Network-error correcting codes using small fields

Recently, Ebrahimi and Fragouli proposed an algorithm to construct scalar network codes using small fields (and vector network codes of small lengths) satisfying multicast constraints in a given single-source, acyclic network. The contribution of this paper is two fold. Primarily, we extend the scalar network coding algorithm of Ebrahimi and Fragouli (henceforth referred to as the EF algorithm) to block network-error correction. Existing construction algorithms of block network-error correcting codes require a rather large field size, which grows with the size of the network and the number of sinks, and thereby can be prohibitive in large networks. We give an algorithm which, starting from a given network-error correcting code, can obtain another network code using a small field, with the same error correcting capability as the original code. Our secondary contribution is to improve the EF Algorithm itself. The major step in the EF algorithm is to find a least degree irreducible polynomial which is coprime to another large degree polynomial. We suggest an alternate method to compute this coprime polynomial, which is faster than the brute force method in the work of Ebrahimi and Fragouli.

Wireless Network-Coded Bidirectional Relaying Using Latin Squares for M-PSK Modulation

The design of modulation schemes for the physical layer network-coded two-way relaying scenario is considered with a protocol which employs two phases: multiple access (MA) phase and broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of MA interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, that this relationship can be used to get the network coding maps satisfying the exclusive law. The channel fade states for which the minimum distance of the effective constellation at the relay become zero are referred to as the singular fade states. For M - PSK modulation (M any power of 2), it is shown that there are (M-2/4 - M/2 + 1) M singular fade states. Also, it is shown that the constraints which the network coding maps should satisfy so that the harmful effects of the singular fade states are removed, can be viewed equivalently as partially filled Latin Squares (PFLS). The problem of finding all the required maps is reduced to finding a small set of maps for M - PSK constellations (any power of 2), obtained by the completion of PFLS. Even though the completability of M x M PFLS using M symbols is an open problem, specific cases where such a completion is always possible are identified and explicit construction procedures are provided. Having obtained the network coding maps, the set of all possible channel realizations (the complex plane) is quantized into a finite number of regions, with a specific network coding map chosen in a particular region. It is shown that the complex plane can be partitioned into two regions: a region in which any network coding map which satisfies the exclusive law gives the same best performance and a region in which the choice of the network coding map affects the performance. The quantization thus obtained analytically, leads to the same as the one obtained using computer search for M = 4-PSK signal set by Koike-Akino et al., when specialized for Simulation results show that the proposed scheme performs better than the conventional exclusive-OR (XOR) network coding and in some cases outperforms the scheme proposed by Koike-Akino et al.

On the feasibility of network alignment for three-source three-destination Multiple Unicast networks with delays

A transform approach to network coding was in-troduced by Bavirisetti et al. (arXiv:1103.3882v3 [cs.IT]) as a tool to view wireline networks with delays as k-instantaneous networks (for some large k). When the local encoding kernels (LEKs) of the network are varied with every time block of length k >1, the network is said to use block time varying LEKs. In this work, we propose a Precoding Based Network Alignment (PBNA) scheme based on transform approach and block time varying LEKs for three-source three-destination multiple unicast network with delays (3-S3-D MUN-D). In a recent work, Menget al. (arXiv:1202.3405v1 [cs.IT]) reduced the infinite set of sufficient conditions for feasibility of PBNA in a three-source three-destination instantaneous multiple unicast network as given by Das et al. (arXiv:1008.0235v1 [cs.IT]) to a finite set and also showed that the conditions are necessary. We show that the conditions of Meng et al. are also necessary and sufficient conditions for feasibility of PBNA based on transform approach and block time varying LEKs for 3-S3-D MUN-D.

Wireless network-coded accumulate-compute and forward two-way relaying

The design of modulation schemes for the physical layer network-coded two way wireless relaying scenario is considered. It was observed by Koike-Akino et al. for the two way relaying scenario, that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA Phase and all these network coding maps should satisfy a requirement called exclusive law. We extend this approach to an Accumulate-Compute and Forward protocol which employs two phases: Multiple Access (MA) phase consisting of two channel uses with independent messages in each channel use, and Broadcast (BC) phase having one channel use. Assuming that the two users transmit points from the same 4-PSK constellation, every such network coding map that satisfies the exclusive law can be represented by a Latin Square with side 16, and conversely, this relationship can be used to get the network coding maps satisfying the exclusive law. Two methods of obtaining this network coding map to be used at the relay are discussed. Using the structural properties of the Latin Squares for a given set of parameters, the problem of finding all the required maps is reduced to finding a small set of maps. Having obtained all the Latin Squares, the set of all possible channel realizations is quantized, depending on which one of the Latin Squares obtained optimizes the performance. The quantization thus obtained, is shown to be the same as the one obtained in [7] for the 2-stage bidirectional relaying.

Wireless network-coded three-way relaying using Latin Cubes

The design of modulation schemes for the physical layer network-coded three-way wireless relaying scenario is considered. The protocol employs two phases: Multiple Access (MA) phase and Broadcast (BC) phase with each phase utilizing one channel use. For the two-way relaying scenario, it was observed by Koike-Akino et al. [4], that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called exclusive law. This paper does the equivalent for the three-way relaying scenario. We show that when the three users transmit points from the same 4-PSK constellation, every such network coding map that satisfies the exclusive law can be represented by a Latin Cube of Second Order. The network code map used by the relay for the BC phase is explicitly obtained and is aimed at reducing the effect of interference at the MA stage.

A matroidal framework for network-error correcting codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. The current work attempts to establish a connection between matroid theory and network-error correcting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of network-error correcting codes to arrive at the definition of a matroidal error correcting network. An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error correcting network code if and only if it is a matroidal error correcting network associated with a representable matroid. Therefore, constructing such network-error correcting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa.

An asymptotically optimal push-pull method for multicasting over a random network

We consider multicast flow problems where either all of the nodes or only a subset of the nodes may be in session. Traffic from each node in the session has to be sent to every other node in the session. If the session does not consist of all the nodes, the remaining nodes act as relays. The nodes are connected by undirected edges whose capacities are independent and identically distributed random variables. We study the asymptotics of the capacity region (with network coding) in the limit of a large number of nodes, and show that the normalized sum rate converges to a constant almost surely. We then provide a decentralized push-pull algorithm that asymptotically achieves this normalized sum rate.

Channel quantization for physical layer network-coded two-way relaying

The design of modulation schemes for the physical layer network-coded two way relaying scenario is considered with the protocol which employs two phases: Multiple access (MA) Phase and Broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA phase. In other words, the set of all possible channel realizations (the complex plane) is quantized into a finite number of regions, with a specific network coding map giving the best performance in a particular region. We obtain such a quantization analytically for the case when M-PSK (for M any power of 2) is the signal set used during the MA phase. We show that the complex plane can be classified into two regions: a region in which any network coding map which satisfies the so called exclusive law gives the same best performance and a region in which the choice of the network coding map affects the performance, which is further quantized based on the choice of the network coding map which optimizes the performance. The quantization thus obtained analytically, leads to the same as the one obtained using computer search for 4-PSK signal set by Koike-Akino et al., for the specific value of M = 4.

Precoding-Based Network Alignment Using Transform Approach for Acyclic Networks With Delay

The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a F(p)m-linear combination of the input symbols across different generations, where F(p)m denotes the field over which the network operates (p is prime and m is a positive integer). We use finite-field discrete Fourier transform to convert the output symbols at the sink nodes, at any given time instant, into a F(p)m-linear combination of the input symbols generated during the same generation without making use of memory at the intermediate nodes. We call this as transforming the acyclic network with delay into n-instantaneous networks (n is sufficiently large). We show that under certain conditions, there exists a network code satisfying sink demands in the usual (nontransform) approach if and only if there exists a network code satisfying sink demands in the transform approach. When the zero-interference conditions are not satisfied, we propose three precoding-based network alignment (PBNA) schemes for three-source three-destination multiple unicast network with delays (3-S 3-D MUN-D) termed as PBNA using transform approach and time-invariant local encoding coefficients (LECs), PBNA using time-varying LECs, and PBNA using transform approach and block time-varying LECs. We derive sets of necessary and sufficient conditions under which throughputs close to n' + 1/2n' + 1, n'/2n' + 1, and n'/2n' + 1 are achieved for the three source-destination pairs in a 3-S 3-D MUN-D employing PBNA using transform approach and time-invariant LECs, and PBNA using transform approach and block time-varying LECs, where n' is a positive integer. For PBNA using time-varying LECs, we obtain a sufficient condition under which a throughput demand of n(1)/n, n(2)/n, and n(3)/n can be met for the three source-destination pairs in a 3-S 3-D MUN-D, where n(1), n(2), and n(3) are positive integers less than or equal to the positive integer n. This condition is also necessary when n(1) + n(3) = n(1) + n(2) = n where n(1) >= n(2) >= n(3).

A Matroidal Framework for Network-Error Correcting Codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of linear network-error detecting codes to arrive at the definition of a matroidal error detecting network (and similarly, a matroidal error correcting network abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource, multicast, and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters, such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, and so on, being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally, we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, whereas there are no linear network-error detecting codes that do the same.

Wireless Network-Coded Accumulate-Compute-and-Forward Two-Way Relaying

For the physical-layer network-coded wireless two-way relaying, it was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to channel conditions greatly reduces the impact of multiple-access interference, which occurs at the relay, and all these network coding maps should satisfy a requirement called exclusive law. We extend this approach to an accumulate-compute-and-forward protocol, which employs two phases: a multiple access (MA) phase consisting of two channel uses with independent messages in each channel use and a broadcast (BC) phase having one channel use. Assuming that the two users transmit points from the same 4-phase-shift keying (PSK) constellation, every such network coding map that satisfies the exclusive law can be represented by a Latin square of side 16, and conversely, this relationship can be used to get the network coding maps satisfying the exclusive law. Two methods of obtaining this network coding map to be used at the relay are discussed. Using the structural properties of the Latin squares for a given set of parameters, the problem of finding all the required maps is reduced to finding a small set of maps for the case. Having obtained all the Latin squares, a criterion is provided to select a Latin square for a given realization of fade state. This criterion turns out to be the same as the one used byMuralidharan et al. for two-stage bidirectional relaying.

SDC: Scalable description coding for adaptive streaming media

Video compression techniques enable adaptive media streaming over heterogeneous links to end-devices. Scalable Video Coding (SVC) and Multiple Description Coding (MDC) represent well-known techniques for video compression with distinct characteristics in terms of bandwidth efficiency and resiliency to packet loss. In this paper, we present Scalable Description Coding (SDC), a technique to compromise the tradeoff between bandwidth efficiency and error resiliency without sacrificing user-perceived quality. Additionally, we propose a scheme that combines network coding and SDC to further improve the error resiliency. SDC yields upwards of 25% bandwidth savings over MDC. Additionally, our scheme features higher quality for longer durations even at high packet loss rates.

On the Study of Network Coded AF Transmission Protocol for Wireless Multiple Access Channels

In this paper, the performance of the network coded amplify-forward cooperative protocol is studied. The use of network coding can suppress the bandwidth resource consumed by relay transmission, and hence increase the spectral efficiency of cooperative diversity. A distributed strategy of relay selection is applied to the cooperative scheme, which can reduce system overhead and also facilitate the development of the explicit expressions of information metrics, such as outage probability and ergodic capacity. Both analytical and numerical results demonstrate that the proposed protocol can achieve large ergodic capacity and full diversity gain simultaneously.

Approximate decoding approaches for network coded correlated data

This paper considers a framework where data from correlated sources are transmitted with the help of network coding in ad hoc network topologies. The correlated data are encoded independently at sensors and network coding is employed in the intermediate nodes in order to improve the data delivery performance. In such settings, we focus on the problem of reconstructing the sources at decoder when perfect decoding is not possible due to losses or bandwidth variations. We show that the source data similarity can be used at decoder to permit decoding based on a novel and simple approximate decoding scheme. We analyze the influence of the network coding parameters and in particular the size of finite coding fields on the decoding performance. We further determine the optimal field size that maximizes the expected decoding performance as a trade-off between information loss incurred by limiting the resolution of the source data and the error probability in the reconstructed data. Moreover, we show that the performance of the approximate decoding improves when the accuracy of the source model increases even with simple approximate decoding techniques. We provide illustrative examples showing how the proposed algorithm can be deployed in sensor networks and distributed imaging applications.

Statistical mechanics analysis of LDPC coding in MIMO Gaussian channels

Using analytical methods of statistical mechanics, we analyse the typical behaviour of a multiple-input multiple-output (MIMO) Gaussian channel with binary inputs under low-density parity-check (LDPC) network coding and joint decoding. The saddle point equations for the replica symmetric solution are found in particular realizations of this channel, including a small and large number of transmitters and receivers. In particular, we examine the cases of a single transmitter, a single receiver and symmetric and asymmetric interference. Both dynamical and thermodynamical transitions from the ferromagnetic solution of perfect decoding to a non-ferromagnetic solution are identified for the cases considered, marking the practical and theoretical limits of the system under the current coding scheme. Numerical results are provided, showing the typical level of improvement/deterioration achieved with respect to the single transmitter/receiver result, for the various cases. © 2007 IOP Publishing Ltd.