Grassmannian Codes with New Distance Measures for Network Coding

TUVI ETZION, Fellow, IEEE HUI ZHANG

Abstract—Grassmannian codes are known to be useful in network has N receivers, each one demands all the h error-correction for random network coding. Recently, they messages of the source to be transmitted in one round of were used to prove that vector network codes outperform a network use. An up-to-date survey on network coding scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which for multicast networks can be found for example in [21]. were used for this purpose are generalized combination Kotter¨ and Medard´ [29] provided an algebraic formulation networks. In both the scalar and the vector network coding for the network coding problem: for a given network, find solutions, the subspace distance is used as the distance coding coefficients for each edge, whose starting vertex measure for the codes which solve the network coding has in-degree greater than one. These coding coefficients problem in the generalized combination networks. In this work we show that the subspace distance can be replaced are multiplied by the symbols received at the starting node with two other possible distance measures which generalize of the edge and these products are added together. These the subspace distance. These two distance measures are coefficients should be chosen in a way that each receiver shown to be equivalent under an orthogonal transformation. can recover the h messages from its received symbols on It is proved that the Grassmannian codes with the new its incoming edges. This sequence of coding coefficients distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the at each such edge is called the local coding vector and strength of the design. Furthermore, optimal Grassmannian the edge is called a coding point. Such an assignment codes with the new distance measures have minimal re- of coding coefficients for all such edges in the network quirements for network coding solutions of some generalized is called a solution for the network and the network is combination networks. The coding problems related to these called solvable. It is easy to verify that the information two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts on each edge is a linear combination of the h messages. it is proved that codes in the Hamming scheme form a The vector of length h of these coefficients of this linear subfamily of the Grassmannian codes. combination is called the global coding vector. From the global coding vectors and the symbols on its incoming edges, the receiver should recover the h messages, by Index Terms—Distance measures, generalized combination networks, Grassmannian codes, network coding. solving a set of h linearly independent equations. The coding coefficients defined in this way are scalars and the solution is a scalar linear solution. Ebrahimi and Fragouli [7] have extended this algebraic approach to I.INTRODUCTION vector network coding. In the setting of vector network ETWORK coding has been attracting increasing coding, the messages of the source are vectors of length ` N attention in the last fifteen years. The seminal work over Fq and the coding coefficients are ` × ` matrices of Ahlswede, Cai, Li, and Yeung [1] and Li, Yeung, over Fq. A set of matrices, which have the role of the and Cai [24] introduced the basic concepts of network coefficients of these vector messages, such that all the arXiv:1801.02329v7 [cs.IT] 8 Feb 2019 coding and how network coding outperforms the well- receivers can recover their requested information, is called known routing. The class of networks which are mainly a vector solution. Also in the setting of vector network studied is the class of multicast networks and these are coding we distinguish between the local coding vectors also the target of this work. A multicast network is a and the global coding vectors. There is a third type of directed acyclic graph with one source. The source has network coding solution, a scalar nonlinear network code. Again, in each coding point there is a function of the h messages, which are scalars over a finite field Fq. The symbols received at the starting node of the coding point. Department of Computer Science, Technion, Haifa 3200003, Israel, This function can be linear or nonlinear. There is clearly a e-mail: [email protected]. hierarchy, where a scalar linear solution can be translated Nanyang Technological University, Singapore, e-mail: [email protected]. Part of the research was performed to a vector solution, and a vector solution can be translated while the author was with the Department of Computer Science, to a scalar nonlinear solution. Technion, Haifa 3200003, supported in part by a fellowship of the Israel Council of Higher Education. The alphabet size of the solution is an important Parts of this work have been presented at the IEEE International Symposium on Information Theory 2018, Vail, Colorado, U.S.A., June parameter that directly influences the complexity of the 2018. calculations at the network nodes and as a consequence the 2 performance of the network. A comparison between the measures on Grassmannian codes which generalize the required alphabet size for a scalar linear solution, a vector Grassmannian distance. We discuss the maximum sizes solution, and a scalar nonlinear solution, of the same of Grassmannian codes with the new distance measures multicast network is an important problem. It was proved and analyse these bounds from a few different point of in [18], [19] that there are multicast networks on which view. We explore the connection between these codes a vector network coding solution with vectors of length and related generalized combination networks. Our ex- ` over Fq outperforms any scalar linear network coding position will derive some interesting properties of these solution, i.e. the scalar solution requires an alphabet of codes with respect to the traditional Grassmannian codes ` size qs, where qs > q . The proof used a family of and some subspace designs. We will show, using a few networks called the generalized combination networks, different approaches, that codes in the Hamming space where the combination networks were defined and used form a subfamily of the Grassmannian codes. Some other in [34]. interesting connections to subspace designs and codes in Kotter¨ and Kschischang [30] introduced a framework the Hamming scheme will be also explored. for error-correction in random network coding. They have The Grassmannian codes (constant dimension codes) shown that for this purpose the codewords (messages) are are the q-analog of the constant weight codes, where taken as subspaces over a finite field Fq. For this purpose q-analogs replace concepts of subsets by concepts of sub- they have defined the subspace distance. This approach spaces when problems on sets are transferred to problems was mainly applied on subspaces of the same dimension. on subspaces over the finite field Fq. For example, the size For given positive integers n and k, 0 ≤ k ≤ n, the n of a set is replaced by the dimension of a subspace, the Grassmannian Gq(n, k) is the set of all subspaces of Fq binomial coefficients are replaced by the Gaussian coef- whose dimension is k. It is well known that ficients, etc. The Grassmann space is the q-analog of the n n−1 n−k+1 n def (q − 1)(q − 1) ··· (q − 1) Johnson space and the subspace distance is the q-analog = |Gq(n, k)| = k k−1 k q (q − 1)(q − 1) ··· (q − 1) of the Hamming distance. The new distance measures are q-analogs of related distances in the Johnson space. n where k q is the q-binomial coefficient (known also as The Johnson scheme J(n, w) consists of all w-subsets the q−ary Gaussian coefficient [45, pp. 325-332]. A code of an n-set (equivalent to binary words on length n and ∈ G (n, k) is called a Grassmannian code or a constant C q weight w). The Johnson distance dJ (x, y) between two dimension code. For two subspaces X,Y ∈ Gq(n, k) w-subsets x and y is half of the related Hamming distance, the subspace distance is reduced to the Grassmannian i.e., dJ (x, y) , |x \ y|. distance defined by The rest of this paper is organized as follows. In def Section II we present the combination network and its gen- dG(X,Y ) = k − dim(X ∩ Y ) . eralization which was defined in [18], [19]. We discuss the Most of the research on Grassmannian codes motivated family of codes which provide network coding solutions by [30] was in two directions – finding the largest codes for these networks. We will make a brief comparison be- with prescribed minimum Grassmannian distance and tween the related scalar coding solutions and vector cod- looking for designs based on subspaces. To this end, the ing solutions. In Section III we further consider this family quantity Aq(n, 2d, k) was defined as the maximum size of of codes, define two dual distance measures on these a code in Gq(n, k) with minimum Grassmannian distance codes, and show how these codes and the new distance d. There has been extensive work on Grassmannian codes measures defined on them generalize the conventional in the last ten years, e.g. [13], [14], [15], [16], [36] and Grassmannian codes with the Grassmannian distance. We references therein. A related concept is a subspace design show the connection of these codes to subspace designs. or a block design t-(n, k, λ)q which is a collection S of We prove that for each such code of the largest size, k-subspaces from Gq(n, k) (called blocks) such that each over Fq, there exists a generalized combination network subspace of Gq(n, t) is contained in exactly λ blocks of S, which is solved only by this code (or another code with where t is called the strength of the design. In particular at least the same number of codewords and possibly more if λ = 1 this subspace design is called a q-Steiner system relaxed parameters). Finally, we discuss which subfamily and is denoted by Sq(t, k, n). Note, that such a q-Steiner of these codes is useful for vector network coding. In system is a Grassmannian code in Gq(n, k) with minimum Section IV basic results on the upper bounds on sizes of Grassmannian distance k − t + 1. Such subspace designs these codes are presented. At this point we note that some were considered for example in [3], [4], [5], [9], [11], of the codes can have repeated codewords. Section IV [20], [27], [28], [32], [33], [38], [39], [40], [41], [42]. concentrates first on the case where there are no repeated The goal of this work is to show that there is a codewords in the code. It continues with upper bounds on tight connection between optimal Grassmannian codes sizes of codes where repeated codewords are considered. and network coding solutions for the generalized com- In Section V we analyse the strength of our bounds and bination networks. We will define two new dual distance the implementation of the codes on specific generalized 3 combination networks. In Section VI we discuss a few of α codewords (k-subspaces) spans a subspace whose approaches to show how codes in the Hamming space dimension is at least h − . For this solution the coding form a subfamily of codes in the Grassmann space. coefficients on a set of s = αk + links (α sets with k- We also discuss other connections of the newly defined parallel links and one set of -direct links) are computed distance measures and codes in the Hamming scheme. as follows. The k links from the source to the i-th node in Section VII provides a conclusion and some directions the middle layer are associated with the i-th codeword of for future research. C. From this codeword any k linearly independent vectors are chosen. The h elements of Fq in each such vector of II.GENERALIZED COMBINATION NETWORKS length h are the h coding coefficients on a related edge from the source to the i-th node in the middle layer. The In this section we will define the generalized combina- middle layer nodes transmit on its outgoing links the exact tion network which is a generalization of the combination information it receives from the source. Given a receiver network [34]. This network defined in [18], [19] was used R, the αk vectors on the edges entering R (formed from to prove that vector network coding outperforms scalar coding coefficients on the edges) span a subspace X linear network coding, in multicast networks, with respect whose dimension is at least h−. On the edges entering to the alphabet size, using Grassmannian codes. R from the source, there are coding coefficients whose The generalized combination network is called the h related vectors complete X to Fq . These coding vectors -direct links k-parallel links Nh,r,s network, in short the form the scalar linear solution for the network. A different (, k)-Nh,r,s network. The network has three layers. In the analysis for the combination network, i.e., = 0 and first layer there is a source with h messages. In the second k = 1 was given by Riis and Ahlswede [34]. For this we layer there are r nodes. The source has k parallel links to need to define an (n, M, d) code over Fq to be a subset of each node in the middle (second) layer. From any α = s− k M words of length n over Fq with minimum Hamming nodes in the middle layer, there are links to one receiver distance d. The functions on the edges related to the scalar r in the third layer, i.e. there are α receivers in the third nonlinear solution implied by the following theorem will layer. From each one of these α nodes there are k parallel be explained (in the general context of the generalized links to the related receiver in the third layer. Additionally, combination networks) in Section VI-A, where they are from the source there are direct parallel links to each relevant. one of the r receivers in the third layer. Therefore, each α Theorem 2 (Proposition 3 [34]). The (0, 1)-N net- receiver has s = αk + incoming links. The (0, 1)-Nh,r,s h,r,s network is the combination network defined in [34]. This work is solvable over Fq if and only if there exists an h network has neither parallel links (between nodes) nor (r, q , r − s + 1) code over Fq. direct links (from the source to the receiver). We will For a vector solution of the (, k)-Nh,r,αk+ network, assume some relations between the parameters h, α, , the h messages are vectors of length ` over Fq. Therefore, and k such that the resulting network does not have a the total number of entries in the messages is h` and the h trivial solution or no solution. messages span an (h`)-space. Hence, each receiver should obtain the space h` from its incoming edges. The source Theorem 1 (Theorem 8 [18]). The (, k)-Nh,r,αk+ net- Fq work has a trivial solution if k + ≥ h, and it has no can send each receiver an (`)-subspace. This implies that solution if αk + < h. Otherwise, the network has a each receiver should obtain a subspace whose dimension nontrivial solution. is at least (h − )` from the related α middle layer nodes. On each set of k parallel links a node can send a (k`)- Which network codes over solve the networks from h` Fq subspace of Fq . Hence, similarly to the scalar linear this family of networks? The answer to this natural coding solution, a vector solution for the network exists if question is quite simple. Since each receiver has direct and only if there exists a Grassmannian code with r (k`)- links from the source, it follows that the source can send h` subspaces of Fq , such that each subset of α codewords h any required -subspace of Fq to the receiver. Hence, ((k`)-subspaces) spans a subspace whose dimension is at the receiver must be able to obtain an (h − )-subspace least (h − )`. h of Fq from the α middle layer nodes connected to it. Each one of these α nodes in the middle layer can send to III.COVERING/MULTIPLE GRASSMANNIAN CODES h the receiver a k-subspace of Fq (we can assume w.l.o.g. In this section we provide the formal definition for that each node of the middle layer holds a k-subspace the codes required to solve the generalized combination h of Fq .). Hence, a scalar linear solution for the network networks. We define two distance measures on these exists if and only if the linear span of the k-subspaces of codes and prove that Grassmannian codes and subspace each subset of α nodes in the middle layer is at least of designs, are subfamilies of the related family of codes. dimension h − . Hence, a scalar linear solution for the We present some basic properties of these codes and network exists if and only if there exists a Grassmannian their connection to the network coding solutions for the h code C with r k-subspaces of Fq , such that each subset generalized combination networks. 4

c An α-(n, k, δ)q covering Grassmannian code (code in short) C is a subset of Gq(n, k) such that each subset of α codewords of C spans a subspace whose dimension is at n least δ+k in Fq . The following theorem is easily veriﬁed.

Theorem 3. C ∈ Gq(n, k) has minimum Grassmannian distance δ if and only if C is a c 2-(n, k, δ)q code. Proof. A code ∈ G (n, k) has minimum Grassmannian C q Fig. 1: the (1, 1)-N network, each one of the r distance δ if for each two distinct codewords X,Y ∈ we 3,r,4 3 C receivers obtains its information from three links of three have δ ≤ dG(X,Y ) = k − dim(X ∩ Y ). Since dim(X ∩ Y ) = dim X +dim Y −dim(X ∪Y ) = 2k −dim(X ∪Y ) distinct nodes in the middle layer and one direct link from the source, where x , x , and x are the three messages. it follows that 1 2 3 dim(X ∪ Y ) = 2k − dim(X ∩ Y ) ≥ k + δ , subspace of 3 is sent. Hence, the largest r for which c Fq and hence C is a 2-(n, k, δ)q code. the network is solvable over is 23 . The solution c Fq 1 q On the other hand if C is a 2-(n, k, δ)q code then for c consists of a 3-(3, 1, 1)q code which contains all one- each two distinct codewords X,Y ∈ C, 3 dimensional subspaces of Fq, each one is contained twice ˜ 3 k + δ ≤ dim(X ∪ Y ) = dim X + dim Y − dim(X ∩ Y ) in the code. This implies that Bq(3, 1, 1; 3) = 2 1 q, while B (3, 1, 1; 3) = 3 . = 2k − dim(X ∩ Y ) , q 1 q which implies that We continue in this section with the assumption that codewords can be repeated, since the network coding δ ≤ k − dim(X ∩ Y ) = dG(X,Y ) , solution can use repeated codewords. In Section IV we will discuss bounds only for codes with non-repeated i.e., C has minimum Grassmannian distance δ. codewords, and the difference when we allow repeated Theorem 3 implies that the Grassmannian codes with codewords. From our discussion it can be readily ver- c the Grassmannian distance form a subfamily of the ified that if C is an α-(n, k, δ)q code which attains α-(n, k, δ)c codes. It also implies that it is natural to define ˜ q Bq(n, k, δ; α) then C solves the (, k)-Nn,r,αk+ network, δ α Grassmannian distance the quantity as the minimum - where ≥ n − δ − k and r ≤ B˜q(n, k, δ; α). Such a code of the code and to define the quantity k+δ as the minimum implies the largest r possible for such a network, given α-Grassmannian covering, where the 2-Grassmannian dis- fixed n, k, α, and . The largest r means the maximum tance is just the Grassmannian distance. In other words, number of nodes that can be taken for the middle layer. the α-Grassmannian covering of α subspaces in Gq(n, k) Clearly from our previous discussion, such a code has n is the dimension of the subspace which they span in Fq , parameters with the minimum requirements which are and the α-Grassmannian distance is the α-Grassmannian necessary to solve such a network. These requirements covering minus k. A code C ∈ Gq(n, k) has minimum are given in the following theorem which generalizes α-Grassmannian distance δ if each subset of α codewords Theorem 2. of C has α-Grassmannian distance at least δ. For such a code the minimum α-Grassmannian covering is k+δ. The Theorem 4. The (, k)-Nh,r,αk+ network is solvable c quantity B˜q(n, k, δ; α) will denote the maximum size of over Fq if and only if there exists an α-(h, k, h − k − )q c code with r codewords. an α-(n, k, δ)q code. For the solution of the generalized combination network one can use two identical codewords Thus, Theorem 4 implies that each such covering Grass- in such a code. But, from a coding theory point of mannian code of the maximum size is exactly what is view it is more interesting to consider such codes in required to solve a certain instance of the generalized which there are no repeated codewords. For this we combination networks. define the quantity B (n, k, δ; α) to be the maximum size q The way in which the code solves the generalized of an α-(n, k, δ)c code in which there are no repeated q combination network, as described, is very natural when codewords. Nevertheless, codes with repeated codewords we consider the definition of the generalized combination are sometimes essential, at least for the network coding network. It implies the generalization for the Grassman- solution, as is demonstrated in the next example. nian distance, namely the α-Grassmannian distance. Since Example 1. Consider the (1, 1)-N3,r,4 network in Fig- some might argue that this generalization is less natural ure 1. In a scalar linear solution, a receiver should get from a point of view of a code definition, we have 3 from its three incoming links a subspace of Fq whose also defined the α-Grassmannian covering which yields dimension is at least two. On each link a one-dimensional a natural interpretation for the α-Grassmannian distance. 5

Now, we will translate this deﬁnition into the requirement ˜ of C. Similarly, let Aq(n, k, t; λ) denote the maximum from a packing point of view. For this purpose we will m size of a t-(n, k, λ)q code. Let Aq(n, k, t; λ) denote need to use the dual subspace V ⊥ of a given subspace V m the maximum size of a t-(n, k, λ)q code, where there n in Fq , and the orthogonal complement of a given code C. are no repeated codewords. Clearly, a subspace design For a code in G (n, k) the orthogonal complement ⊥ m C q C t-(n, k, λ)q is a t-(n, k, λ)q code. It should be noted that is deﬁned by also in combinatorial designs repeated blocks are usually

⊥def ⊥ not considered in the literature. One can easily verify that C = {V : V ∈ C} . Theorem 6. C ∈ Gq(n, k) has minimum Grassmannian It is well-known [16, Lemma 13] that the minimum sub- m distance k − t + 1 if and only if C is a t-(n, k, 1)q code. space distance of C and the minimum subspace distance of C⊥ are equal. The following lemma is also well known. Proof. If C has minimum Grassmannian distance k − t + 1, then for each two distinct codewords X,Y ∈ C Lemma 1 (Lemma 12 [16]). For any two subspaces U, V we have k − t + 1 ≤ dG(X,Y ) = k − dim(X ∩ Y ). n ⊥ ⊥ ⊥ of Fq we have that U ∩ V = (U + V ) . It implies that dim(X ∩ Y ) ≤ t − 1 and hence each n Clearly, by induction we have the following conse- t-subspace of Fq is contained in at most one codeword m quence from Lemma 1. of C, i.e., C is a t-(n, k, 1)q code. m On the other hand, if C is a t-(n, k, 1)q code, then any Corollary 1. For any given set of α subspaces n t-subspace of Fq is contained in at most one codeword V ,V ,...,V of n we have 1 2 α Fq of C. It follows that for any X,Y ∈ C we have dim(X ∩ α α !⊥ Y ) ≤ t − 1. Hence, \ X V ⊥ = V . i i dG(X,Y ) = k − dim(X ∩ Y ) ≥ k − (t − 1) = k − t + 1 , i=1 i=1 Corollary 1 induces a new definition of a distance mea- and therefore, C has minimum Grassmannian distance sure for the orthogonal complements of the Grassmannian k − t + 1. codes which solve the generalized combination networks. Clearly, by Theorem 6 we have that the λ-multiple For a Grassmannian code C ∈ Gq(n, k), the minimum Grassmannian distance is a generalization of the Grass- λ-multiple Grassmannian distance is k − t + 1, where mannian distance (which is a 1-multiple Grassmannian t is the smallest integer such that each t-subspace of distance). The generalization implied by Theorem 6 is for n is contained in at most λ k-subspaces of . In the m Fq C the packing interpretation of a t-(n, k, 1)q code. This is sequel it will be explained why this definition is a natural also a generalization of block design over Fq (a subspace generalization of the Grassmannian distance. Moreover, design). If each t-subspace is contained exactly once we will present it later in a more natural way, which is m in a t-(n, k, 1)q code C, then C is a q-Steiner system similar to the definition of a subspace design. These will Sq(t, k, n). If each t-subspace is contained exactly λ times be understood from the results given in the rest of this m in a t-(n, k, λ)q code C, then C is a t-(n, k, λ)q subspace section. design. Similarly to Theorem 5 we have c Theorem 5. If ∈ Gq(n, k) is an α-(n, k, δ) code m C q Theorem 7. If C ∈ Gq(n, k) is a t-(n, k, λ)q code then ⊥ ∈ G (n, n − k) has minimum (α − 1)- ⊥ C q multiple then C has minimum (λ + 1)-Grassmannian covering Grassmannian distance δ. n − t + 1 and minimum (λ + 1)-Grassmannian distance c k − t + 1. Proof. By the definition of an α-(n, k, δ)q code it follows that for each α subspaces V1,V2,...,Vα of m C Proof. If C is a t-(n, k, λ)q code then for each λ + 1 dimPα V ≥ δ + k we have that i=1 i and hence subspaces X1,X2,...,Xλ+1 ∈ we have that ⊥ C dimPα V ≤ n − δ − k i=1 i . Therefore, by Corollary 1 λ+1 Tα ⊥ we have that dim Vi ≤ n − δ − k. This implies \ i=1 dim Xi < t . that each subspace of dimension n − δ − k + 1 of n can Fq i=1 be contained in at most α − 1 codewords of C⊥. Thus, ⊥ Therefore, since C ∈ Gq(n, n − k), it follows by definition that the ⊥ ⊥ minimum (α − 1)-multiple Grassmannian distance of C  λ+1 !⊥ λ+1 !⊥ is (n − k) − (n − δ − k + 1) + 1 = δ. \ X ⊥ t − 1 ≥ dim  Xi  = dim Xi , Theorem 5 leads to a new definition for Grassmannian i=1 i=1 c codes (based on orthogonal complements of α-(n, k, δ)q and hence, codes). A t-(n, k, λ)m multiple Grassmannian code (code q λ+1 G (n, k) X in short) C is a subset of q such that each dim X⊥ ≥ n − t + 1 . t-subspace of n is contained in at most λ codewords i Fq i=1 6

⊥ Thus, C ∈ Gq(n, n − k) has minimum (λ + 1)- in [18], [19] and other codes are a subject for further Grassmannian covering n − t + 1 and minimum (λ + 1)- research. Grassmannian distance (n−t+1)−(n−k) = k−t+1. IV. UPPER BOUNDSONTHE SIZESOF CODES Combining Theorems 5 and 7 yields the following results. In this section we will present some upper bounds on the sizes of codes. Upper bounds on the sizes of Corollary 2. codes with no repeated codewords are considered first and c (1) C ∈ Gq(n, k) is an α-(n, k, δ)q code if and only if later a short discussion is given for codes with repeated ⊥ m C is an (n − k − δ + 1)-(n, n − k, α − 1)q code. codewords. (2) For any feasible δ, k, n, and α, B˜q(n, k, δ; α) = A˜q(n, n − k, n − k − δ + 1; α − 1). A. Bounds on Sizes of Codes with no Repeated Codewords (3) For any feasible δ, k, n, and α, Bq(n, k, δ; α) = Aq(n, n − k, n − k − δ + 1; α − 1). Clearly, there is a huge ground for research since the m (4) C ∈ Gq(n, k) is a t-(n, k, λ)q code if and only if parameters of the codes are in a very large range and our ⊥ c C is a (λ + 1)-(n, n − k, k − t + 1)q code. knowledge is very limited. We will give some ideas and (5) For any feasible t, k, n, and λ, A˜q(n, k, t; λ) = some insight about the difficulty of obtaining new bounds B˜q(n, n − k, k − t + 1; λ + 1). and especially the exact size of optimal codes. The bounds (6) For any feasible t, k, n, and λ, Aq(n, k, t; λ) = are on Aq(n, k, t; λ) and on Bq(n, k, δ; α) and clearly Bq(n, n − k, k − t + 1; λ + 1). by Corollary 2(3) and 2(6), bounds are required only on one of them since they are equivalent (when the related Theorem 4 provides a necessary and sufficient condition parameters are taken). There is a duality between the two for the requirements to solve the (, k)-Nh,r,s network types of codes which were considered with the two dual with a scalar linear network code over Fq. Theorem 4 distance measures. Hence, it is sometimes simpler and is generalized for a solution with vector network coding more convenient to analyse or construct a large code with whose vectors have length `, as follows. one of the two distance measures. As mentioned before,

Theorem 8. The (, k)-Nh,r,αk+ network is solvable with the case of λ = 1 was considered in the last ten years, vectors of length ` over Fq if and only if there exists an and the following simple equality reduced the range for c the search of such bounds. α-(h`, k`, h` − k` − `)q code with r codewords. Proof. Since there are h messages, each one is a vector Theorem 9 (Lemma 13 [16]). If n, k, t, are positive of length `, it follows that the source has a subspace of integers such that 1 ≤ t < k < n, then Aq(n, k, t; 1) = dimension h`. The source can send to each receiver in Aq(n, n − k, n − 2k + t; 1). the direct links any subspace of dimension `. Therefore, Theorem 9 implies that if λ = 1 then it is enough to any α nodes in the middle layer should send to the related A (n, k, t; 1) k ≤ n−k h` ﬁnd q for . This is not the case when receiver an (h` − `)-subspace of Fq . The source sends λ > 1, where an analysis will be considered in Section V. to each one of the r nodes in the middle layer a (k`)- In this analysis for λ > 1 we will consider the case where subspace and hence for a vector solution to the (, k)- n ≥ 2k as well as the case where n < 2k. In this section Nh,r,αk+ network, a Grassmannian code C ∈ Gq(h`, k`) we will consider only the basic upper bounds. of size r, with minimum α-Grassmannian covering h`−`, We start by considering the duality between the two i.e., minimum α-Grassmannian distance h` − ` − k`, is c distance measures and related different simple approaches required. Hence, C is an α-(h`, k`, h` − k` − `)q code to obtain bounds on the maximum sizes of codes. To sim- with r codewords. plify and emphasize the properties on which the bounds Can the scalar linear solution and the vector solution are analysed we summarize them. Our starting point will c be compared only on the basis of Theorems 4 and 8? be an α-(n, k, δ)q code. c Assume we are given the (, k)-Nh,r,αk+ network. A (c.1) In an α-(n, k, δ)q code C, each subset of α code- n scalar linear solution over Fq` requires by Theorem 4 an words (k-subspaces) spans a subspace of Fq whose c α-(h, k, h − k − )q` code with r codewords. The related dimension is at least δ + k. n vector network coding with vectors of length ` over Fq re- (c.2) Each (δ + k − 1)-subspace of Fq contains at most c quires an α-(h`, k`, h`−k`−`)q code with r codewords. α − 1 codewords of C (by (c.1)). ⊥ m One can construct the vector network code from the scalar (c.3) C is an (n − k − δ + 1)-(n, n − k, α − 1)q code linear network code by using companion matrices and (see Corollary 2(1)). In such a code each (n − δ − n their powers [7], [18], [19]. But, the important question is k + 1)-subspace of Fq is contained in at most α − 1 whether we can ﬁnd a Grassmannian code for the vector codewords. network coding larger than the largest one for scalar linear (c.4) Any α codewords from C⊥ intersect in a subspace network coding. Some examples of such codes are given whose dimension is at most n − δ − k (by (c.3)). 7

Bounds on the maximum size of related codes can be Corollary 3. If n, k, δ, and α are positive integers such k+δ−1 obtained based on any one of these four observations and that 1 < k < n, 1 ≤ δ ≤ n−k and 2 ≤ α ≤ k q +1, properties. Each one of these four properties can give then another direction to obtain related bounds.    n  If our starting point is a t-(n, k, λ)m code then the four  δ+k−1 q  q Bq(n, k, δ; α) ≤ (α − 1)  . dual properties are as follows. n−k δ−1 q m n (m.1) In a t-(n, k, λ)q code C, each t-subspace of Fq is contained in at most λ codewords. The rest of this section is devoted to the q-analog of the Johnson bounds [16], [47]. We should remark that there is (m.2) Any λ + 1 codewords of C intersect in a subspace whose dimension is at most t − 1 (by (m.1)). an improvement of the Johnson bound for Grassmannian ⊥ c codes [26]. The same improvement can be applied also for (m.3) C is a (λ + 1)-(n, n − k, k − t + 1)q code (see Corollary 2(4)). In such a code each subset of λ + 1 Grassmannian codes with the new distance measures [12]. n We omit this improvement since we concentrate in this codewords ((n − k)-subspaces) of Fq spans a subspace whose dimension is at least (n − t + 1). paper only on the basic bounds. Also the improvement is n for some parameters, it is not dramatic improvement, and (m.4) Each (n − t)-subspace of Fq contains at most λ far from being simple. codewords of C⊥ (by (m.3)). The classic bounds for the cases λ = 1 (or α = 2, Theorem 11. If n, k, t, and λ are positive integers such t (n, k, λ)m α (n, k, δ)c that 1 ≤ t < k < n and 1 ≤ λ ≤ n−t , then respectively) for a - q code (or an - q k−t q code, respectively) can be easily generalized for larger λ n (α, respectively), where the simplest ones are the packing q − 1 Aq(n, k, t; λ) ≤ Aq(n − 1, k − 1, t − 1; λ) . bound and the Johnson bounds [16]. It might be easier to qk − 1 m generalize these bounds when we consider t-(n, k, λ)q m Proof. Let C be a t-(n, k, λ)q code for which |C| = codes and two proofs based on two of the four given qk−1 M = Aq(n, k, t; λ). Each codeword of C contains q−1 properties can be given. The following well-known lemma n one-dimensional subspaces of Fq . Since the number of will be used frequently in our results. n n q −1 one-dimensional subspaces of Fq is q−1 , it follows that Lemma 2. A t-subspace of n is contained in n−t n Fq k−t q there exists a one-dimensional subspace X of Fq which n k distinct k-subspaces of Fq . q −1 is contained in at least qn−1 M codewords of C. Let Y n n Remark 1. Note, that by Lemma 2 we have that if λ = be an (n − 1)-subspace of Fq such that Fq = Y + X. n−t then A (n, k, t; λ) = n . Hence, the largest λ 0def 0 k−t q q k q Deﬁne C = {C ∩ Y : C ∈ C,X ⊂ C}. Clearly, C is that we should consider is n−t . a subspace code with (k − 1)-subspaces of Y . Note, that k−t q for two distinct codewords C1,C2 ∈ C such that X ⊂ C1 The ﬁrst few results are the q-analog of the packing 0 0 and X ⊂ C2 we have C1 = C1 ∩ Y 6= C2 ∩ Y = C2 bound. 0 0 since C1 = C1 + X and C2 = C2 + X. Therefore, the 0 qk−1 Theorem 10. If n, k, t, and λ are positive integers such size of C is at least qn−1 M. n−t Z0 (t − 1) Y Z = Z0 + X that 1 ≤ t < k < n and 1 ≤ λ ≤ k−t , then Let be a -subspace of . is a q n t-subspace of Fq and hence it is a subspace of at most $ n % t λ codewords of C. Let C1,C2,...,Cs, s ≤ λ, be the A (n, k, t; λ) ≤ λ q . q k only s distinct codewords of C which contain Z. Since t q 0 Z contains X it follows that Ci = Ci ∩ Y , 1 ≤ i ≤ s, m n are distinct codewords in 0 which contain Z0 (note that Proof. Let C be a t-(n, k, λ)q code. There are t q C n Z0 = Z ∩ Y ). If there exists another codeword C0 ∈ 0 distinct t-subspaces of Fq and hence by (m.1) there are C n such that Z0 ⊂ C0 then C = C0 + X is a codeword of at most λ t pairs (X,Y ), where X is a t-subspace and C q which contains Z, a contradiction to the fact that only Y is a codeword in C which contains X. The number k C1,C2,...,Cs are the codewords of C which contain Z. of t-subspaces in a codeword (a k-subspace) is t q, and hence the theorem follows. Hence, each (t−1)-subspace of Y is contained in at most s ≤ λ codewords of C0. Assume that α = k+δ−1 + 1 and we 0 m Remark 2. k q Therefore, C is a (t−1)-(n−1, k−1, λ)q code whose consider to be the Grassmannian code which contains qk−1 qk−1 C size is at least n M. Hence, n Aq(n, k, t; λ) ≤ all the n subspaces of G (n, k). If we take all the k- q −1 q −1 k q q | 0| ≤ A (n − 1, k − 1, t − 1; λ), and thus, n C q subspaces in a (k + δ − 1)-subspace X of Fq then they n span only X, but with one more subspace a (k + δ)- q − 1 Aq(n, k, t; λ) ≤ Aq(n − 1, k − 1, t − 1; λ) . subspace will be spanned. Hence, the largest α that we qk − 1 k+δ−1 should consider is k q + 1. 8

Corollary 4. If n, k, δ, and α are positive integers such Theorem 13. If n, k, t, and λ are positive integers such that 1 < k < n, 1 ≤ δ ≤ n−k and 2 ≤ α ≤ k+δ−1 +1, that 1 ≤ t < k < n and 1 ≤ λ ≤ n−t , then k q k−t q then ˜ qn − 1 Aq(n, k, t; λ) ≥ Aq(n, k, t; λ) . Bq(n, k, δ; α) ≤ Bq(n − 1, k, δ; α) . qn−k − 1 Corollary 6. If n, k, δ, and α are positive integers such that 1 < k < n, 1 ≤ δ ≤ n−k and 2 ≤ α ≤ k+δ−1 +1, Theorem 12. If n, k, t, and λ are positive integers such k q that 1 ≤ t < k < n and 1 ≤ λ ≤ n−1−t , then then k−t q B˜q(n, k, δ; α) ≥ Bq(n, k, δ; α) . qn − 1 A (n, k, t; λ) ≤ A (n − 1, k, t; λ) . Finally, since we allow repeated codewords we have the q qn−k − 1 q following trivial lower bound. Proof. Let be a t-(n, k, λ)m code for which | | = C q C Theorem 14. If n, k, t, λ, and λ0 are positive integers M = Aq(n, k, t; λ). For each (n − 1)-subspace Y let def such that 1 ≤ t < k < n, then CY = {C : C ∈ C,C ⊂ Y }. Clearly, CY is a m n A˜ (n, k, t; λ0λ) ≥ λ0A˜ (n, k, t; λ) . t-(n−1, k, λ)q code. By Lemma 2, each k-subspace of Fq q q qn−k−1 is contained in q−1 subspaces of Gq(n, n − 1). There One can immediately infer from Theorem 14 that qn−1 exist q−1 subspaces in Gq(n, n−1) and hence there exists there are many examples like Example 1 for which ˜ n one (n − 1)-subspace Z such that Bq(n, k, δ; α) > Bq(n, k, δ; α), e.g., (α − 1) k q = n n−k ˜ q − 1 Bq(n, k, 1; α) > Bq(n, k, 1; α) = k q. To end this | | ≥ M. CZ qn − 1 section we remind the reader that for the solution of the generalized combination network repeated codewords are m Since also CZ is a t-(n−1, k, λ)q code (see the deﬁnition allowed. On the other hand, usually in coding theory and qn−k−1 of CY ), it follows that qn−1 Aq(n, k, t; λ) ≤ |CZ | ≤ block design, repeated codewords are not allowed, or more Aq(n − 1, k, t; λ), and thus, precisely, the related code or design with repeated codeword is considered to be not simple and less interesting (in qn − 1 the literature bounds are considered only for codes with Aq(n, k, t; λ) ≤ n−k Aq(n − 1, k, t; λ) . q − 1 no repeated codewords).

Corollary 5. If n, k, δ, and α are positive integers such V. ANALYSIS OF THE RELATED CODES that 1 < k < n, 1 ≤ δ ≤ n−k and 2 ≤ α ≤ k+δ−2 +1, How good are the upper bounds given in Section IV? k−1 q then Can the bound of Theorem 10 be attained? If λ = 1 then in view of Theorem 9 we have to consider A (n, k, t; 1) only qn − 1 q B (n, k, δ; α) ≤ B (n − 1, k − 1, δ; α) . for k ≤ n − k, where good constructions and asymptotic q qk − 1 q bounds are known [2], [14]. If λ > 1 then the situation is We would also like to remind and to mention that some quite different. In this section we present a short analysis bounds can be obtained from known results on arcs and of the lower bounds compared to the upper bounds. We caps in projective geometry [23]. Discussion on these is do this analysis from a few point of view which are given in [12]. described in related subsections. We do not intend to give any speciﬁc construction as this is a topic for another B. Bounds on Sizes of Codes with Repeated Codewords research work (see [12]). Note, that most constructions for λ = 1 can be generalized for larger λ if k ≤ n − k. In this subsection we will discuss the differences be- As for λ = 1, also for larger λ it is not difﬁcult to tween codes with repeated codewords on those with no design constructions based on projective geometry [12], repeated codewords. This will be done with respect to the on Ferrers diagrams [10], [13], [35], and on rank-metric sizes of codes. We note that usually the difference in the codes [36]. size of the codes is minor and in most cases this difference We also take into account the generalized combination can be ignored. Some cases where this difference is networks solved by some of the given codes. Since each not minor will be considered in this subsection. Packing t-subspace is contained in n−t different k-subspaces bounds for codes with repeated codewords are obtained k−t q of n we have that exactly as in Theorem 10 and Corollary 3, respectively. Fq Similar results can be obtained for the Johnson bounds. Theorem 15. If n, k, and t are positive integers such that We will omit these repetitions. Although, the bounds seem 1 ≤ t < k < n, then to be the same there might be a small difference in the ! n − t n sizes of the codes, as we have the following trivial results. Aq n, k, t; = . k − t q k q 9

In view of the equality that Aq(n, k, t; 1) = Aq(n, n − A. Complements k, n − 2k + t; 1) (see Theorem 9) and Theorem 15, it is ⊥ By Corollary 2 the orthogonal complement code C interesting to have a hierarchy of the values Aq(n, k, t; λ), of a t-(n, k, λ)m code is a (λ + 1)-(n, n − k, k − t + n−t q C where λ = 1, 2,..., . It should be noted that c k−t q 1)q code. By Theorem 17, the complement code of C, similarly to the claim in Theorem 9 we have for 2k ≤ n, cdef C = Gq(n, k) \ C is a covering code with multiplicity n−t n−t c Aq(n, k, t; λ) ≤ Aq(n, n − k, n − 2k + t; λ) (1) − λ. If − λ = 1 then is a covering code, k−t q k−t q C n by considering orthogonal complement codes. But, except i.e. a set of k-subspaces of Fq such that each t-subspace n c for λ = 1 such an inequality in (1) is unlikely to be of Fq is contained in at least one codeword of C . It was c ⊥ equality. For example, we have for 2k ≤ n that proved in [17] that (C ) is a subspace q-Turan´ design, i.e. a set of (n−k)-subspaces of n such that each (n−t)- ! Fq n − t n subspace of n contains at least one codeword of ( c)⊥. Aq n, k, t; = , Fq C k − t q k q This implies various connections between all these types of codes. We would not go into detailed specific connec- while for the dual dimension we have tions between codes with given parameters. But, we will ! 2k − t n mention one specific code. The most interesting specific Aq n, n − k, n − 2k + t; = . k − t q k q code which was heavily studied is the q-analog of the Fano plane, i.e. the q-Fano plane [3], [5], [11]. Such a When 2k < n we have code is a 2-(7, 3, 1) code with 7 /3 codewords. It 2k − t n − t 2 q 2 q < , is not known whether such a code exists and there was k − t q k − t q an extensive search and research for the largest code with which implies that these parameters. It was shown in [22] that for q = 2 there ! exists a code with such parameters and 333 codewords. 2k − t Aq n, k, t; The size of the smallest known related covering code is k − t q 396 [8]. The size of the related q-Fano plane (if exists) ! is 381. These results coincide with the common belief 2k − t n < Aq n, n − k, n − 2k + t; = . that a construction of a small covering code is simpler k − t q k q than a construction of related large packing code. In view 7 We have proved a duality between the two distance of Theorem 17 we have that A2(7, 3, 2; 30) ≥ 3 2 − 396, 7 measures and their codes, but this is not the only duality while C2(7, 3, 2; 30) ≤ 3 2 − 333. This implies that that can be obtained. Let Cq(n, k, t; λ) be the minimum the current known lower bound on the packing number size of a code (with distinct codewords) in Gq(n, k) such is closer to the packing bound (Theorem 10) than the that each t-subspace of Gq(n, t) is contained in at least λ related known upper bound on the covering number. codewords. While Aq(n, k, t; λ) represents the maximum A construction which might lead to a q-Fano plane for size of a multiple packing, the quantity Cq(n, k, t; λ) rep- larger q was recently given in [11]. Related results on resents the minimum size of a multiple covering. Clearly similar structures were proved recently in [6]. we have

Theorem 16. If n, k, and t are positive integers such that B. Aq(n, k, t; λ) vs. Aq(n, n − k, n − 2k + t; λ) 1 ≤ t < k < n and 1 ≤ λ ≤ n−t , then k−t q The gap between the size of the largest code and the packing bound (Theorem 10) might be small or large, Aq(n, k, t; λ) ≤ Cq(n, k, t; λ) , depending on the parameters. Consider the case where with equality if and only if a subspace design t-(n, k, λ)q n = 6, k = 4, t = 3, λ = 1, and q = 2. By exists, i.e. the bound of Theorem 10 is attained with Theorem 10 we have that A2(6, 4, 3; 1) ≤ 93, while the equality. actual value is A2(6, 4, 3; 1) = 21. This is not unique n for these parameters, and it occurs since Aq(n, k, t; 1) = By Lemma 2, each t-subspace of Fq is contained in n−t n Aq(n, n − k, n − 2k + t; 1), i.e. we should consider k-subspaces of Fq and since the total number of k−t q only k ≤ n/2, when λ = 1. For λ > 1 there is no k-subspaces of n is n we have a duality between a Fq k q similar connection between codes of G (n, k) and codes packing of k-subspaces and the k-subspaces in G (n, k) q C q of G (n, n−k). Interesting phenomena might occur when which are not contained in . Hence, we have q C k > n/2. A good example can be given by considering Theorem 17. If n, k, and t are positive integers such that n = 6, k = 4, t = 3, and λ = 2. By Theorem 10 we n−t 1 ≤ t < k < n and 1 ≤ λ ≤ k−t , then have that A2(6, 4, 3; 2) ≤ 186. By Theorem 11 we have q 63 that A2(6, 4, 3; 2) ≤ A2(5, 3, 2; 2) . Hence, we have ! 15 n n − t to consider the value of A2(5, 3, 2; 2). It is proved in [12] Aq(n, k, t; λ) = − Cq n, k, t; − λ . k q k − t q that A2(5, 3, 2; 2) = 32 and hence A2(6, 4, 3; 2) ≤ 134. 10

This bound was improved by a certain linear program- for a scalar linear network coding solution we need a c ming and a related construction was given, so finally 3-(3, 1, 1)4 code. The largest such code consists of the 21 3 we have 121 ≤ A2(6, 4, 3; 2) ≤ 126 [12]. This is an one-dimensional subspaces of F4, each one is contained indication that when 2k > n and λ > 1 the value of twice in the code. Therefore, the number of nodes in A2(n, k, t; λ > 1) is relatively much larger than the one the middle layer can be at most 42, while for vector of A2(n, n−k, n−2k+t; λ > 1) (A comprehensive work network coding 121 nodes in the middle layer can be used on bounds and constructions related to Aq(n, k, t; λ) is (since 121 ≤ A2(6, 4, 3; 2)). The smallest alphabet size discussed in [12]). for which a scalar solution with 121 nodes in the middle layer exists is 8. There are 73 one-dimensional subspaces 3 C. Subspace Designs of F8 and each one can be used twice in the code, but There is a sequence of subspace designs in which only 121 codewords out of these 146 possible codewords each t-subspace is contained in exactly λ k-subspaces. are required. This is the indication on the superiority of These designs form optimal generalized Grassmannian vector network coding on scalar linear network coding. codes. These subspace designs have small block length and very large λ. Hence, they are restricted for solutions VI.HAMMING CODESVS.GRASSMANNIAN CODES of generalized combination networks for which the sub- In this section we will examine codes in the Hamming spaces have dimension close to the one of the ambient scheme related to the new distance measures. This will space. But, of course these subspace designs are opti- be done in two different directions. In Section VI-A sets mal as Grassmannian codes. For example, Thomas [41] of codes in the Hamming scheme will be considered as has constructed a 2-(n, 3, 7)2 design, where n ≥ 7 subfamilies of the Grassmannian codes. In Section VI-B and n ≡ 1 or 5 (mod 6). This is the same as a 2- we will consider the related distance measures in the m (n, 3, 7)2 code and its orthogonal complement is an Hamming scheme. Finally, in Section VI-C we show how c 8-(n, n−3, 2)2 code. By Theorem 4, such a code provides the concepts of generalized Hamming weights can be a solution for the (1, n−3)-Nn,(2n−1)(2n−1−1)/3,8(n−3)+1 used in the Grassmannian scheme using the new distance network. Some interesting codes and related networks measures. exist for small parameters. For example, consider the m 2-(6, 3, 3)2 design presented in [4], which is a 2-(6, 3, 3)2 A. Hamming Codes as a Subfamily of Grassmannian c code and its orthogonal complement is a 4-(6, 3, 2)2 code. Codes By Theorem 4, such a code provides a solution for the It should be noted that codes in the Hamming space (1, 3)-N6,279,13 network. Other designs which lead to with the Hamming distance form a subfamily of the optimal codes for related networks can be found in many Grassmannian codes. This can be shown using a few recent papers on this emerging topic, e.g. [3], [4], [5], [6], different approaches. The first one was pointed out in [9], [11], [20], [27], [28], [32], [33], [38], [39], [40], [41], 1957 by Tits [43] who suggested that combinatorics of [42]. sets could be regarded as the limiting case q → 1 of combinatorics of vector spaces over the finite field Fq. D. Network Coding Solutions This can be viewed as the traditional approach, but it How do the results affect the gaps between vector seems that this is not the approach that we need for our network coding and scalar linear network coding? A good purpose. The goal is to show that Hamming codes form example can be given by considering the (1, 1)-N3,r,4 a subfamily of the Grassmannian codes based on network network which was used in [18] to show that there is coding solutions for the generalized combination networks a network with three messages for which vector network and the new distance measures. coding outperforms scalar linear network coding. The first approach that we suggest is based on the net- By Theorem 8, for a vector solution of this network work coding solutions for the combination networks. By c a 3-(3`, `, `)q code is required. For simplicity and for Theorem 2, the (0, 1)-Nh,r,α network has a scalar solution explaining the problems in obtaining lower and upper over Fq, if and only if there exists a code C, over Fq, of h bounds on Aq(n, k, t; λ) we consider the case of t = 3, length r, q codewords, and minimum Hamming distance c ` = 2, and q = 2, i.e. a 3-(6, 2, 2)2 code or its orthogonal r − α + 1. This code can be either linear or nonlinear. If m complement, a 3-(6, 4, 2)2 code. In [19] a code with the code is nonlinear then some of the coding functions c 51 codewords was presented. When a 3-(6, 2, 2)2 code on the r edges between the first layer and the second was considered a code of size 121 (related to the bound layer in the (0, 1)-Nh,r,α network are nonlinear and our c 121 ≤ A2(6, 4, 3; 2) ≤ 126 mentioned before) was framework using α-(h, k, δ)q codes is not the appropriate obtained [12]. one. Such nonlinear codes for nonlinear network coding For a related scalar linear coding solution the related will be considered in the next paragraph. If the code C alphabet size is 4 (since for the vector coding binary is linear then our framework is indeed a q-analog for the vectors of length 2 were considered). By Theorem 4, solution with C. For such a code C we have an h × r 11 generator matrix G for which any set of α columns from G α nodes in the middle layer, appears at most q times has a subset of h linearly independent columns. The r as such a sub-codeword in the projection of these αk columns, considered as one-subspaces of G, form an α- coordinates. In this way, the source can send symbols, c (h, 1, h − 1)q code as implied by Theorem 4. The number to the related receiver, that will distinguish between the of codewords in the largest such a code is the same as codewords which have the same values in these αk coor- the largest length r of a code C with dimension h and dinates. Such codes which have larger values of r than the minimum Hamming distance r − α + 1. We note that if linear ones exist. For example, the vector network codes h and α are fixed then the minimum Hamming distance which outperform the scalar linear network codes can be depends on this largest length r. Hence, it is more natural translated to scalar nonlinear network codes. But, to find in c to search for the largest α-(h, 1, h−1)q code for which the this way nonlinear codes with larger length than the ones minimum Hamming distance is not a parameter, when the obtained by the vector network codes is an interesting and r codewords are taken as the columns of an h×r generator intriguing question for future research. When we consider matrix. a linear code the last requirement is slightly stronger (and weaker in the sense of obtaining larger codes). Each In the second approach, a solution for the (, k)- subset of α nodes from the middle layer have in the Nh,r,αk+ network based on a code in the Hamming projection of their αk coordinates, in the codeword, at scheme is considered. In other words, let us consider a least h − linearly independent vectors. This corresponds scalar solution for the network based on a code over Fq exactly to the fact that each subset of α codewords in in the Hamming scheme rather than a solution based on h the Grassmannian code spans a subspace of Fq whose a code in the Grassmannian Gq(h, k) as was done in the dimension is at least h−. This analysis simulates exactly previous sections. The code can be linear or nonlinear the requirements from a scalar linear solution for the and we distinguish between these two cases. There will (, k)-Nh,r,αk+ network. To end this analysis we have be another distinction in the formulation of the network to explain what are the nonlinear functions written on the coding solutions. For a code in the Grassmannian scheme, edges. For each h messages and any given k links from the the coefficients (which form the local coding vectors) of source to a node v in the middle layer, the source should the linear functions on the edges of the network are the send a symbol from Fq. This clearly define a function entries of the vectors related to a basis of the subspaces h with h variables in Fq. Each one of the q different from the Grassmannian code. If the k columns of each combinations of the h messages implies k symbols to the such basis were written in an h × (rk) matrix G (each k edges from the source to the ith nodes in the middle node in the middle layer yields k consecutive columns for layer. This implies some functions of the h messages. this matrix), then G would have been the generator matrix These functions are written on the k links which connect of a code in the Hamming scheme which form a solution the source to the ith node of the middle layer. for the network. In fact an optimal solution (maximum number of nodes in the middle layer) with a code in Linear codes in the Hamming scheme can be used also the Grassmannian scheme will induce an optimal linear in other ways as solutions for the generalized combination solution (largest length) with a code in the Hamming network. Let C be a linear code of length r, dimension k, scheme and vice versa. But, we are interested in analysing and minimum Hamming distance d, over Fq. For such a the solution for the network using a code in the Hamming code, there exists an (r − k) × r parity-check matrix H. In H, each d−1 columns are linearly independent. Hence, scheme as this might give us extra information and maybe c codes with smaller alphabet size (clearly only in the case the r columns of H form a (d−1)-(r −k, 1, d−2)q code. By Theorem 4, this code solves the (r − k − d + 1, 1)- of nonlinear codes). Such a code (nonlinear over Fq) must have length rk as for the linear code. The source Nr−k,r,r−k network. The number of codewords in the transmit k consecutive symbols of a codeword to each largest such a Grassmannian code is the same as the different node from the middle layer, where the i-th node largest length r of a code of dimension k, and minimum will receive the i-th set of k consecutive symbols of the Hamming distance d. Also, in this case we note that in codeword. The code should have qh codewords since the general we would like to obtain the largest r when r − k and d are fixed. Hence, it is more natural to search for source must send a codeword to the nodes in the middle c layer for each one of the different qh messages. This is the largest (d − 1)-(h, 1, d − 2)q code for which d and h the point to make it clear that some codewords sent by are fixed. There are related results from arcs in projective the source might be identical. This would not cause any planes. Since these results are limited for a space of problem since identical codewords can be distinguished dimension three we omit these results. The interested at the receivers by the information sent from the source reader is referred to [23]. to the receiver through the direct links. For simplicity The conclusion from this discussion is that the codes the reader can assume first that k = 1, although the in the Hamming scheme can be viewed as a subfamily description will be for any k ≥ 1. Each sub-codeword of Grassmmannian codes. But, it is still interesting to of length αk in a projection of αk coordinates, related to find nonlinear codes which outperform vector codes as 12 solutions for the generalized combination networks. coordinates which are nonzero in some of the α nonzero codewords. The minimum α-Hamming covering of a code C is the minimum α-Hamming covering on any set B. Related Codes in the Hamming Scheme of α distinct nonzero codewords of C. We can form a We have defined two new distance measures and their hierarchy of the α-Hamming covering for C, from α = 1, related codes in the Grassmann space. What are the related α = 2, up to α = |C|. This hierarchy will be denoted codes in the Hamming scheme (or more precisely in by c1, c2, . . . , c|C|. This hierarchy, for the Grassmannian the Johnson scheme)? This family of codes was defined codes, can be viewed as a q-analog for the generalized before, but usually was considered for limited number of Hamming weights [46]. Previously a geometric approach parameters. A t-(n, k, λ) packing for 1 ≤ t ≤ k ≤ n, for these generalized weights was given in [44]. Let C be is a collection of k-subsets (called blocks) from an n- a linear code of length n and dimension k and A be a set Q, such that each t-subset of Q is contained in at linear subcode of C. The support of A, denoted by χ(A), most λ blocks. These packings were considered for small is defined by t and k and mainly optimal packings were considered. def For the known results on this topic the reader is referred χ(A) = {i : ∃(a1, a2, . . . , an) = a ∈ A, ai 6= 0}. to [31], [37] and references therein. Generally, each such packing can be viewed as a code of length n. If the n- The rth generalized Hamming weight of a linear code C, set is {1, 2, . . . , n} then a k-subset X is translated into a denoted by dr(C) (dr in short), is the minimum support codeword of length n and weight k, where the ones are of any r-dimensional subcode of C, 1 ≤ r ≤ k, namely, in the coordinates indexed by the k-subset X. def dr = dr(C) = min{|χ(A)| : Let A(n, k, t; λ) be the maximum number of k-subsets A in a t-(n, k, λ) packing. For these new families of codes A is a linear subcode of C, dim(A) = r}. only codes with no repeated codewords are considered.

When k > n/2 the bounds on the sizes of such codes are Clearly, dr ≤ dr+1 for 1 ≤ r ≤ k − 1. The related to the Turan’s´ problem [25], in a similar way to set {d1, d2, . . . , dk} is called the generalized Hamming the complements in Section V-A. weight hierarchy of C [46]. For example we consider the value of A(n, n − 2, n − It is not difficult to verify that for a binary linear code C 3; λ) (n − 3) [n] . An -subset of is contained in exactly we have, c1 = d1, c2 = c3 = d2, c4 = c5 = c6 = c7 = d3, three (n − 2)-subsets of [n]. Hence, λ can have only the and in general c2i = c2i+1 = ··· = c2i+1−1 = di+1. values λ = 1, λ = 2, or λ = 3. If λ = 1 then the This direction of research is interesting, especially when value is just the packing number A(n, n − 2, n − 3; 1) = we consider the q-analog of the generalized Hamming n A(n, 2, 1; 1) = 2 . If λ = 3 then there exists a trivial weights in the Grassmann scheme and a straightforward code which contains all the (n − 2)-subsets of [n]. This generalization to subspace codes which are not necessarily n code attains the value A(n, n − 2, n − 3; 3) = 2 . The constant dimension. This possible q-analog for the gener- only nontrivial value to consider is A(n, n − 2, n − 3; 2) alized Hamming weights can be of great interest and it is which is a well-known Turan´ number. Hence, we have a topic for future research. for n ≥ 2, A(2n, 2n − 2, 2n − 3; 2) = n2 and for n ≥ 2, A(2n+1, 2n−1, 2n−2; 2) = n(n+1). Note, that by the n(n−1) VII.CONCLUSIONSAND OPEN PROBLEMS packing bound we have A(n, n − 2, n − 3; 1) ≤ 6 , n(n−1) We have introduced a new family of Grassmannian A(n, n−2, n−3; 2) ≤ 3 , and A(n, n−2, n−3; 3) ≤ n codes with two new distance measures which generalize 2 . While A(n, n−2, n−3; 3) attains the packing bound, the value of A(n, n − 2, n − 3; 1) is far from the bound. the traditional Grassmannian codes with the Grassmannian As for A(n, n − 2, n − 3; 2) we have an asymptotic ratio distance. There is a correspondence between the set of of 3/4 between the exact values and the upper bound of these codes of maximum size and the optimal scalar linear the packing bound. The behavior, of these ratios, when solution for the set of generalized combination networks. there are many more possible values for λ is an interesting Based on the generalized combination networks and the question. new distance measures we have proved that codes in the Hamming scheme can be viewed as a subfamily of the Grassmannian codes from a few different angles. The C. Generalized Weights research we have started for bounds on the sizes of such The definition of α-Hamming covering codes is very preliminary and there are many obvious codes (which is the related definition for coding questions related to these codes, some of which α-Grassmannian covering codes) for constant weight are currently under research [12] and will provide lots codes can be straightforward generalized to any code in of ground for future research. Other interesting problems the Hamming scheme, where the α-Hamming covering were suggested throughout our exposition and are sum- of α nonzero codewords in a code C is the number of marized as follows. 13

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Tuvi Etzion (M’89–SM’94–F’04) was born in Tel Aviv, Israel, in 1956. He received the B.A., M.Sc., and D.Sc. degrees from the Technion – Israel Institute of Technology, Haifa, Israel, in 1980, 1982, and 1984, respectively. From 1984 he held a position in the Department of Computer Science at the Technion, where he now holds the Bernard Elkin Chair in Computer Science. During the years 1985–1987 he was Visiting Research Professor with the Department of Electrical Engineering – Systems at the University of Southern California, Los Angeles. During the summers of 1990 and 1991 he was visiting Bellcore in Morristown, New Jersey. During the years 1994–1996 he was a Visiting Research Fellow in the Computer Science Department at Royal Holloway University of London, Egham, England. He also had several visits to the Coordinated Science Laboratory at University of Illinois in Urbana-Champaign during the years 1995–1998, two visits to HP Bristol during the summers of 1996, 2000, a few visits to the Department of Electrical Engineering, University of California at San Diego during the years 2000–2017, several visits to the Mathematics Department at Royal Holloway University of London, during the years 2007–2017, and a few visits to the School of Physical and Mathematical Science (SPMS), Nanyang Technological University, Singapore, during the years 2016– 2019. His research interests include applications of discrete mathematics to problems in computer science and information theory, coding theory, network coding, and combinatorial designs. Dr. Etzion was an Associate Editor for Coding Theory for the IEEE TRANSACTIONSON INFORMATION THEORY from 2006 till 2009. From 2004 to 2009, he was an Editor for the Journal of Combinatorial Designs. From 2011 he is an Editor for Designs, Codes, and Cryp- tography, and from 2013 an Editor for Advances of Mathematics in Communications.

Hui Zhang received the Ph.D. degree in Applied Mathematics from Zhejiang University, Hangzhou, Zhejiang, P. R. China in 2013. During 2012–2015, she used to work as a Project Ofﬁcer and a Research Fellow at the School of Physical and Mathematical Sciences at Nanyang Techno- logical University, Singapore. During 2015–2017, she was a Postdoctoral Researcher at the Computer Science Department at Technion - Israel Institute of Technology. Currently, she is a Research Fellow at the School of Physical and Mathematical Sciences at Nanyang Technological Uni- versity, Singapore. Her research interests include combinatorial theory, coding theory and cryptography, and their intersections.