Network-Coding Solutions for Minimal Combination Networks and Their Sub-networks

TL;DR

This paper investigates the structure and solutions of minimal multicast networks, especially combination networks, focusing on the gap between vector-linear and scalar-linear solutions, and introduces Kneser networks to analyze this gap.

Contribution

It characterizes the gap in alphabet size between vector-linear and scalar-linear solutions for minimal combination networks and introduces Kneser networks as a key tool.

Findings

Maximum gap identified for networks with two source messages.

Kneser networks attain the upper bound on the gap.

Nearly no gap in full minimal combination networks, with bounds derived.

Abstract

Minimal multicast networks are fascinating and efficient combinatorial objects, where the removal of a single link makes it impossible for all receivers to obtain all messages. We study the structure of such networks, and prove some constraints on their possible solutions. We then focus on the combination network, which is one of the simplest and most insightful network in network-coding theory. Of particular interest are minimal combination networks. We study the gap in alphabet size between vector-linear and scalar-linear network-coding solutions for such minimal combination networks and some of their sub-networks. For minimal multicast networks with two source messages we find the maximum possible gap. We define and study sub-networks of the combination network, which we call Kneser networks, and prove that they attain the upper bound on the gap with equality. We also prove that the study of this gap may be limited to the study of sub-networks of minimal combination networks, by using graph homomorphisms connected with the $q$-analog of Kneser graphs. Additionally, we prove a gap for minimal multicast networks with three or more source messages by studying Kneser networks. Finally, an upper bound on the gap for full minimal combination networks shows nearly no gap, or none in some cases. This is obtained using an MDS-like bound for subspaces over a finite field.

Figures

Equations

$$q_s(\mathcal{N})⩾q_v(\mathcal{N}),$$

$$\mathrm{gap}(\mathcal{N})\triangleq q_s(\mathcal{N})-q_v(\mathcal{N}).$$

$$\left|\left(\genfrac{}{}{0ex}{}{W}{t}\right)\right|=\left(\genfrac{}{}{0ex}{}{n}{t}\right)\triangleq \frac{n!}{t!(n-t)!}.$$

$$\left|\left[\genfrac{}{}{0ex}{}{V}{t}\right]\right|={\left[\genfrac{}{}{0ex}{}{n}{t}\right]}_q\triangleq \frac{{\prod }_{i=1}^n(q^i-1)}{{\prod }_{i=1}^t(q^i-1){\prod }_{i=1}^{n-t}(q^i-1)}.$$

$$\psi (x)\triangleq \min \left\{q\in \mathbb{P}:q⩾x\right\}.$$

$$0⩽\psi (n)-n⩽n,$$

$$0⩽\psi (n)-n⩽n^{21/40}.$$

$$\mathcal{M}(e)=\left\{G_e\cdot (x_1|\dots |x_h{)}^T:x_i\in {\mathbb{F}}_q^t\right\},$$

$$0⩽\dim \mathcal{M}(e)=\mathrm{rank}(G_e)⩽t.$$

$$\mathcal{M}(\nu )\triangleq \left\{G_{\nu }\cdot (x_1|\dots |x_h{)}^T:x_i\in {\mathbb{F}}_q^t\right\},$$

$$G_{\nu }\triangleq \left(\begin{array}{c}{G}_{e_1}\\ G_{e_2}\\ ⋮\\ G_{e_{d_{\mathrm{in}}(\nu )}}\end{array}\right),$$

$$\dim \mathcal{M}(\nu )=\mathrm{rank}(G_{\nu })⩽\min (d_{\mathrm{in}}(\nu )\cdot t,ht).$$

$$\dim \mathcal{M}(\tau )=\mathrm{rank}(G_{\tau })=ht.$$

$$\mathrm{gap}(\mathcal{N})\triangleq q_s(\mathcal{N})-q_v(\mathcal{N}).$$

$$\dim \mathcal{M}(\nu )=d_{\mathrm{in}}(\nu )\cdot t$$

$$d_{\mathrm{in}}(\nu )\cdot t=\dim \mathcal{M}(\nu )⩽ht.$$

$${\mathcal{V}}_{\ne 1}$$

$${\mathcal{E}}_{\ne 1}$$

$${\mathcal{V}}^{\prime }=\left\{T(e):e\in {\mathcal{E}}_{\ne 1}\right\}.$$

$${\mathcal{E}}^{\prime }$$

$$\exists ({\nu }_1,\nu ),({\nu }_2,\nu )\in \mathcal{E}\text{s.t.}({\nu }_1,\nu )\in T(e_1),({\nu }_2,\nu )\in T(e_2)\}.$$

$$\mathcal{E}=\left\{(\sigma ,\nu ):\nu \in {\mathcal{V}}^{\prime }\right\}\cup \bigcup _{e=\left\{\nu ,{\nu }^{\prime }\right\}\in {\mathcal{E}}^{\prime }}\left\{(\nu ,e),({\nu }^{\prime },e)\right\}.$$

$${\mathcal{V}}_{\ne 1}=\left\{\sigma ,{\nu }_3,{\tau }_1,{\tau }_2\right\}.$$

$${\mathcal{E}}_{\ne 1}=\left\{e_1,e_2,e_6\right\},$$

$$T(e_1)$$

$$T(e_2)$$

$$T(e_6)$$

$$q_v(\mathcal{N})$$

$$q_s(\mathcal{N})$$

$$q_s(\mathcal{N})=\psi (\chi (\mathcal{H})-1),$$

$$\chi (q{K}_{2t:t})⩽q^t+q^{t-1}.$$

$$\chi (q{K}_{2t:t})=q^t+q^{t-1}.$$

$$q{K}_{2t:t}\nrightarrow \overline{q}{K}_{2\overline{t}:\overline{t}}.$$

Full text

We then focus on the combination network, which is one of the simplest and most insightful network in network-coding theory. Of particular interest are minimal combination networks. We study the gap in alphabet size between vector-linear and scalar-linear network-coding solutions for such minimal combination networks and some of their sub-networks.

For minimal multicast networks with two source messages we find the maximum possible gap. We define and study sub-networks of the combination network, which we call Kneser networks, and prove that they attain the upper bound on the gap with equality. We also prove that the study of this gap may be limited to the study of sub-networks of minimal combination networks, by using graph homomorphisms connected with the $q$-analog of Kneser graphs. Additionally, we prove a gap for minimal multicast networks with three or more source messages by studying Kneser networks.

Finally, an upper bound on the gap for full minimal combination networks shows nearly no gap, or none in some cases. This is obtained using an MDS-like bound for subspaces over a finite field.

I Introduction

Network coding has been attracting increased attention for almost two decades since the seminal papers [1, 18]. Multicast networks have received most of this attention. A recent survey on the foundations of multicast network coding may be found in [11]. The multicast network-coding problem can be formulated as follows: given a network with one source which has $h$ messages, for each edge find a function of the messages received at the starting node of the edge, such that each terminal can recover all the messages from its received messages. Such an assignment of a function to each edge is called a solution for the network.

Obviously, received messages on an edge can be expressed as functions of the source messages. If these functions are linear, we obtain a linear solution. Otherwise, we have a nonlinear solution. In linear network coding, each linear function on an edge consists of coding coefficients for each incoming message. If the coding coefficients and the messages are scalars, it is called a scalar solution. If the messages are vectors and the coding coefficients are matrices then it is called a vector solution. A network which has a solution is called a solvable network. It is well-known that a multicast network with one source, $h$ messages, and $N$ terminals, is solvable if and only if the min-cut between the source and each terminal is at least $h$ [11].

The minimal alphabet size, and in the linear setting, field size, of a solution is an important parameter that directly influences the complexity of the calculations at the network nodes. An efficient algorithm to find a field size (not necessarily minimal) that allows a linear solution, and the related linear network code was given in [16]. It is known that any field size $q\geqslant N$ suffices for a linear solution, but it is conjectured that the smallest field size allowing a linear solution is much smaller [10, 11]. This, however, was proved only for two messages [10].

In this work we distinguish between scalar and vector linear solutions. Given a network $\mathcal{N}$, we define $q_{s}(\mathcal{N})$ to be the smallest field size $q$ for which $\mathcal{N}$ has a scalar linear solution. Similarly, we define $q_{v}(\mathcal{N})$ to be the smallest value $q^{t}$, $q$ a prime power, such that $\mathcal{N}$ has a vector solution with vectors of length $t$ over a field of size $q$. By definition,

[TABLE]

and we define the gap by

[TABLE]

One of the most celebrated families of networks is the family of combination networks [24], which has been used for various topics in network coding, e.g., [13, 27, 22, 20, 14]. The $\mathcal{N}_{h,r,s}$ combination network, where $s\geqslant h$, is shown in Fig. 1. The network has three layers: the first layer consists of a single source with $h$ messages. The source transmits $r$ messages to the $r$ nodes of the middle layer. Any $s$ nodes in the middle layer are connected to a terminal, and each one of the $\binom{r}{s}$ terminals wants to recover all the $h$ messages. Since we shall also examine sub-networks of the combination network, we will sometimes stress that no part of the network has been removed by saying that the network is a full combination network.

It was proved in [24] that a solution for a combination network exists if and only if a related error-correcting code exists. This network was also generalized to compare scalar and vector network coding [9]. Its sub-networks were used to prove that finding the minimum required field size of a (linear or nonlinear) scalar network code for a certain multicast network is NP-complete [17].

Of particular interest are minimal multicast networks. In such a network, the removal of even a single link reduces the cut between the source and at least one of the terminals, which in turn, makes the network unable to transmit all of its messages to all the terminals, i.e., the network becomes unsolvable. In the case of combination networks, only the $\mathcal{N}_{h,r,h}$ networks are minimal. Minimal networks are not only a fascinating extremal combinatorial objects, but also of practical importance since they require the least amount of network resources to enable the multicast operation. Minimal networks have been studied in the past, e.g., [17, 8], but only in the context of scalar network coding.

The goal of this work is to consider two problems which are related to vector coding solutions for minimal combination networks and their sub-networks. Our main contributions are the following: we first study general multicast networks which are minimal. We extend the scalar setting of [17, 8] to the vector setting, and we show minimality entails some structural properties of the graph, as well as some constraints on linear solutions.

We then focus on the gap in general minimal multicast networks. The first to demonstrate a gap exists in some network was [26]. However, the gap there is only $1$, and requires at least $h\geqslant 8190$ source messages. This was significantly improved by [9], which showed much larger gaps for as little as $h\geqslant 4$ source messages, in non-minimal networks. In this paper we show that large gaps exist already for $h=2$ messages, the lowest number of messages for which a gap is possible. Not only that, but the networks we construct are minimal, and we show that they attain the highest possible gap of all minimal multicast networks with two source messages. We further prove that studying the gap in general minimal multicast networks with two source messages is equivalent to studying it only in minimal multicast networks which are sub-networks of combination networks, thus, further motivating the study of combination networks. The main tool we use is a connection with $q$-Kneser graphs (which are $q$-analogs of Kneser graphs), as well as a generalization to a new $q$-Kneser hypergraph.

Finally, we also prove an upper bound on the gap in full minimal combination networks. To the best of our knowledge, this is the first such upper bound on the gap. The bound is obtained using certain longest MDS array codes (e.g., see [4]) or a combinatorial structure which we call an independent configuration. While an upper bound on the length of such codes already exists, we present a different proof approach based on the properties of subspaces in the configuration.

The paper is organized as follows. In Section II we provide the basic notation and definitions used in the paper. Section III studies fundamental properties of general minimal multicast networks. In Section IV we study the gap in networks with two source messages, whereas Section V is devoted to three or more source messages. Section VI focuses on an upper bound on the gap for full minimal combination networks. We conclude in Section VII with a summary of the results and some open problems.

II Preliminaries

We now provide the basic notation and definitions used throughout the paper. If $W$ is some finite set of $n$ elements, we use $\binom{W}{t}$ to denote the set of all subsets of $W$ of size $t$. Obviously, using the binomial coefficients,

[TABLE]

Now, let ${\mathbb{F}}_{q}$ denote the finite field of size $q$, where $q\in\mathbb{P}$ and $\mathbb{P}\subset\mathbb{N}$ denotes the set of prime powers. Taking the $q$-analog of sets, if $V$ is a vector space over ${\mathbb{F}}_{q}$, with $n=\dim V$, we use $\genfrac{[}{]}{0.0pt}{}{V}{t}$ to denote the set of all vector subspaces of $V$ of dimension $t$. It is well known that the size of $\genfrac{[}{]}{0.0pt}{}{V}{t}$ is given by the Gaussian coefficient, namely,

[TABLE]

We omit the subscript $q$ whenever the field size is understood from the context, and just write $\genfrac{[}{]}{0.0pt}{}{n}{t}$.

For any $x\in\mathbb{R}$, $x>0$ we use $\psi(x)$ to denote the smallest power of a prime that is greater or equal to $x$, i.e.,

[TABLE]

By Bertrand’s postulate (e.g., see [2]),

[TABLE]

for all $n\in\mathbb{N}$. We mention that much stronger results may be obtained at the cost of working only for large enough $n$. For example, [3] showed that the interval $[x,x+x^{21/40}]$ contains a prime for all large enough $x$. Thus, for all large enough $n$,

[TABLE]

Our main objects of interest are networks. We shall always assume that our network consists of a finite directed acyclic graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. Nodes in the graph will be denoted using lower-case Greek letters. The network contains a single node that is designated as the source, $\sigma\in\mathcal{V}$. It also contains terminal nodes $\mathcal{T}=\left\{\tau_{1},\tau_{2},\dots,\tau_{N}\right\}\subseteq\mathcal{V}$. Finally, the source has $h$ messages, denoted $x_{1},\dots,x_{h}\in{\mathbb{F}}_{q}^{t}$. We therefore denote this network by $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$.

In the network-coding model, each edge $e\in\mathcal{E}$ carries a value from ${\mathbb{F}}_{q}^{t}$. For a node $\nu\in\mathcal{V}$, let $d_{\mathrm{in}}(\nu)$ denote the in-degree of $\nu$, and let $\operatorname{In}(\nu)$ denote the set of incoming edges into $\nu$, hence $\left|\operatorname{In}(\nu)\right|=d_{\mathrm{in}}(\nu)$. For each node $\nu$ and each edge $e$ outgoing from $\nu$, the value $e$ carries is a function of the values on edges incoming into $\nu$. In a linear setting, this value is a linear combination of the incoming values, where the coefficients are $t\times t$ matrices over ${\mathbb{F}}_{q}$.

The goal of the terminals is to gain knowledge of source messages. In the multicast setting, every terminal wants to recover all the source messages. If each of the terminals may recover all the source messages then a linear solution to the network is possible, and we say $\mathcal{N}$ has a $(q,t)$-linear solution. When $t=1$ we call it a scalar linear solution, and in general, a vector linear solution.

The source messages naturally form a vector space over ${\mathbb{F}}_{q}$ of dimension $ht$, namely, ${\mathbb{F}}_{q}^{ht}$. In the linear case, for each edge $e\in\mathcal{E}$, the values carried by $e$ also form a vector space over ${\mathbb{F}}_{q}$, which we denote by $\mathcal{M}(e)$. More precisely, this vector space is given by

[TABLE]

where $G_{e}$ is a $t\times ht$ matrix over ${\mathbb{F}}_{q}$. The matrix $G_{e}$ is called the global coding matrix (and when $t=1$, the global coding vector). Obviously,

[TABLE]

We also say each vertex $\nu\in\mathcal{V}$ has access to

[TABLE]

where $G_{\nu}$ is the $(d_{\mathrm{in}}(\nu)\cdot t)\times ht$ matrix over ${\mathbb{F}}_{q}$ defined by

[TABLE]

with $\operatorname{In}(\nu)=\left\{e_{1},\dots,e_{d_{\mathrm{in}}(\nu)}\right\}$. Thus,

[TABLE]

Combining these facts together, a terminal $\tau\in\mathcal{T}$ is successful in the linear multicast setting if and only if

[TABLE]

A node in the network is said to be essential if it is on a path from the source to some terminal. We assume throughout the paper that all nodes are essential, since non-essential nodes may be removed without affecting the solution.

A cut in the network is defined by a partition $\mathcal{V}=S\cup T$, with the source $\sigma\in S$, and some terminal $\tau_{i}\in T$. We say the size of the cut is $m$ if there are exactly $m$ edges crossing it from $S$ to $T$. It is well known (see [11]) that in a multicast network there exists a $(q,t)$-linear solution if and only if every cut has size at least $h$.

Given a network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$, we define $q_{s}(\mathcal{N})$ to be the smallest field size $q$ for which $\mathcal{N}$ has a $(q,1)$-scalar linear solution. Such a solution is said to be scalar-optimal. We also define $q_{v}(\mathcal{N})$ to be the smallest value $q^{t}$ such that $\mathcal{N}$ has a $(q,t)$-linear solution for some $t$, and such a solution is said to be vector-optimal. By definition, $q_{s}(\mathcal{N})\geqslant q_{v}(\mathcal{N})$, and so we define the (vector) gap by

[TABLE]

Finally, as mentioned in the previous section, a celebrated family of networks that has been studied extensively is the family of combination networks. We briefly recall its definition: for $h,r,s\in\mathbb{N}$, $r\geqslant s$, the $\mathcal{N}_{h,r,s}$ combination network (depicted in Fig. 1) is a multicast network with a single source $\sigma$, connected to $r$ nodes in the middle layer, which we denote $\mathcal{L}=\left\{\lambda_{1},\dots,\lambda_{r}\right\}$. We then have $\binom{r}{s}$ terminal nodes, which we may think of as indexed by $\binom{\mathcal{L}}{s}$. Each of the terminals is connected to a unique subset of $s$ nodes from the middle layer. These networks and their sub-networks will be the focus of this work.

III Minimal Multicast Networks

A minimal multicast network can deliver $h$ messages from the source to each of the terminals while each of its proper sub-networks (containing all the original terminals) can deliver at most $h-1$ messages to at least one of the terminals. From a practical point of view, considering such minimal networks is interesting as it minimizes the used network resources. From a theoretical perspective, minimal networks are a fascinating extremal combinatorial object. Minimal networks have been considered in the past [17, 8], however only in the scalar case. The results of this section may be considered as a generalization of these works.

Definition** 1**

. A multicast network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ is said to be minimal if every edge crosses a cut of size $h$.

Thus, in a minimal network, the removal of any edge from $\mathcal{E}$ makes at least one cut have size strictly less than $h$, and therefore unsolvable.

Given a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, and $m\in\mathbb{N}$, we define the $m$-parallelized version of it as $m\mathcal{G}=(\mathcal{V},m\mathcal{E})$, where $m\mathcal{E}$ denotes the multiset obtained from $\mathcal{E}$ by having each element appear $m$ times. In essence, in the $m$-parallelized graph we keep the same nodes, but duplicate each edge $m$ times.

Lemma** 2**

.

If $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ is a minimal multicast network with a $(q,t)$-linear solution, then $\mathcal{N}^{\prime}=(t\mathcal{G},\sigma,\mathcal{T},ht)$ is a minimal multicast network with a $(q,1)$-scalar linear solution.

Proof:

The claim is entirely trivial: each transmitted vector on an edge $\nu_{1}\xrightarrow{e}\nu_{2}$ in $\mathcal{N}$ is broken up into its $t$ components which are then transmitted separately over the corresponding $t$ parallel edges in $\mathcal{N}^{\prime}$. These are clearly linear combinations (with scalar coefficients from ${\mathbb{F}}_{q}$) of the values entering the node $\nu_{1}$ in $\mathcal{N}^{\prime}$. Finally, the size of each cut in $\mathcal{N}^{\prime}$ is obviously $t$ times the size of the same cut in $\mathcal{N}$, hence the minimality of $\mathcal{N}^{\prime}$.

Lemma** 3**

.

Consider a minimal multicast network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ with a $(q,t)$-linear solution. Then

[TABLE]

for each $\nu\in\mathcal{V}\setminus\left\{\sigma\right\}$, where $\mathcal{M}(\nu)$ was defined in (4).

Proof:

Define $\mathcal{N}^{\prime}=(t\mathcal{G},\sigma,\mathcal{T},ht)$, and construct a $(q,1)$-scalar linear solution to $\mathcal{N}^{\prime}$ from the solution to $\mathcal{N}$, as described in the proof of Lemma 2. It is obvious that $\mathcal{M}(\nu)$ remains unchanged for all $\nu\in\mathcal{V}$. To avoid confusion, we let $d_{\mathrm{in}}^{\mathcal{G}}$ denote the in-degree in the graph $\mathcal{G}$, and $d_{\mathrm{in}}^{t\mathcal{G}}$ the in-degree in the $t$-parallelized graph $t\mathcal{G}$. Assume to the contrary that $\dim\mathcal{M}(\nu)<t\cdot d_{\mathrm{in}}^{\mathcal{G}}(\nu)=d_{\mathrm{in}}^{t\mathcal{G}}(\nu)$. Then, in the network $\mathcal{N}^{\prime}$ there exists an edge $e\in\operatorname{In}(\nu)$ that always carries some fixed linear combination of other edges in $\operatorname{In}(\nu)$. Removing $e$ still allows a solution to $\mathcal{N}^{\prime}$ since the original $\mathcal{M}(\nu)$ may still be computed from the surviving edges, and therefore the original values on edges leaving $\nu$ may be computed as a linear combination as well. However, this contradicts the minimality of $\mathcal{N}^{\prime}$ obtained by Lemma 2.

The following corollary is trivial in the scalar regime, appearing as a side note in [8]. Using the previous lemmas it may also be proved for the general case.

Corollary** 4**

.

In a minimal multicast network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ with a $(q,t)$-linear solution we have $d_{\mathrm{in}}(\nu)\leqslant h$ for all $\nu\in\mathcal{V}$.

Proof:

Combining Lemma 3 with (5) we get

[TABLE]

The claim follows immediately.

IV Minimal Networks with Two Source Messages

In this section we focus on the case of two source messages, i.e., $h=2$. Our goal is to show that there exist networks with two source messages with a positive gap. Such networks have not been demonstrated in the past, and a gap was shown to exist only in networks with at least three messages [9]. A part of the method we describe here bears some resemblance to the ones described in [17, 10, 8]. The resulting networks are in fact minimal, as well as sub-networks of the (minimal) combination network $\mathcal{N}_{h,r,h}$. Additionally, we show that the networks we construct attain the largest possible gap of any minimal multicast network with two source messages.

Of particular interest in what follows, will be the $q$-analog of the Kneser graph, denoted $qK_{n:m}$, whose set of vertices is $\genfrac{[}{]}{0.0pt}{}{V}{m}$, where $V={\mathbb{F}}_{q}^{n}$, and an undirected edge connects two vertices iff the corresponding $m$-dimensional subspaces have a trivial intersection (e.g., see [7, 6, 15] and references therein).

We also recall graph homomorphisms, which act as a generalization of graph coloring. Given two undirected graphs, $\mathcal{G}_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ and $\mathcal{G}_{2}=(\mathcal{V}_{2},\mathcal{E}_{2})$, we say $\phi:\mathcal{V}_{1}\to\mathcal{V}_{2}$ is a homomorphism, denoted by abuse of notation $\phi:\mathcal{G}_{1}\to\mathcal{G}_{2}$, if $\left\{\nu,\nu^{\prime}\right\}\in\mathcal{E}_{1}$ implies $\left\{\phi(\nu),\phi(\nu^{\prime})\right\}\in\mathcal{E}_{2}$ for all $\nu,\nu^{\prime}\in\mathcal{E}_{1}$. This is a generalization of coloring since a homomorphism $\mathcal{G}_{2}\to K_{n}$ (where $K_{n}$ denotes the complete graph on $n$ vertices) is equivalent to a coloring of $\mathcal{G}_{2}$ with $n$ colors (think of each of the vertices of $K_{n}$ representing a distinct color, which is assigned to the vertices in the reverse image of the homomorphism). Additionally, it is easy to see that $\chi(\mathcal{G}_{2})$ (the chromatic number of $\mathcal{G}_{2}$) is the minimum $n$ such that there exists a homomorphism $\mathcal{G}_{2}\to K_{n}$. Since homomorphisms are easily seen to be transitive, i.e., $\mathcal{G}_{1}\to\mathcal{G}_{2}\to\mathcal{G}_{3}$ implies $\mathcal{G}_{1}\to\mathcal{G}_{3}$, we therefore have that $\mathcal{G}_{1}\to\mathcal{G}_{2}$ implies $\chi(\mathcal{G}_{1})\leqslant\chi(\mathcal{G}_{2})$.

We now continue with the case of two source messages. Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a finite directed acyclic graph. We describe the following construction of an undirected graph, $\operatorname{skel}(\mathcal{G})$, which we call the skeleton of $\mathcal{G}$.

Construction** A**

.

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a finite directed acyclic graph. Define

[TABLE]

Additionally, for each $e\in\mathcal{E}_{\neq 1}$ we define $T(e)$ to be the set of all edges $e^{\prime}\in\mathcal{E}$ that may be reached from $e$ by directed paths in $\mathcal{G}$ that pass only through vertices in $\mathcal{V}\setminus\mathcal{V}_{\neq 1}$. Namely, assuming $e=(\nu_{1},\nu_{2})$ and $e^{\prime}=(\nu_{m-1},\nu_{m})$, then $e^{\prime}\in T(e)$ if there exist $\nu_{3},\dots,\nu_{m-2}\in\mathcal{V}$ such that $(\nu_{i},\nu_{i+1})\in\mathcal{E}\setminus\mathcal{E}_{\neq 1}$ for all $2\leqslant i\leqslant m-1$.

We now define the skeleton graph $\operatorname{skel}(\mathcal{G})=(\mathcal{V}^{\prime},\mathcal{E}^{\prime})$. The vertex set is defined as

[TABLE]

The edge set is defined by

[TABLE]

$\Box$

In [17] only the reverse process, i.e., mapping what we call $\operatorname{skel}(\mathcal{G})$ back into a network, is used. This is later also cited and used in [8]. The forward process of mapping a network to its skeleton bears some resemblance to the procedure described in [10]. There, however, a variation on the line graph is used and not the graph. Additionally, the tree decomposition described in [10] creates a directed graph unlike the undirected skeleton we have here. The trees in the decomposition of [10] form a subset (sometimes a proper subset) of $\left\{T(e)~{}:~{}e\in\mathcal{E}_{\neq 1}\right\}$ which was defined here.

A simple observation is proved in the next lemma.

Lemma** 5**

. In Construction A, the set $\mathcal{P}\triangleq\left\{T(e)~{}:~{}e\in\mathcal{E}_{\neq 1}\right\}$ is a partition of $\mathcal{E}$.

Proof:

Since $\mathcal{G}$ is directed and acyclic, find a topological ordering of its vertices. Assume to the contrary that $\mathcal{P}$ is not a partition of $\mathcal{E}$, which implies an edge $e_{1}=(\nu_{1},\nu_{2})\in\mathcal{E}$ is not a member of any of the sets in $\mathcal{P}$ or a member of two sets in $\mathcal{P}$. Assume further, without loss of generality, that among all such edges, $e_{1}$ denotes an edge for which $\nu_{1}$ is minimal under the topological ordering.

If $\nu_{1}\in\mathcal{V}_{\neq 1}$, then by construction, $e_{1}\in T(e_{1})\in\mathcal{P}$ uniquely. Otherwise, there exists exactly a single edge $e_{0}=(\nu_{0},\nu_{1})\in\mathcal{E}$. By the minimality of our choice of $e_{1}$, we have $e_{0}\in T(e)\in\mathcal{P}$ uniquely, for some $e\in\mathcal{E}_{\neq 1}$. By construction, $e_{1}\in T(e)\in\mathcal{P}$ uniquely as well.

The following lemma connects the existence of a $(q,t)$-linear solution to the existence of a certain graph homomorphism when we have $h=2$ source messages.

Lemma** 6**

.

Let $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$ be a minimal multicast network. Then $\mathcal{N}$ has a $(q,t)$-linear solution if and only if $\operatorname{skel}(\mathcal{G})\to qK_{2t:t}$, i.e., there is a homomorphism from $\operatorname{skel}(\mathcal{G})$ to $qK_{2t:t}$.

Proof:

In the first direction, assume $\mathcal{N}$ has a $(q,t)$-linear solution. We assume without loss of generality that nodes with in-degree $1$ simply forward their incoming message. Thus, given any $e\in\mathcal{E}_{\neq 1}$, all the edges $e^{\prime}\in T(e)$ carry the same message, i.e., $\mathcal{M}(e^{\prime})=\mathcal{M}(e)$. We construct the homomorphism $\phi:\operatorname{skel}(\mathcal{G})\to qK_{2t:t}$ by setting $\phi(T(e))=\mathcal{M}(e)$ for each $e\in\mathcal{E}_{\neq 1}$.

We verify this is indeed a homomorphism, since an edge $\left\{T(e_{1}),T(e_{2})\right\}$ in $\operatorname{skel}(\mathcal{G})$ corresponds to edges $(\nu_{1},\nu)\in T(e_{1})$ and $(\nu_{2},\nu)\in T(e_{2})$ in $\mathcal{G}$. By Corollary 4, $d_{\mathrm{in}}(\nu)=2$. By Lemma 3, $\dim\mathcal{M}(\nu)=2t$, hence $\dim\mathcal{M}(e_{1})=\dim\mathcal{M}(e_{2})=t$ and $\mathcal{M}(e_{1}),\mathcal{M}(e_{2})$ must intersect trivially. Thus, $\mathcal{M}(e_{1})$ and $\mathcal{M}(e_{2})$ are joined by an edge in $qK_{2t:t}$.

In the other direction, assume we have a homomorphism $\phi:\operatorname{skel}(\mathcal{G})\to qK_{2t:t}$. It is easy to verify that setting $\mathcal{M}(e^{\prime})=\phi(T(e))$, for all $e^{\prime}\in T(e)$, $e\in\mathcal{E}_{\neq 1}$, is indeed a $(q,t)$-linear solution, since nodes with in-degree $1$ simply forward their incoming message, whereas nodes with in-degree two receive two trivially intersecting subspaces, and may therefore reconstruct the entire space ${\mathbb{F}}_{q}^{2t}$. Such nodes can linearly create any subspace as an outgoing message.

We now prove that the mapping $\operatorname{skel}(\cdot)$ of minimal multicast networks with two messages to undirected graphs, is surjective. This already appears in part in [17].

Lemma** 7**

.

Let $\mathcal{G}^{\prime}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime})$ be a finite undirected graph with no isolated vertices. Then there exists a minimal multicast network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$ such that $\operatorname{skel}(\mathcal{G})=\mathcal{G}^{\prime}$.

Proof:

Construct the desired network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$ as follows. Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and fix $\mathcal{V}=\mathcal{V}^{\prime}\cup\mathcal{E}^{\prime}\cup\left\{\sigma\right\}$. Additionally, the directed edges are defined as

[TABLE]

The source is $\sigma$, and the terminals are $\mathcal{T}=\mathcal{E}^{\prime}$.

Intuitively, we construct the network by adding a new source node $\sigma$, which is connected to each of the original nodes of $\mathcal{G}^{\prime}$. Every edge $e=\left\{\nu,\nu^{\prime}\right\}$ in the original graph $\mathcal{G}^{\prime}$ defines a new terminal node into which an edge from $\nu$ and an edge from $\nu^{\prime}$ are connected. It is easy to verify that $\operatorname{skel}(\mathcal{G})=\mathcal{G}^{\prime}$, and that $\mathcal{N}$ is minimal. The requirement that there exist no isolated vertices ensures all nodes in the middle layer are essential.

We note that the network constructed in Lemma 7 is in fact a sub-network of the combination network $\mathcal{N}_{2,r,2}$. Thus, as a corollary we obtain that the study of minimal multicast networks with two source messages may be restricted to sub-networks of combination networks of the form $\mathcal{N}_{2,r,2}$.

Corollary** 8**

. For any minimal multicast network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$ there exists a minimal multicast network $\mathcal{N}^{\prime}=(\mathcal{G}^{\prime},\sigma^{\prime},\mathcal{T}^{\prime},2)$ such that $\mathcal{G}^{\prime}$ is a subgraph of a combination network, and $\operatorname{gap}(\mathcal{N})=\operatorname{gap}(\mathcal{N}^{\prime})$.

Proof:

We first note that, by construction, $\operatorname{skel}(\mathcal{G})$ does not have isolated nodes as these would imply non-essential nodes in $\mathcal{G}$. We then use Lemma 7 to construct $\mathcal{N}^{\prime}$ such that $\operatorname{skel}(\mathcal{G}^{\prime})=\operatorname{skel}(\mathcal{G})$. By Lemma 6, the networks $\mathcal{N}$ and $\mathcal{N}^{\prime}$ have $(q,t)$-linear solutions for exactly the same pairs $(q,t)\in\mathbb{P}\times\mathbb{N}$. Thus, $\operatorname{gap}(\mathcal{N})=\operatorname{gap}(\mathcal{N}^{\prime})$. Finally, as observed, $\mathcal{N}^{\prime}$ is a sub-network of a minimal combination network $\mathcal{N}_{2,r,2}$.

Example** 9**

. Figure 2(a) shows the famous butterfly network, $\mathcal{N}=(\mathcal{G},\sigma,\left\{\tau_{1},\tau_{2}\right\},2)$. For the graph $\mathcal{G}$ we have (in terms of Construction A) that

[TABLE]

Note that we must always have $\sigma\in\mathcal{V}_{\neq 1}$. Additionally,

[TABLE]

i.e., the set of edges whose source has in-degree that is not $1$. We also have

[TABLE]

Each of these sets becomes a vertex in $\operatorname{skel}(\mathcal{G})$, shown in Figure 2(b). Finally, Lemma 7 constructs a sub-network of a combination network whose skeleton is also $\operatorname{skel}(\mathcal{G})$. This is done by adding a source node $\sigma$, each vertex in $\operatorname{skel}(\mathcal{G})$ becomes a node in the middle layer of the constructed graph, and each edge in $\operatorname{skel}(\mathcal{G})$ becomes a terminal node. The result is depicted in Figure 2(c). $\Box$

We can now restate the definition of the gap for minimal multicast networks $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$. First, we define $\mathcal{H}=\operatorname{skel}(\mathcal{G})$. Then, by Lemma 6,

[TABLE]

However, since $qK_{2:1}\cong K_{q+1}$, i.e., the complete graph on $q+1$ vertices, we have

[TABLE]

where we use $\psi$ from (1), since $q_{s}(\mathcal{N})$ is required to be in $\mathbb{P}$.

We recall some known results on the chromatic number of certain $q$-Kneser graphs.

Theorem** 10**

.

([7, 6, 15]) Let $q\in\mathbb{P}$ and $t\in\mathbb{N}$. Then,

[TABLE]

If $q\geqslant 5$ or $t\leqslant 3$, then

[TABLE]

We also prove the following lemma which shall become useful soon.

Lemma** 11**

.

Let $q,\overline{q}\in\mathbb{P}$ and $t,\overline{t}\in\mathbb{N}$. If $q^{t}>\overline{q}^{\overline{t}}$ then

[TABLE]

Proof:

An upper bound on the size of any clique in $qK_{2t:t}$ is given by $(q^{2t}-1)/(q^{t}-1)=q^{t}+1$, describing a scenario where (except for the zero vector) the subspaces corresponding to the nodes in the clique tile ${\mathbb{F}}_{q}^{2t}$. This scenario is indeed always possible, and is attained by a $t$-spread of ${\mathbb{F}}_{q}^{2t}$ (e.g., see [25]).

Assume to the contrary that there exists a homomorphism $qK_{2t:t}\to\overline{q}K_{2\overline{t}:\overline{t}}$. Since $q$-Kneser graphs never contain self loops, the nodes in a clique in $qK_{2t:t}$ are mapped to distinct nodes of a clique in $\overline{q}K_{2\overline{t}:\overline{t}}$. But by assumption, $q^{t}+1>\overline{q}^{\overline{t}}+1$, namely, the size of largest clique in $qK_{2t:t}$ is strictly larger than the size of the largest clique in $\overline{q}K_{2\overline{t}:\overline{t}}$, a contradiction.

We are now in a position to provide minimal multicast networks with two messages with a positive gap. The key to proving this claim is to build networks whose skeleton is a $q$-Kneser graph. We define these networks in more generality than required here, since they will be used in the following section as well.

Definition** 12**

. Let $q\in\mathbb{P}$, and $h,t\in\mathbb{N}$, $h\geqslant 2$. We define the Kneser network, $\mathcal{K}_{q,t;h}$ in the following way: the network has a single source node $\sigma$. Denote $V\triangleq{\mathbb{F}}_{q}^{ht}$. The source node is connected to $\genfrac{[}{]}{0.0pt}{}{ht}{t}$ nodes in the middle layer, which are indexed by $\genfrac{[}{]}{0.0pt}{}{V}{t}=\left\{V_{1},V_{2},\dots\right\}$. Finally, if we have indices $1\leqslant i_{1}<i_{2}<\dots<i_{h}\leqslant\genfrac{[}{]}{0.0pt}{}{ht}{t}$ such that

[TABLE]

then their corresponding nodes in the middle layer are connected to a unique terminal node.

We observe that for $h=2$,

[TABLE]

We can therefore determine its $q_{v}$ exactly, which we do in the following lemma.

Lemma** 13**

.

For all $q\in\mathbb{P}$, $t\in\mathbb{N}$,

[TABLE]

Proof:

By (7), and since the identity homomorphism $qK_{2t:t}\to qK_{2t:t}$ always exists, we have

[TABLE]

By Lemma 11, this becomes an equality, as claimed.

We now show a gap exists in many Kneser networks with two source messages.

Theorem** 14**

.

For all $q\in\mathbb{P}$ and $t\in\mathbb{N}$, with $q\geqslant 5$ or $t\leqslant 3$,

[TABLE]

Proof:

By Lemma 13,

[TABLE]

However, by Theorem 10, the chromatic number of $qK_{2t:t}$ is

[TABLE]

for $q\geqslant 5$ or $t\leqslant 3$. Thus, for these cases

[TABLE]

and the equality part of the claim is proved. The claimed inequality follows from $\psi(n)\geqslant n$ for all $n\in\mathbb{N}$.

This result is matched by the following upper bound.

Theorem** 15**

.

If $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},2)$ is a minimal multicast network with a $(q,t)$-vector-optimal linear solution, then

[TABLE]

Proof:

By definition, a $(q,t)$-vector-optimal linear solution implies $q_{v}(\mathcal{N})=q^{t}$, and we have

[TABLE]

By Theorem 10 we know that $\chi(qK_{2t:t})\leqslant q^{t}+q^{t-1}$. We denote $q^{\prime}=\psi(q^{t}+q^{t-1}-1)$, and therefore

[TABLE]

where the third homomorphism follows trivially from the fact that $q^{\prime}+1\geqslant q^{t}+q^{t-1}$. By the transitivity of homomorphisms we have

[TABLE]

and then by Lemma 6, the network $\mathcal{N}$ has a $(q^{\prime},1)$-linear solution, namely,

[TABLE]

Finally, the last inequality follows from (2).

Corollary** 16**

. Kneser networks with two source messages, $\mathcal{K}_{q,t;2}$, $q\in\mathbb{P}$, $t\in\mathbb{N}$, and $q\geqslant 5$ or $t\leqslant 3$, attain the maximum possible gap among all minimal networks with two source messages and $(q,t)$-vector-optimal linear solutions.

A few cases remain uncovered by our previous treatment, namely, $q\in\left\{2,3,4\right\}$ and $t\geqslant 4$. We can show a gap for these cases as well, albeit a much smaller guaranteed gap of merely $1$.

Theorem** 17**

. For all $q\in\mathbb{P}$ and all $t\in\mathbb{N}$, $t\geqslant 2$,

[TABLE]

Proof:

Again, by Lemma 13, $q_{v}(\mathcal{K}_{q,t;2})=q^{t}$. We now contend that $\chi(qK_{2t:t})>q^{t}+1$. Recall that the vertex set of $qK_{2t:t}$ is $\genfrac{[}{]}{0.0pt}{}{V}{t}$, where $V={\mathbb{F}}_{q}^{2t}$. Assume a coloring of $qK_{2t:t}$ with $c$ colors. Let $U_{i}\subset\genfrac{[}{]}{0.0pt}{}{V}{t}$, $1\leqslant i\leqslant c$, be the set of vertices colored with color $i$. Then each $U_{i}$ is a $1$-intersecting family in the language of [12]111If $V\triangleq{\mathbb{F}}_{q}^{n}$, then $\mathcal{F}\subseteq\genfrac{[}{]}{0.0pt}{}{V}{\ell}$ is an $m$-intersecting family if for all $W,W^{\prime}\in\mathcal{F}$, $\dim(W\cap W^{\prime})\geqslant m$., and an anticode of diameter $t-1$ in the language of [25]222If $V\triangleq{\mathbb{F}}_{q}^{n}$, the distance between $W,W^{\prime}\in\genfrac{[}{]}{0.0pt}{}{V}{\ell}$ is defined as $d(W,W^{\prime})\triangleq\ell-\dim(W\cap W^{\prime})$. An anticode of diameter $m$ is a set $\mathcal{F}\subseteq\genfrac{[}{]}{0.0pt}{}{V}{\ell}$ such that $W,W^{\prime}\in\mathcal{F}$ implies $d(W,W^{\prime})\leqslant m$.. Also, the set $\left\{U_{i}\right\}_{1\leqslant i\leqslant c}$ forms a tiling (partition) of $\genfrac{[}{]}{0.0pt}{}{V}{t}$.

By [12], for all $1\leqslant i\leqslant c$,

[TABLE]

and $U_{i}$ is either

[TABLE]

or

[TABLE]

where $V_{1},V_{2t-1}$ are subspaces of $V$ of dimensions $1$ and $2t-1$, respectively. However, by [25], there is no tiling of $V$ by $U_{i}$ of these shapes. Thus,

[TABLE]

It follows that

[TABLE]

as claimed.

V Minimal Networks with More Than Two Source Messages

Prior to this work, a gap was shown to exist only in networks with at least four source messages [9], and the networks there are not minimal. An additional ad-hoc example with three source messages was also given in [9] but not in the form of a gap. Instead, it was shown that for the same field size, a different number of nodes in the middle layer of the network is required.

In the previous section we showed a gap exists for $h=2$ source messages. We first contend this immediately shows a gap exists in networks with any $h\geqslant 3$ source messages. We show this by giving a general construction that translates any network $\mathcal{N}$ with $h$ source messages to a network $\mathcal{N}^{\prime}$ with $h^{\prime}>h$ source messages while keeping the parameters of the solution.

Construction** B**

.

Let $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ be a multicast network, with $\mathcal{G}=(\mathcal{V},\mathcal{E})$, and $h\in\mathbb{N}$. For any $h^{\prime}\in\mathbb{N}$, $h^{\prime}>h$, we construct the network $\mathcal{N}^{\prime}=(\mathcal{G}^{\prime},\sigma^{\prime},\mathcal{T}^{\prime},h^{\prime})$, with $\mathcal{G}^{\prime}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime})$, as follows.

We set $\mathcal{V}^{\prime}=\mathcal{V}\cup\left\{\sigma^{\prime}\right\}$, and $\mathcal{T}^{\prime}=\mathcal{T}$. We keep all of the edges of $\mathcal{E}$ in $\mathcal{E}^{\prime}$, and we add $h$ parallel edges from $\sigma^{\prime}$ to $\sigma$. Finally, for each terminal $\tau\in\mathcal{T}^{\prime}$, we add $h^{\prime}-h$ parallel edges from $\sigma^{\prime}$ to $\tau$. $\Box$

Intuitively, Construction B keeps the network $\mathcal{N}$ in its entirety. The terminals also remain as they were but a new source is added. The new source is connected to the original source by $h$ parallel edges, and to each of the terminals by $h^{\prime}-h$ parallel edges. We also make the observation that if $\mathcal{N}$ is minimal, then so is $\mathcal{N}^{\prime}$.

Lemma** 18**

.

Let $h,h^{\prime}\in\mathbb{N}$, $h^{\prime}>h$. If $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ and $\mathcal{N}^{\prime}=(\mathcal{G}^{\prime},\sigma^{\prime},\mathcal{T}^{\prime},h^{\prime})$ are as in Construction B, then there exists a $(q,t)$-linear solution to $\mathcal{N}$ if and only if there exists a $(q,t)$-linear solution to $\mathcal{N}^{\prime}$.

Proof:

In the first direction, assume we have a $(q,t)$-linear solution to $\mathcal{N}$. We build a simple $(q,t)$-linear solution to $\mathcal{N}^{\prime}$ by using the same messages over the edges in $\mathcal{E}$, sending the $h$ source messages $x_{1},\dots,x_{h}$ from $\sigma^{\prime}$ to $\sigma$ over the new $h$ edges, and sending $x_{h+1},\dots,x_{h^{\prime}}$ over the $h^{\prime}-h$ edges connecting $\sigma^{\prime}$ to each of the terminals.

In the second direction, assume we have a $(q,t)$-linear solution to $\mathcal{N}^{\prime}$. By change of bases we can assume without loss of generality that $x_{1},\dots,x_{h}$ are sent over the $h$ edges connecting $\sigma^{\prime}$ to $\sigma$. Consider any terminal $\tau\in\mathcal{T}$. Since only $h^{\prime}-h$ new parallel edges connect $\sigma^{\prime}$ to $\tau$ directly, the remaining incoming edges into $\tau$, namely, $\operatorname{In}(\tau)\cap\mathcal{E}$, must contain messages that enable the recovery of $x_{1},\dots,x_{h}$. Hence, restricting ourselves to the original network, there is a $(q,t)$-linear solution allowing $\sigma$ to convey all of its $h$ messages to the terminals.

Corollary** 19**

. For any network $\mathcal{N}=(\mathcal{G},\sigma,\mathcal{T},h)$ there exists a network $\mathcal{N}^{\prime}=(\mathcal{G}^{\prime},\sigma^{\prime},\mathcal{T}^{\prime},h^{\prime})$ with $h^{\prime}>h$ such that

[TABLE]

It now follows that there exist networks with $h>2$ source messages that exhibit the same gap as the networks we constructed in the previous section with $h=2$ source messages. Moreover, the minimality of the networks is preserved. However, the networks resulting from Construction B are not combination networks, nor are they sub-networks of combination networks. Under the assumption that such a structure might be of possible interest, theoretical or practical, we proceed by generalizing the previous section’s results to more than two source messages.

A full-fledged generalization of the case of $h=2$ to $h>2$ via the skeleton-graph approach, seems highly intricate. Instead, we study Kneser networks $\mathcal{K}_{q,t;h}$ with $h>2$. Their analysis is made possible by replacing their skeleton, a $q$-analog of the Kneser graph, with a generalization of it to a hypergraph.

We assume throughout this section that $h>2$. We again observe that $\mathcal{K}_{q,t;h}$ is a minimal multicast network which is a sub-network of a minimal combination network of the type $\mathcal{N}_{h,r,h}$. We also observe that trivially,

[TABLE]

since a simple linear solution is for the source $\sigma$ to send the vector space $V_{i}\in\genfrac{[}{]}{0.0pt}{}{V}{t}$ to the node in the middle layer indexed by $V_{i}$, and all the nodes in the middle layer just forward their incoming message.

It remains to consider a scalar solution for $\mathcal{K}_{q,t;h}$. To that end we define the $q$-analog Kneser hypergraph. The $q$-analog Kneser hypergraph, denoted $qK^{h}_{ht:t}$ has $\genfrac{[}{]}{0.0pt}{}{V}{t}=\left\{V_{1},V_{2},\dots\right\}$ as vertices, and an $h$-hyperedge $\left\{V_{i_{1}},V_{i_{2}},\dots,V_{i_{h}}\right\}$ exists, if and only if (6) holds. We note that our generalization to the $q$-analog Kneser hypergraph is different from other generalizations, e.g., [21] and the references therein.

Several definitions exist for colorings of hypergraphs. In our case, a coloring is an assignment of a color to each of the vertices of the hypergraph such that no hyperedge contains two vertices of the same color. The minimal number of colors required to color a given hypergraph $\mathcal{G}$ is called its chromatic number, and is denoted by $\chi(\mathcal{G})$.

Lemma** 20**

.

For all $q\in\mathbb{P}$, $t,h\in\mathbb{N}$, and $h\geqslant 3$,

[TABLE]

Proof:

We prove every coloring of $qK^{h}_{ht:t}$ is a coloring of $qK_{ht:t}$, and vice versa. First, consider a coloring $c$ of $qK_{ht:t}$. We contend it is also a coloring of $qK^{h}_{ht:t}$. Indeed, for any $h$ subspaces of dimension $t$, $V_{i_{1}},\dots,V_{i_{h}}\subseteq V={\mathbb{F}}_{q}^{ht}$, such that (6) holds, each pair of them is trivially intersecting, hence $c$ gives them all different colors.

In the other direction, assume $c$ is a coloring of $qK^{h}_{ht:t}$ and we prove it is a coloring of $qK_{ht:t}$. Given two trivially intersecting subspaces of dimension $t$, $V_{i_{1}},V_{i_{2}}$, we can easily build $h-2$ more subspaces of dimension $t$, $V_{i_{3}},\dots,V_{i_{h}}$, such that (6) holds. Thus, $V_{i_{1}}$ and $V_{i_{2}}$ are elements in an hyperedge of $qK^{h}_{ht:t}$, hence $c$ colors them distinctly.

We recall more results on the chromatic number of $qK_{n:m}$.

Theorem** 21**

. [7, 5]

Let $q\in\mathbb{P}$, and $n,m\in\mathbb{N}$, $n\geqslant 2m+1$. Except for the case of $n=2m+1$ and $q=2$, in all other cases,

[TABLE]

We can now state the existence of a gap for Kneser networks with more than two source messages.

Theorem** 22**

. For all $q\in\mathbb{P}$, $t,h\in\mathbb{N}$, $t\geqslant 2$, and $h\geqslant 3$,

[TABLE]

Proof:

As already observed in (8),

[TABLE]

For the scalar solution, our choice of $\mathcal{K}_{q,t;h}$ allows us to follow in a manner similar to the previous section. To avoid tedious notation, denote $q_{s}\triangleq q_{s}(\mathcal{K}_{q,t;h})$. The source $\sigma$ sends a one-dimensional subspace of ${\mathbb{F}}_{q_{s}}^{h}$ on each of the links to the nodes in the middle layer. We may think of the choice of subspace as a color which we assign to the nodes in the middle layer. The total number of colors at our disposal is

[TABLE]

The structure of $\mathcal{K}_{q,t;h}$ implies that a scalar solution must induce a valid coloring of $qK^{h}_{ht:t}$. Therefore, using Lemma 20 and Theorem 21,

[TABLE]

If we define

[TABLE]

then, since $f(x)$ is strictly increasing,

[TABLE]

If $t\geqslant h$ we observe that for all $1\leqslant i\leqslant h-1$,

[TABLE]

which follows from

[TABLE]

and $i\leqslant h-1$. Thus,

[TABLE]

where the last inequality uses the fact that $t\geqslant h$ and hence no power of $q$ repeats in the last double summation. It follows that when $t\geqslant h$ we have

[TABLE]

The case of $t<h$ requires a slightly different treatment. For this case, when $1\leqslant i\leqslant h-1$,

[TABLE]

where (a) follows from

[TABLE]

since $i\leqslant h-1$, and (b) follows again from $i\leqslant h-1$. Thus,

[TABLE]

It follows that when $t<h$ we have

[TABLE]

By combining the bounds on $q_{v}$ and $q_{s}$ in all the cases we obtain the desired result.

VI Minimal Full Combination Networks

We turn to consider, in this section, an upper bound on the gap. In particular, we study the full minimal combination network $\mathcal{N}_{h,r,h}$, and show that it has a very small gap (if at all) compared to its sub-networks, which were considered in the previous section.

The key result we use is the following theorem proved in [24]. Let $(r,q^{h},r-s+1)_{q}$ denote a code over ${\mathbb{F}}_{q}$ of length $r$ with $q^{h}$ codewords and minimum Hamming distance $r-s+1$. If this code is linear, it is denoted by $[r,h,r-s+1]_{q}$.

Theorem** 23**

. ([24])

The $\mathcal{N}_{h,r,s}$ combination network is solvable over ${\mathbb{F}}_{q}$ if and only if there exists an $(r,q^{h},r-s+1)_{q}$ code.

In view of Theorem 23, what are the functions on the edges of the $\mathcal{N}_{h,r,s}$ combination network in the three types of solutions – nonlinear, vector linear, and linear?

For the (scalar) nonlinear solution, take an $(r,q^{h},r-s+1)_{q}$ code. We can feed the $h$ source messages to an arbitrary encoder for the code to obtain a codeword. The $r$ symbols of the codeword are then transmitted on the $r$ links leaving the source node. In the middle layer, each node simply repeats its incoming message on all of its outgoing links. Finally, each terminal obtains $s$ symbols from the codeword, and since the code has minimum distance of $r-s+1$ it may recover the entire codeword from the surviving $s$ symbols, and reverse the encoding process to obtain the $h$ source messages.

For the scalar linear solution, an $[r,h,r-s+1]_{q}$ code is required. We use the same approach as the one for nonlinear solutions, but using a linear encoder for the code results in a scalar linear solution. Namely, the code has an $r\times h$ generator matrix and the $h$ entries of its $i$th row are the coding coefficients of the linear function on the link from the source to the $i$th node in the middle layer.

The vector-linear case differs. A fundamental combinatorial structure that underpins the vector solutions to minimal combination networks is a structure we call a $(t;h,\alpha)_{q}$-independent configuration.

Definition** 24**

. Let $q\in\mathbb{P}$, $t,h,\alpha\in\mathbb{N}$, $\alpha\leqslant h$, and denote $V={\mathbb{F}}_{q}^{ht}$. A $(t;h,\alpha)_{q}$-independent configuration (IC) is a set $\mathcal{C}=\left\{V_{1},V_{2},\dots,V_{m}\right\}\subseteq\genfrac{[}{]}{0.0pt}{}{V}{t}$, such that for all $1\leqslant i_{1}<i_{2}<\dots<i_{\alpha}\leqslant m$,

[TABLE]

We say $\left|\mathcal{C}\right|=m$ is the size of the IC.

We now make the connection between ICs and full minimal combination networks, $\mathcal{N}_{h,r,h}$.

Lemma** 25**

.

The $\mathcal{N}_{h,r,h}$ combination network has a $(q,t)$-solution over ${\mathbb{F}}_{q}$ if and only if there exists a $(t;h,h)_{q}$-IC of size $r$.

Proof:

In the first direction assume that a $(q,t)$-solution over ${\mathbb{F}}_{q}$ exists. We note that by construction, any node $i$ in the middle layer of $\mathcal{N}_{h,r,h}$ receives a subspace $V_{i}\subseteq V\triangleq{\mathbb{F}}_{q}^{ht}$, with $\dim(V_{i})\leqslant t$. If the terminal $R_{j}$ gets from the middle layer the subspaces $V_{i_{1}},\dots,V_{i_{h}}$, then

[TABLE]

which implies that $\dim(V_{i})=t$. Thus, $\left\{V_{i}\right\}_{1\leqslant i\leqslant r}$ is a $(t;h,h)_{q}$-IC.

In the other direction, assume $\mathcal{C}=\left\{V_{1},\dots,V_{r}\right\}$ is a $(t;h,h)_{q}$-IC. We can easily construct a vector network coding solution to the $\mathcal{N}_{h,r,h}$ combination network. Simply send $V_{i}$ to the $i$th middle layer node. Since $\mathcal{C}$ is a $(t;h,h)_{q}$-IC it follows that each terminal has a full rank $(ht)\times(ht)$ transfer matrix from which it can recover the $h$ messages.

We now bound the size of ICs, which will later enable us to upper bound the gap in minimal combination networks.

Lemma** 26**

.

Let $\mathcal{C}$ be a $(t;h,\alpha)_{q}$-IC. If $\alpha\geqslant 2$ then

[TABLE]

Proof:

If $\alpha=2$ the claim is immediate by considering the size of a $t$-spread (e.g., see [25]).

Assume now $\alpha>2$, and denote $V\triangleq{\mathbb{F}}_{q}^{ht}$. Let us write $\mathcal{C}=\left\{V_{1},V_{2},\dots,V_{m}\right\}$, and define

[TABLE]

where $\dim(W_{1})=(\alpha-2)t$. By the definition of an IC, ${\mathbb{F}}_{q}^{ht}=W_{1}+W_{2}$, where $W_{2}\in\genfrac{[}{]}{0.0pt}{}{V}{(h-\alpha+2)t}$. It follows that any vector $v\in V_{j}$, $\alpha-1\leqslant j\leqslant m$, may be written uniquely as $v=v_{1}+v_{2},$ where $v_{1}\in W_{1}$ and $v_{2}\in W_{2}$. We now define

[TABLE]

for all $\alpha-1\leqslant j\leqslant m$. It is easily seen that $\dim(V^{\prime}_{j})=t$.

Furthermore, for any $\alpha-1\leqslant j_{1}<j_{2}\leqslant m$,

[TABLE]

Thus, the set $\left\{V^{\prime}_{i}\right\}_{\alpha-1\leqslant i\leqslant m}$ contains $\left|\mathcal{C}\right|-\alpha+2$ pairwise disjoint $t$-subspaces of $W_{2}$. Thus,

[TABLE]

When $t=1$, bounding the size of $(1;h,h)_{q}$-IC is equivalent to finding the longest MDS codes, and hence related to the MDS conjecture. Thus, Lemma 26 forms a generalization for an upper bound on the length of MDS code. The related results for (scalar, $t=1$) linear codes are given in [19]. Corollary 7 [19, p. 321] asserts that for an $[n,k,n-k+1]_{q}$ MDS code, we have that $n\leqslant q+k-1$. This result is strengthened in Theorem 11 [19, p. 326] by using a more complicated proof based on projective geometry. The theorem asserts that if $k\geqslant 3$ and $q$ is odd then $n\leqslant q+k-2$. Another involved proof for the same result is given for nonlinear codes in [23, pp. 12-13].

We can now give an upper bound on the gap for minimal combination networks.

Theorem** 27**

.

For all $h,r\in\mathbb{N}$, $r\geqslant h\geqslant 2$,

[TABLE]

and for all large enough $r$,

[TABLE]

Proof:

By [24], a $(q,1)$-scalar linear solution to $\mathcal{N}_{h,r,h}$ exists if and only if an $[r,h,r-h+1]_{q_{s}}$ MDS code exists. Thus, we certainly have

[TABLE]

(e.g., see [19, pp. 317–331]). On the other hand, by Lemma 25, and Lemma 26

[TABLE]

Thus, by (2), for all $r\in\mathbb{N}$,

[TABLE]

and by (3), for all large enough $r$,

[TABLE]

Loosely speaking, for the minimal combination network, $\mathcal{N}_{h,r,h}$, the gap is upper bounded by $h-2$ plus the maximum distance between powers of primes. Since it is conjectured that $\psi(n)-n\leqslant O(\log n)$, the upper bound may be further reduced in the future if this conjecture is proved.

VII Conclusion

In this paper we studied scalar and vector linear solutions to the combination network and its sub-networks. We first found the maximal possible gap for minimal multicast networks with $h=2$ source messages, and showed that the Kneser network, which was defined in this work, attains this bound on the gap with equality. Furthermore, we studied Kneser networks with $h\geqslant 3$ source messages and proved they also exhibit a gap. Finally, we provided an upper bound on the gap of full minimal combination networks, showing the gap is relatively small.

It is interesting to compare the gap results with those of [9]. In [9] it was proved that for even $h\geqslant 4$ there exists a multicast network $\mathcal{N}$ for which $\operatorname{gap}(\mathcal{N})=q^{(h-2)t^{2}/h+o(t)}$, and for odd $h\geqslant 5$ there exists a multicast network $\mathcal{N}$ for which $\operatorname{gap}(\mathcal{N})=q^{(h-3)t^{2}/(h-1)+o(t)}$, and in any case, the gap is of order $q^{\Theta(t^{2})}$, where $q$ and $h$ are considered constant. In comparison with the results of Section IV and Section V, the gap we present is only of the order of $q^{\Theta(t)}$, which we also prove is optimal in the case of minimal multicast networks with $h=2$. Two important points distinguish between the two papers: the networks in [9] are not minimal, and [9] uses $h\geqslant 4$. Which of the two, minimality or the number of source messages, contributes to this difference is still an open question. The results of Section V may hint at the former, but we are still missing an upper bound for $h\geqslant 3$.

We also want to highlight the significance of Section VI. If we take for example the case of $h=2$ source messages, by Theorem 27, the full minimal combination network $\mathcal{N}_{2,r,2}$ has [math] gap, for any $r\in\mathbb{N}$ (a fact that was noticed in [9] using Latin squares). If we pick $q\in\mathbb{P}$, $t\in\mathbb{N}$, $r=\genfrac{[}{]}{0.0pt}{}{2t}{t}$, keep the source and all of the middle layer of nodes, but remove a few of the terminals, we obtain the Kneser network $\mathcal{K}_{q,t;2}$, the network from Theorem 14, that has the maximal possible gap, $\psi(q^{t}+q^{t-1}-1)-q^{t}$. Thus, it appears that $\mathcal{N}_{2,r,2}$ is equally difficult for a scalar solution as it is for a vector solution. However, by pruning some terminals, the resulting Kneser network $\mathcal{K}_{q,t;2}$ becomes easy for a vector solution but difficult for a scalar solution. This upper bound on the gap by Theorem 27 grows very slowly. As another example, if $r-2$ is not a prime power then $\mathcal{N}_{3,r,3}$ has no gap.

A full characterization of the gap in linear multicast networks is still an open question. We also suggest minimality of such networks may play a role in limiting the gap, a problem which we leave for future work. Finally, the gap between non-linear and linear solutions for multicast networks is still unknown.

Acknowledgment

The authors thank Yiwei Zhang for helpful discussions.