed as , where . It can be easily shown that for , and hence . Therefore,