Grassmannian Codes With New Distance Measures for Network Coding

Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). There are classical exemplifying multi-source networks that have simple vector linear solutions but no scalar linear solutions over any field. For (single-source) multicast networks, a popular conjecture characterizes the following benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist multicast networks that are vector linearly solvable of dimension L over GF(q) but not scalar linearly solvable over any field of size q' ≤ q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sup> . This paper introduces a general method to construct such a network, and subsequently constructs the first examples to affirm the positive answer of this conjecture. Moreover, among these exemplifying networks vector linearly solvable of dimension L over GF(q), there are instances with the additional property that even for some extremely large q' > q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sup> , they are still not scalar linearly solvable over GF(q').

네트워크 코딩(Network Coding)은 통신망의 성능 향상에 도움을 줄 수 있으나 이의 소프트웨어적 구현은 부호화/복호화 단계에서 매우 큰 지연시간을 유발할 수 있어 이를 줄일 수 있는 병렬화된 구현이 필수적이라 할 수 있다. 본 논문에서는 랜덤 리니어 네트워크 코딩(Random Linear Network Coding)과 랜덤 리니어 네트워크 코딩의 단점을 보완하고자 최근 제안된 파이프라인 네트워크 코딩(Pipeline Network Coding)의 병렬처리 성능을 비교한다. 또한, 네트워크 코딩의 CPU에서의 병렬처리 기법과 GPGPU(General Purpose Graphics Processing Units)에서의 병렬처리 기법을 비교하여 네트워크 코딩의 사용 시 그 파라미터에 따라 적절한 병렬처리 기법을 선택할 필요성이 있음을 보여준다. Network coding has been shown to improve various performance metrics in network systems. However, if network coding is implemented as software a huge time delay may be incurred at encoding/decoding stage so it is imperative for network coding to be parallelized to reduce time delay when encoding/decoding. In this paper, we compare the performance of parallelized decoders for random linear network coding (RLC) and pipeline network coding (PNC), a recent development in order to alleviate problems of RLC. We also compare multi-threaded algorithms on multi-core CPUs and massively parallelized algorithms on GPGPU for PNC/RLC.

Linear network coding is a promising technology that can maximize the throughput capacity of communication network. Despite this salient feature, there are still many challenges to be addressed, and security is clearly one of the most important challenges. In this paper, we will address the design of secure linear network coding. Specifically, we will investigate the network coding design that can both satisfy the weakly secure requirements and maximize the transmission data rate of multiple unicast streams between the same source and destination pair, which has not been addressed in the literature. In our study, we first prove that the secure unicast routing problem is equivalent to a constrained link-disjoint path problem. We then develop efficient algorithm that can find the optimal unicast topology in a polynomial amount of time. Based on the topology, we design deterministic linear network code that is weakly secure and can be constructed at the source node. And finally, we investigate the potential of random linear code for weakly secure unicast and prove the low bound of the probability that a random linear code is weakly secure.

In this paper we consider the per-node run-time complexity of network multicast codes. We show that the randomized algebraic network code design algorithms described extensively in the literature result in codes that on average require a number of operations that scales quadratically with the block-length m of the codes. We then propose an alternative type of linear network code whose complexity scales linearly in m and still enjoys the attractive properties of random algebraic network codes. We also show that these codes are optimal in the sense that any rate-optimal linear network code must have at least a linear scaling in run-time complexity

Network coding is a promising generalization of routing which allows a network node to generate output messages by encoding its received messages to reduce the bandwidth consumption in the network. An important application where network coding offers unique advantages is the multicast network where a source node generates messages and multiple receivers collect the messages. Previous network coding schemes primarily considered encoding the messages in a single multicast session. In this paper, we consider the linear inter-session network coding for multicast. The basic idea is to divide the sessions into different groups and construct a linear network coding scheme for each group. We use two metrics to guide the group division: overlap ratio and overlap width. These two metrics measure the benefit that a system can achieve by inter-session network coding with different considerations. The overlap ratio mainly characterizes the network bandwidth while the overlap width characterizes the system throughput. Our simulation results show that the proposed inter-session network coding scheme can achieve about 30% higher throughput than intra-session network coding.

In this paper we are interested in improving the performance of constructive network coding schemes for video transmission over packet lossy networks. A novel unequal packet loss protection scheme R2NC based on low-triangular global coding matrix with ladder-shaped partition will be presented, which combines redundant and random network coding for robust H.264/SVC video transmission. Firstly, the error-correcting capabilities of redundant network coding make our scheme resilient to loss. Secondly, the implementation of random network coding at the intermediate nodes with multiple input links can reduce the cost of network bandwidth, thus reducing the end to end delay for video transmission. Thirdly, the low-triangular global coding matrix with ladder-shaped partition is maintained throughout R2NC process to provide unequal erasure protection for H.264/SVC priority layers. The redundant network coding avoids the retransmission of lost packets and improves error correcting capabilities of lost packets. Based only on the knowledge of the loss rates on the output links, the source node and intermediate nodes can make decisions for redundant network coding and random network coding (i.e., how much redundancy to add at this node). However, the redundancy caused by redundant network coding makes the network load increases, in order to improve network throughput, we perform random network coding at the intermediate nodes. Our approach is grounded on the overall distortion of reconstructed video minimization by optimizing the amount of redundancy assigned to each layer. Experimental results are shown to demonstrate the significant improvement of H.264/SVC video reconstruction quality with R2NC over packet lossy networks.

In multihop wireless networks, many applications require end-to-end reliable multicast communication with 100% packet delivery ratio. Recently, network coding has been applied to the reliable link-level multicast in wireless networks, where multiple lost packets with distinct intended receivers are XOR-ed together as one packet and forwarded via single retransmission, resulting in a significant reduction of bandwidth consumption in the scenarios of many receivers. However, the simple XOR operation cannot fully exploit the potential coding opportunities. Moreover, for the reliable multicast in multihop wireless networks, the number of receivers of link-level multicast is usually not large and thus this coding-based link-level multicast cannot provide a promising gain. In this work, we propose a new network coding-based reliable multicast scheme which exploits both network coding and the physical-layer broadcast nature. Specifically, first, at each link-level multicast the source node performs the more general coding operations instead of XOR and conducts the random linear packet coding such that more coding gain can be achieved. Second, each node snoops on all communications over the wireless medium and stores the overheard packets for packet decoding, such that an additional coding gain can be obtained. Extensive simulations demonstrate that the proposed scheme can greatly improve the transmission efficiency, especially in the case of large-group multicast.

In this paper, two-sender unicast index coding problem (TUICP) is studied, where the senders possibly have some messages in common, and each receiver requests a unique message. It is analyzed using three independent sub-problems (which are single-sender unicast index coding problems (SUICPs)) and the interactions among them. These sub-problems are described by three disjoint vertex-induced subgraphs of the side-information graph of the TUICP respectively, based on the availability of messages at the senders. The TUICP is classified based on the type of interactions among the sub-problems. Optimal scalar linear index codes for a class of TUICP are obtained using those of the sub-problems. For two classes, we identify a sub-class for which scalar linear codes are obtained using the notion of joint extensions of SUICPs. An SUICP ${{\mathcal{I}}_E}$ is said to be a joint extension of l SUICPs if the fitting matrices of all the l SUICPs are disjoint submatrices of that of ${{\mathcal{I}}_E}$. Joint extensions generalize the notion of rank-invariant extensions. Scalar linear codes and a condition for optimality of the codes are given for a class of joint extensions. Using this result, scalar linear codes and the conditions for their optimality are obtained for two classes of the TUICP.

In this paper, we consider a wireless multicast network with multiple sources, relays, and destinations. We adopt a multi-hop decode-and-forward relay protocol such that two canonical subnetworks are relevant, namely broadcast channel with receiver side information (BC-RSI) and orthogonal multiple access channel with correlated sources and receiver side information (MAC-CS-RSI). A joint network and channel coding (JNCC) strategy is proposed by exploiting ARQ, RSI, and correlated sources. The proposed JNCC strategy does not require the knowledge of RSI nor any transmit channel state information, such that each transmitter simply performs retransmissions until the intended receivers have accumulated enough mutual information to successfully decode all desired messages. This successful decoding is then conveyed from the receivers to the respective transmitters via an acknowledgement message. To measure the performance of the proposed JNCC strategy with ARQ, we derive closed-form expressions of network throughput by applying the renewal-reward theorem. Analytical results show that the proposed JNCC strategy outperforms, in terms of network throughput, the conventional separate network and channel coding strategy with random linear network coding.

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by the network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e., nodes that generate new packets by combining the packets received over incoming links. We present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph G(V, E) in time O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (k+h)), where k is the number of destinations and h is the number of packets. This improves the best known running time of O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (k+h)) of Jaggi et al. (2005) in the typical case of large communication graphs. In addition, our algorithm guarantees that the number of encoding nodes in the obtained network code is bounded by O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Next, we address the problem of finding a network code with the minimum number of encoding nodes in both integer and fractional coding networks. We prove that in the majority of settings this problem is NP-hard. However, we show that if h=O(1) and k=O(1) and the underlying communication graph is acyclic, then there exists an algorithm that solves this problem in polynomial time.

Network coding has been shown to be useful for throughput and reliability in various network topologies, under a fixed-rate, point-to-multipoint wireless network model. We study the effect of introducing a wireless network model where link capacity depends on the network geometry and the signal to interference and noise ratio. In particular, we compare strategies with and without network coding on a multicast network with and without fading, and on single-user multiple path networks with fading. For the multicast network without fading, we find that the network geometry affects which scheme attains higher throughput. For the case with fading, we compare the throughput-outage probability curves achieved by network coding and repetition schemes. For the multiple path networks, we further consider the case where multiple simultaneous transmissions of identical information signals can be combined at a receiver. We find that the relative performance of the schemes we consider depends on the network geometry, the ratio of signal to noise power, whether multiple simultaneous transmissions can be combined, and the operating point on the throughput-outage probability curve

Chapter 1 - An Introduction to Network Coding

With the gestation and evolution of the Network Coding paradigm and its application on multicast networks, there is a tendency to increase the network performance, by sending multiple packets simultaneously to multiple destinations from a source node. Using the concepts to explain the Multicast Linear Network Coding on symbols belonging to finite field F_2, a program is presented to obtain to the level of each of the intermediate links in the network, the packets size m-bits belonging to a finite field F_q, where q=2^m, for any value of m and, in the sink nodes, the original packets sent from the source node.

We propose an efficient Adaptive Random Convolutional Network Coding (ARCNC)\nalgorithm to address the issue of field size in random network coding. ARCNC\noperates as a convolutional code, with the coefficients of local encoding\nkernels chosen randomly over a small finite field. The lengths of local\nencoding kernels increase with time until the global encoding kernel matrices\nat related sink nodes all have full rank. Instead of estimating the necessary\nfield size a priori, ARCNC operates in a small finite field. It adapts to\nunknown network topologies without prior knowledge, by locally incrementing the\ndimensionality of the convolutional code. Because convolutional codes of\ndifferent constraint lengths can coexist in different portions of the network,\nreductions in decoding delay and memory overheads can be achieved with ARCNC.\nWe show through analysis that this method performs no worse than random linear\nnetwork codes in general networks, and can provide significant gains in terms\nof average decoding delay in combination networks.\n

This paper presents an efficient data gathering framework of network coding (NC) coupled compressed sensing (CS) in wireless sensor networks (WSNs), called network compressed coding (NCC). We employ network coding to adapt to the dynamic nature of WSNs, such as moving obstacles and link failure. Measurements from sensor networks are often correlated because sensors nodes nearby observe the contiguous phenomenon, and the operation of the random linear network coding (RLNC) scheme is similar to that of the random projection in compressed sensing. Therefore, we introduce compressed sensing into the network coding (NC), to prevent all-or-nothing impact on NC. NCC simultaneously transmits and encodes specific packets of sensor measurements to form random projections for CS recovery. CS technology guarantees that the data gathered at all nodes are accurately recovered with a high probability from a small number of projections, which is less than the total number of source nodes in the network. Our simulation results show that, only less than half number of packets are required to reconstruct measurements with reasonable quality compared with the traditional network coding schemes. Also, NCC increases the data gathering efficiency by over 20% compared to the conventional NC scheme.