Network coding can significantly improve the transmission rate of communication networks with packet loss compared with routing. However, using network coding usually incurs high computational and storage costs in the network devices and terminals. For example, some network coding schemes require the computational and/or storage capacities of an intermediate network node to increase linearly with the number of packets for transmission, making such schemes difficult to be implemented in a router-like device that has only constant computational and storage capacities. In this paper, we introduce BATched Sparse code (BATS code), which enables a digital fountain approach to resolve the above issue. BATS code is a coding scheme that consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code. It works with the inner code that comprises random linear coding at the intermediate network nodes. BATS codes preserve such desirable properties of fountain codes as ratelessness and low encoding/decoding complexity. The computational and storage capacities of the intermediate network nodes required for applying BATS codes are independent of the number of packets for transmission. Almost capacity-achieving BATS code schemes are devised for unicast networks, two-way relay networks, tree networks, a class of three-layer networks, and the butterfly network. For general networks, under different optimization criteria, guaranteed decoding rates for the receiving nodes can be obtained.Comment: 51 pages, 12 figures, submitted to IEEE Transactions on Information Theor

Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for error-free transmission and the other makes use of classical error-correcting codes. The implication of the tightness of the refined Singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities. Index TermsNetwork error correction, network coding, Hamming bound, Singleton bound, Gilbert-Varshamov bound, network code construction. I. INTRODUCTIONNetwork coding has been extensively studied for multicasting information in a directed communication network when the communication links in the network are error free. It was shown by Ahlswede et al.[1] that the network capacity for multicast satisfies the max-flow min-cut theorem, and this capacity can be achieved by network coding.Li, Yeung, and Cai [2] further showed that it is sufficient to consider linear network codes only. Subsequently, Koetter and Médard [3] developed a matrix framework for network coding. Jaggi et al. [4] proposed a deterministic polynomial-time algorithm to construct linear network codes. Ho et al. [5] showed that optimal linear network codes can be efficiently constructed by a randomized algorithm with an exponentially decreasing probability of failure. DRAFT Fig. 1 shows one special case of network error correction with two nodes, one source node and one sink node, which are connected by parallel links. This is the model studied in classical algebraic coding theory [8], [9], a very rich research field for the past 50 years. Cai and Yeung [6], [10], [11] extended the study of algebraic coding from classical error correction to network error correction. They generalized the Hamming bound (sphere-packing bound), the Singleton bound and the Gilbert-Varshamov bound (sphere-covering bound) in classical error correction coding to network coding. Zhang studied network error correction in packet networks [12], where an algebraic definition of the minimum distance for linear network codes was introduced and the decoding problem was studied. The relation between network codes and maximum distance separation (MDS) codes in classical algebraic coding [13] was clarified in [14].In [6], [10], [11], the common assumption is that the sink nodes know the network topology as well as the network code used in transmission. This kind of network error correction is referred to as coherent network error correct...

As a variation of random linear network coding, segmented network coding (SNC) has attracted great interest in data dissemination over lossy networks due to its low computational cost. In order to guarantee the success of decoding, SNC can adopt a feedbackless FEC (forward error correction) approach by applying a linear block code to the input packets before segmentation at the source node. In particular, if the empirical rank distribution of transfer matrices of segments is known in advance, several classes of coded SNC can achieve close-to-optimal decoding performance. However, the empirical rank distribution in the absence of feedback has been little investigated yet, making the whole performance of the FEC approach unknown. To close this gap, in this paper, we present the first comprehensive study on the transmission scheduling issue for the FEC approach, aiming at optimizing the rank distribution of transfer matrices with little control overhead. We propose an efficient adaptive scheduling framework for coded SNC in lossy unicast networks. This framework is one-sided (i.e., each network node forwards the segments adaptively only according to its own state) and scalable (i.e., its buffer cost will not keep on growing when the number of input packets goes to infinity). The performance of the framework is further optimized based on a linear programming approach. Extensive numerical results show that our framework performs near-optimally with respect to the empirical rank distribution.Index Terms-Random linear network coding, segmented network coding, forward error correction, scheduling strategy. ! 0018-9340 (c)

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Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver.There are NO constraints on the distribution of the transformation matrix and the field size.Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant.The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated.Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix.The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.