A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear network coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under non-linear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.
A single-letter lower bound on the sum rate of multiple description coding with tree-structured distortion constraints is established by generalizing Ozarow's celebrated converse argument through the introduction of auxiliary random variables that form a Markov tree. For the quadratic vector Gaussian case, this lower bound is shown to be achievable by an extended version of the El Gamal-Cover scheme, yielding a complete sum-rate characterization.
We consider a network coding problem where the destination wants to recover the sum of the signals (Gaussian random variables or random finite field elements) at all the source nodes, but the sum must be kept secret from an eavesdropper that can wiretap on a subset of edges. This setting arises naturally in sensor networks and federated learning, where the secrecy of the sum of the signals (e.g. weights, gradients) may be desired. While the case for finite field can be solved, the case for Gaussian random variables is surprisingly difficult. We give a simple conjecture on the necessary and suf…
In this paper, we propose a Polar coding scheme for parallel Gaussian channels. The encoder knows the sum rate of the parallel channels but does not know the rate of any channel. By using the nesting property of Polar code, we design a coding/decoding scheme to achieve the sum rates.
Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a vector linear solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number $k/n$, there exists a sum-network which has capacity equal to $k/n$. It is a natural question whether, for any positive rational number $k/n$, and for any finite set/co-finite set of primes $\{p_1,p_2,\ldots,p_l\}$, t…
和网络是有向非循环网络，其中每个源独立地从给定字段$\mathbb F$生成一个符号，并且每个终端想要接收源符号的$\mathbb F)$上的和$($。对于具有两个源或两个终端的和网络，可解性由每个源-终端对的连接条件来表征[3]。Shenvi和Dey [6]给出了$3$ -源$3$ -端$(3$ s $/3$ t $)$和网络可解的充要条件。然而，任意源/汇的一般情况仍然是开放的。本文利用区域分解方法研究了具有三个源和$n$汇的和网络。给出了一类$3$ s $/n$ t和网络的充要条件.作为这个结果的一个直接应用，得到了$3$ s $/3$ t和网络可解的一个充要条件.
We present the problem of two-terminal source coding with Common Sum Reconstruction (CSR). Consider two terminals, each with access to one of two correlated sources. Both terminals want to reconstruct the sum of the two sources under some average distortion constraint, and the reconstructions at two terminals must be identical with high probability. In this paper, we develop inner and outer bounds to the achievable rate distortion region of the CSR problem for a doubly symmetric binary source. We employ existing achievability results for Steinberg's common reconstruction and Wyner-Ziv's source…
In this paper, we first propose an universal polar coding scheme for parallel Gaussian channels with non-binary inputs. It is assumed that the encoder knows only the sum capacity of M parallel channels instead of the capacity of any single channel. By decomposing each parallel channel into T = [log2r] sub channels, we therefore obtain MT binary sub-channels. A super polar coding scheme that across all sub-channels is then proposed. This scheme can achieve the sum capacity when the block length is sufficiently large. We have also discussed the applications of parallel polar coding design for bo…
This paper considers vector network coding based on rank-metric codes and subspace codes. Our main result is that vector network coding can significantly reduce the required field size compared to scalar linear network coding in the same multicast network. The achieved gap between the field size of scalar and vector network coding is in the order of $q^{(\ell-1)t^2/\ell}-q^t$, for any $q \geq 2$, where $t$ denotes the dimension of the vector solution, and the number of messages is $2 \ell$, ${\ell \geq 2}$. Previously, only a gap of constant size had been shown. This implies also the same gap …
A code equivalence between index coding and network coding was established, which shows that any index-coding instance can be mapped to a network-coding instance, for which any index code can be translated to a network code with the same decoding-error performance, and vice versa. Also, any network-coding instance can be mapped to an index-coding instance with a similar code translation. In this paper, we extend the equivalence to secure index coding and secure network coding, where eavesdroppers are present in the networks, and any code construction needs to guarantee security constraints in …
In this paper, we consider the application of Reed- Solomon (RS) channel coding for joint error correction and cooperative network coding on non-binary phase shift keying (PSK) modulated signals. The relay first decodes the RS channel coded messages received each in a time slot from all sources before applying network coding (NC) by the use of bit-level exclusive OR (XOR) operation. The network coded resulting message is then channel encoded before its transmission to the next relay or to the destination according to the network configuration. This scenario shows superior performance in compar…
We extend the equivalence between network coding and index coding by Effros, El Rouayheb, and Langberg to the secure communication setting in the presence of an eavesdropper. Specifically, we show that the most general versions of secure network-coding setup by Chan and Grant and the secure index-coding setup by Dau, Skachek, and Chee, which also include the randomised encoding setting, are equivalent.
In contrast to the network coding problem wherein the sinks in a network demand subsets of the source messages, in a network computation problem the sinks demand functions of the source messages. Similarly, in the functional index coding problem, the side information and demands of the clients include disjoint sets of functions of the information messages held by the transmitter instead of disjoint subsets of the messages, as is the case in the conventional index coding problem. It is known that any network coding problem can be transformed into an index coding problem and vice versa. In this …
Network coding is an efficient means to improve the spectrum efficiency of satellite communications. However, its resilience to eavesdropping attacks is not well understood. This paper studies the confidentiality issue in a bidirectional satellite network consisting of two mobile users who want to exchange message via a multibeam satellite using the XOR network coding protocol. We aim to maximize the sum secrecy rate by designing the optimal beamforming vector along with optimizing the return and forward link time allocation. The problem is non-convex, and we find its optimal solution using se…
We develop a network coding technique based on flags of subspaces and a corresponding network channel model. To define error correcting codes we introduce a new distance on the flag variety, the Grassmann distance on flags and compare it to the commonly used gallery distance for full flags.
We propose a piggybacking scheme for network coding where strong source inputs piggyback the weaker ones, a scheme necessary and sufficient to achieve the cut-set upper bound at high/low-snr regime, a new asymptotically optimal operational regime for the multihop Amplify and Forward (AF) networks.
In this work, we derive the random coding error exponent for the uplink phase of a two-way relay system where physical layer network coding (PNC) is employed. The error exponent is derived for the practical (yet sub-optimum) XOR channel decoding setting. We show that the random coding error exponent under optimum (i.e., maximum likelihood) PNC channel decoding can be achieved even under the sub-optimal XOR channel decoding. The derived achievability bounds provide us with valuable insight and can be used as a benchmark for the performance of practical channel-coded PNC systems employing low co…
We consider a network coding setting where some of the messages and edges have fixed alphabet sizes, that do not change when we increase the common alphabet size of the rest of the messages and edges. We prove that the problem of deciding whether such network admits a coding scheme is undecidable. This can be considered as a partial solution to the conjecture that network coding (without fixed-size messages/edges) is undecidable. The proof, which makes heavy use of analogies with digital circuits, is essentially constructing a digital circuit of logic gates and flip-flops within a network codi…
Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of networks over a field $\mathbb{F}_q,$ with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a \textit{discrete polymatroidal} network and show that a linear fractional solution over a field $\mathbb{F}_q,$ exists for a network if and only if the network is discrete polymatroidal with respect to a…
It is known that there exists a multiple-unicast network which has a rate $1$ linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we show that for any non-zero positive rational number $\frac{k}{n}$, there exists a multiple-unicast network which has a rate $\frac{k}{n}$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes.
Performance Comparison between Network Coding in Space and Routing in Space
几何空间中的网络编码是一个新兴的研究方向，也被称为空间信息流（Space Information Flow），是一个很有前途的研究领域，它显示了空间中的网络编码相对于空间中的路由的优越性。现有文献证明，对于规则（5+1）模型中的单源组播，在给定6个终端节点的情况下，空间网络编码严格优于空间路由，其中5个终端节点形成一个以一个终端节点为中心的正五边形。为了比较空间网络编码和空间路由的性能，以代价优势为度量，定量研究了任意终端节点在二维欧氏空间中任意移动时的两类空间网络编码和空间最优路由.进一步，给出了空间网络编码优于空间路由的代价优势上界和区域。给出了空间信息流的若干性质。
Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this work we show that the subspace distance can…
Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t…
