Robust Wireless Network Coding – An Overview
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paper_title: Embracing wireless interference: analog network coding

paper_content:

Traditionally, interference is considered harmful. Wireless networks strive to avoid scheduling multiple transmissions at the same time in order to prevent interference. This paper adopts the opposite approach; it encourages strategically picked senders to interfere. Instead of forwarding packets, routers forward the interfering signals. The destination leverages network-level information to cancel the interference and recover the signal destined to it. The result is analog network coding because it mixes signals not bits. So, what if wireless routers forward signals instead of packets? Theoretically, such an approach doubles the capacity of the canonical 2-way relay network. Surprisingly, it is also practical. We implement our design using software radios and show that it achieves significantly higher throughput than both traditional wireless routing and prior work on wireless network coding.
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paper_title: Linear network coding

paper_content:

Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node.
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paper_title: Linear Network Error Correction Codes in Packet Networks

paper_content:

In this paper, we study basic properties of linear network error correction codes, their construction and error correction capability for various kinds of errors. Our discussion is restricted to the single-source multicast case. We define the minimum distance of a network error correction code. This plays the same role as it does in classical coding theory. We construct codes that can correct errors up to the full error correction capability specified by Singleton bound for network error correction codes recently established by Cai and Yeung. We propose a decoding principle for network error correction codes, based on which we introduce two decoding algorithms and analyze their performance. We formulate the global kernel error correction problem and characterize the error correction capability of codes for this kind of error.
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paper_title: A Random Linear Network Coding Approach to Multicast

paper_content:

We present a distributed random linear network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves capacity with probability exponentially approaching 1 with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error exponents for linear Slepian-Wolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two examples of practical scenarios: distributed network operation and networks with dynamically varying connections. Our derivation of these results also yields a new bound on required field size for centralized network coding on general multicast networks
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paper_title: Iterative Network and Channel Decoding for the Two-Way Relay Channel

paper_content:

We introduce an extension of the relay channel that we call two-way relay channel. The two-way relay channel consists of two users which want to communicate to each other with the help of one relay. We consider the time-division two-way relay channel without power control, where the broadcast channels are orthogonalized in time and where the two users and the relay use the same transmission power. We describe a joint network-channel coding method for this channel model, where channel codes are used at both users and a network code is used at the relay. The channel code of one user and the network code form a distributed turbo code which we call turbo network code and which can be iteratively decoded at the other user. Moreover, we conjecture closed expressions for lower bounds for the channel capacities of the time-division relay and two-way relay channel without power control and deliver simulation results of the proposed turbo network code.
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paper_title: The benefits of coding over routing in a randomized setting

paper_content:

A novel randomized network coding approach for robust, distributed transmission and compression of information in networks is presented, and its advantages over routing-based approaches is demonstrated.
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paper_title: An algebraic approach to network coding

paper_content:

We consider the problem of information flow in networks. In particular, we relate the question whether a set of desired connections can be accommodated in a network to the problem of finding a point on a variety defined over a suitable field. This approach lends itself to the derivation of a number of theorems concerning the feasibility of a communication scenario involving failures.
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paper_title: Coding for Errors and Erasures in Random Network Coding

paper_content:

The problem of error-control in a "noncoherent" random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomon-like code construction is provided and decoding algorithm given.
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paper_title: XORs in the air: practical wireless network coding

paper_content:

This paper proposes COPE, a new architecture for wireless mesh networks. In addition to forwarding packets, routers mix (i.e., code) packets from different sources to increase the information content of each transmission. We show that intelligently mixing packets increases network throughput. Our design is rooted in the theory of network coding. Prior work on network coding is mainly theoretical and focuses on multicast traffic. This paper aims to bridge theory with practice; it addresses the common case of unicast traffic, dynamic and potentially bursty flows, and practical issues facing the integration of network coding in the current network stack. We evaluate our design on a 20-node wireless network, and discuss the results of the first testbed deployment of wireless network coding. The results show that using COPE at the forwarding layer, without modifying routing and higher layers, increases network throughput. The gains vary from a few percent to several folds depending on the traffic pattern, congestion level, and transport protocol.
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paper_title: Network Information Flow

paper_content:

We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.
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paper_title: The benefits of coding over routing in a randomized setting

paper_content:

A novel randomized network coding approach for robust, distributed transmission and compression of information in networks is presented, and its advantages over routing-based approaches is demonstrated.
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paper_title: Coding for Errors and Erasures in Random Network Coding

paper_content:

The problem of error-control in a "noncoherent" random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomon-like code construction is provided and decoding algorithm given.
---
paper_title: Linear Network Error Correction Codes in Packet Networks

paper_content:

In this paper, we study basic properties of linear network error correction codes, their construction and error correction capability for various kinds of errors. Our discussion is restricted to the single-source multicast case. We define the minimum distance of a network error correction code. This plays the same role as it does in classical coding theory. We construct codes that can correct errors up to the full error correction capability specified by Singleton bound for network error correction codes recently established by Cai and Yeung. We propose a decoding principle for network error correction codes, based on which we introduce two decoding algorithms and analyze their performance. We formulate the global kernel error correction problem and characterize the error correction capability of codes for this kind of error.
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paper_title: On Randomized Linear Network Codes and Their Error Correction Capabilities

paper_content:

Randomized linear network code for single source multicast was introduced and analyzed in Ho et al. (IEEE Transactions on Information Theory, October 2006) where the main results are upper bounds for the failure probability of the code. In this paper, these bounds are improved and tightness of the new bounds is studied by analyzing the limiting behavior of the failure probability as the field size goes to infinity. In the linear random coding setting for single source multicast, the minimum distance of the code defined in Zhang, (IEEE Transactions on Information Theory, January 2008) is a random variable taking nonnegative integer values that satisfy the inequality in the Singleton bound recently established in Yeung and Cai (Communications in Information and Systems, 2006) for network error correction codes. We derive a bound on the probability mass function of the minimum distance of the random linear network code based on our improved upper bounds for the failure probability. Codes having the highest possible minimum distance in the Singleton bound are called maximum distance separable (MDS). The bound on the field size required for the existence of MDS codes reported in Zhang, (IEEE Transactions on Information Theory, January 2008) and Matsumoto (arXiv:cs.IT/0610121, Oct. 2006) suggests that such codes exist only when field size is large. Define the degradation of a code as the difference between the highest possible minimum distance in the Singleton bound and the actual minimum distance of the code. The bound for the probability mass function of the minimum distance leads to a bound on the field size required for the existence of network error correction codes with a given maximum degradation. The result shows that allowing minor degradation reduces the field size required dramatically.
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paper_title: Error-Correcting Codes in Projective Spaces Via Rank-Metric Codes and Ferrers Diagrams

paper_content:

Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper, we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant-weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant-dimension code. The union of these codes is our final constant-dimension code. In particular, the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant-weight codes and the rank-metric codes. Finally, we use puncturing on our final constant-dimension codes to obtain large codes in the projective space which are not constant-dimension.
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paper_title: Network coding and error correction

paper_content:

We introduce network error-correcting codes for error correction when a source message is transmitted to a set of receiving nodes on a network. The usual approach in existing networks, namely link-by-link error correction, is a special case of network error correction. The network generalizations of the Hamming bound and the Gilbert-Varshamov bound are derived.
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paper_title: NETWORK ERROR CORRECTION, PART II: Lower Bounds

paper_content:

In Part I of this paper, we introduced the paradigm of network error correction as a generalization of classical link-by-link error correction. We also obtained the network generalizations of the Hamming bound and the Singleton bound in classical algebraic coding theory. In Part II, we prove the network generalization of the Gilbert-Varshamov bound and its enhancement. With the latter, we show that the tightness of the Singleton bound is preserved in the network setting. We also discuss the implication of the results in this paper. Definition 2. An etwork code ist-error-correcting if it can correct all τ -errors for τ ≤ t, i.e., if the total number of errors in the network is at most t, then the source message can be recovered by all the sink nodes u ∈U.A network code is Y-error-correcting if it can correct E-errors for all E ∈ Y. In Part I, we have proved the network generalizations of the Hamming bound and the Singleton bound. In this part, we will prove a network generalization of the Gilbert-Varshamov bound and its enhancement. With the latter, we will show that the tightness of the Singleton bound is preserved in the network setting. The rest of Part II is organized as follows. In Section 2, we prove the Gilbert bound and the Varshamov bound for network error-correcting codes. In Section 3, we sharpen the Varshamov bound obtained in Section 2 to the strengthened Varshamov bound. By means of the latter, we prove the tightness of the Singleton bound for
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paper_title: Coding for Errors and Erasures in Random Network Coding

paper_content:

The problem of error-control in a "noncoherent" random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomon-like code construction is provided and decoding algorithm given.
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paper_title: Construction algorithm for network error-correcting codes attaining the Singleton bound

paper_content:

We give a centralized deterministic algorithm for constructing linear network error-correcting codes that attain the Singleton bound of network error-correcting codes. The proposed algorithm is based on the algorithm by Jaggi et al. We give estimates on the time complexity and the required symbol size of the proposed algorithm. We also estimate the probability of a random choice of local encoding vectors by all intermediate nodes giving a network error-correcting codes attaining the Singleton bound. We also clarify the relationship between the robust network coding and the network error-correcting codes with known locations of errors.
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paper_title: The benefits of coding over routing in a randomized setting

paper_content:

A novel randomized network coding approach for robust, distributed transmission and compression of information in networks is presented, and its advantages over routing-based approaches is demonstrated.
---
paper_title: Coding for Errors and Erasures in Random Network Coding

paper_content:

The problem of error-control in a "noncoherent" random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomon-like code construction is provided and decoding algorithm given.
---
paper_title: Rank metric decoder architectures for noncoherent error control in random network coding

paper_content:

While random network coding has proved to be a powerful tool for disseminating information in networks, it is highly susceptible to errors caused by various sources. Recently, constant-dimension codes (CDCs), especially Kotter-Kschischang (KK) codes, have been proposed for error control in random network coding. It has been shown that KK codes can be constructed from Gabidulin codes, an important class of rank metric codes used in storage and cryptography. Although rank metric decoders have been proposed for both Gabidulin and KK codes, it is not clear whether such decoders are feasible and suitable for hardware implementations. In this paper, we propose novel decoder architectures for both codes. The synthesis results of our decoder architectures for Gabidulin and KK codes over small fields and with limited error-correcting capabilities not only are affordable, but also achieve high throughput.
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paper_title: Error-Correcting Codes in Projective Spaces Via Rank-Metric Codes and Ferrers Diagrams

paper_content:

Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper, we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant-weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant-dimension code. The union of these codes is our final constant-dimension code. In particular, the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant-weight codes and the rank-metric codes. Finally, we use puncturing on our final constant-dimension codes to obtain large codes in the projective space which are not constant-dimension.
---
paper_title: A Rank-Metric Approach to Error Control in Random Network Coding

paper_content:

The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Rotter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if mu erasures and delta deviations occur, then errors of rank t can always be corrected provided that 2t les d - 1 + mu + delta, where d is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application, where n packets of length M over F(q) are transmitted, the complexity of the decoding algorithm is given by O(dM) operations in an extension field F(qn).
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paper_title: Codes for network coding

paper_content:

In [4] a metric for error correction in network coding is introduced. Also constant-dimension codes were introduced and investigated. Nevertheless little is known on codes in this metric in general. In this paper, several classes of codes are defined and investigated.
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paper_title: Construction of Large Constant Dimension Codes With a Prescribed Minimum Distance

paper_content:

In this paper we construct constant dimension codes with prescribed minimum distance. There is an increased interest in subspace codes in general since a paper [13] by Kotter and Kschischang where they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to construct constant dimension codes. Using this approach we found many new constant dimension codes with a larger number of codewords than previously known codes. We finally give a table of the best constant dimension codes we found.
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paper_title: On Metrics for Error Correction in Network Coding

paper_content:

The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the injection metric, which is closely related to, but different than, the subspace metric of KOumltter and Kschischang. In particular, in the case of a non-constant-dimension code, the decoder associated with the injection metric is shown to correct more errors then a minimum-subspace-distance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding.
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paper_title: Bounds on covering codes with the rank metric

paper_content:

In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an upper bound on the volume of the union of multiple balls with rank radii. Using these geometrical properties, we derive both upper and lower bounds on the minimum cardinality of a code with a given rank covering radius. The geometrical properties and bounds proposed in this paper are significant to the design, decoding, and performance analysis of rank metric codes.
---
paper_title: Coding for Errors and Erasures in Random Network Coding

paper_content:

The problem of error-control in a "noncoherent" random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomon-like code construction is provided and decoding algorithm given.
---
paper_title: Projective space codes for the injection metric

paper_content:

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. In this paper, the metric used is the so-called "injection distance", introduced by Silva and Kschischang. A Gilbert-Varshamov bound for such codes is derived. Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric.
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paper_title: Joint Network-Channel Coding for the Multiple-Access Relay Channel

paper_content:

We propose to use joint network-channel coding based on turbo codes for the multiple-access relay channel. Such a system can be used for the cooperative uplink for two mobile stations to a base station with the help of a relay. We compare the proposed system with a distributed turbo code for the relay channel and with a system which uses separate network-channel coding for the multiple-access relay channel. Simulation results confirm that the systems with network coding for the multiple-access relay channel gain cooperative diversity compared to the system with the distributed turbo code for the relay channel. Moreover, the results show that joint network-channel coding outperforms separate network-channel coding. The reason for this is that the redundancy which is contained in the transmission of the relay can be exploited more efficiently with joint network-channel coding
---
paper_title: Iterative Network and Channel Decoding for the Two-Way Relay Channel

paper_content:

We introduce an extension of the relay channel that we call two-way relay channel. The two-way relay channel consists of two users which want to communicate to each other with the help of one relay. We consider the time-division two-way relay channel without power control, where the broadcast channels are orthogonalized in time and where the two users and the relay use the same transmission power. We describe a joint network-channel coding method for this channel model, where channel codes are used at both users and a network code is used at the relay. The channel code of one user and the network code form a distributed turbo code which we call turbo network code and which can be iteratively decoded at the other user. Moreover, we conjecture closed expressions for lower bounds for the channel capacities of the time-division relay and two-way relay channel without power control and deliver simulation results of the proposed turbo network code.
---
paper_title: MIXIT: The Network Meets the Wireless Channel

paper_content:

The traditional contract between the network and the lower layers states that the network does routing and the lower layers deliver correct packets. In a wireless network, however, different nodes may hear most bits in a transmission, yet none of them receives the whole packet uncorrupted. The current approach imposes fate sharing on the bits, dropping a whole packet because of a few incorrect bits. In contrast, this paper proposes MIXIT, a new architecture that performs opportunistic routing on groups of correctly received symbols. We show using simulations driven with Software Radios measurements that MIXIT provides 4x throughput improvement over state-of-the-art opportunistic routing.
---
paper_title: A Random Linear Network Coding Approach to Multicast

paper_content:

We present a distributed random linear network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves capacity with probability exponentially approaching 1 with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error exponents for linear Slepian-Wolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two examples of practical scenarios: distributed network operation and networks with dynamically varying connections. Our derivation of these results also yields a new bound on required field size for centralized network coding on general multicast networks
---
paper_title: A Practical Joint Network-Channel Coding Scheme for Reliable Communication in Wireless Networks

paper_content:

In this paper, we propose a practical scheme, Non-Binary Joint Network-Channel Coding (NB-JNCC), for reliable multi-path multi-hop communication in arbitrary large-scale wireless networks. NB-JNCC seamlessly couples channel coding and network coding to effectively combat the detrimental effect of fading of wireless channels. Specifically, NB-JNCC combines non-binary irregular low-density parity-check (LDPC) channel coding and random linear network coding through iterative joint decoding, which helps to fully exploit the spatial diversity and redundancy residing in both channel codes and network codes. In addition, since it operates over a high order Galois field, NB-JNCC can be directly combined with high order modulation without the need of any bit-to-symbol conversion nor its inverse. Through both analysis and simulation, we demonstrate the significant performance improvement of NB-JNCC over other schemes.
---
paper_title: Joint Network-Channel Coding for the Multiple-Access Relay Channel

paper_content:

We propose to use joint network-channel coding based on turbo codes for the multiple-access relay channel. Such a system can be used for the cooperative uplink for two mobile stations to a base station with the help of a relay. We compare the proposed system with a distributed turbo code for the relay channel and with a system which uses separate network-channel coding for the multiple-access relay channel. Simulation results confirm that the systems with network coding for the multiple-access relay channel gain cooperative diversity compared to the system with the distributed turbo code for the relay channel. Moreover, the results show that joint network-channel coding outperforms separate network-channel coding. The reason for this is that the redundancy which is contained in the transmission of the relay can be exploited more efficiently with joint network-channel coding
---
paper_title: Iterative Network and Channel Decoding for the Two-Way Relay Channel

paper_content:

We introduce an extension of the relay channel that we call two-way relay channel. The two-way relay channel consists of two users which want to communicate to each other with the help of one relay. We consider the time-division two-way relay channel without power control, where the broadcast channels are orthogonalized in time and where the two users and the relay use the same transmission power. We describe a joint network-channel coding method for this channel model, where channel codes are used at both users and a network code is used at the relay. The channel code of one user and the network code form a distributed turbo code which we call turbo network code and which can be iteratively decoded at the other user. Moreover, we conjecture closed expressions for lower bounds for the channel capacities of the time-division relay and two-way relay channel without power control and deliver simulation results of the proposed turbo network code.
---
paper_title: Design of joint network-low density parity check codes based on the EXIT charts

paper_content:

For the multiple-access relay network where two sources communicate with one destination in the presence of one relay, a practical joint network-low density parity check code is designed to help the relay jointly re-encode the messages from both sources. A bilayer extrinsic information transfer chart is developed based on which a design methodology is proposed to iteratively improve the degree distribution of the proposed code. Simulations illustrate that the gap between the convergence threshold and the performance of the coding scheme is less than 0.6dB at BER=10-5.
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paper_title: A Unified Channel-Network Coding Treatment for User Cooperation in Wireless Ad-Hoc Networks

paper_content:

We propose a combined channel-network coding solution for efficient user cooperation in wireless ad-hoc networks that comprise a host of terminals communicating to a common destination. The proposed framework, termed generalized adaptive network coded cooperation or GANCC, addresses the challenge of inter-user outage, which widely persists in practical cooperation scenarios, by adaptively matching code graphs to instantaneous network graphs (topologies). Additionally, GANCC treats channel codes as an integral part of the network code, and in doing so not only extracts the most benefit from these codes but also provides a live example supporting the notion that network codes are generalization of channel codes (as well as source codes).
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paper_title: A network coding approach to cooperative diversity

paper_content:

This paper proposes a network coding approach to cooperative diversity featuring the algebraic superposition of channel codes over a finite field. The scenario under consideration is one in which two ldquopartnersrdquo - node A and node B - cooperate in transmitting information to a single destination; each partner transmits both locally generated information and relayed information that originated at the other partner. A key observation is that node B already knows node A's relayed information (because it originated at node B) and can exploit that knowledge when decoding node A's local information. This leads to an encoding scheme in which each partner transmits the algebraic superposition of its local and relayed information, and the superimposed codeword is interpreted differently at the two receivers i.e., at the other partner and at the destination node, based on their different a priori knowledge. Decoding at the destination is then carried out by iterating between the codewords from the two partners. It is shown via simulation that the proposed scheme provides substantial coding gain over other cooperative diversity techniques, including those based on time multiplexing and signal (Euclidean space) superposition.
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Title: Robust Wireless Network Coding – An Overview

Section 1: Introduction
Description 1: Provide an introduction to the concept of Network Coding (NC), its potential advantages over traditional routing, and its applications and challenges in lossy networks.

Section 2: Fundamentals of Network Coding
Description 2: Explain the basics of Network Coding, including linear NC, matrix representation of transmitted and received packets, and the Random Linear Network Coding Channel (RLNCC) model for lossy networks.

Section 3: Error-Correcting Codes in Projective Spaces
Description 3: Summarize the main contributions on the analysis and design of error-correcting codes, particularly the concept of code design in projective spaces, and describe recent developments in this area.

Section 4: How It Works
Description 4: Describe the theoretical motivation and working principles behind the design of error-correcting codes in projective spaces, using the RLNCC model for illustration.

Section 5: Recent Developments
Description 5: Discuss recent advancements and contributions to the field of error-correcting codes in projective spaces, including new constructions, bounds, and decoding algorithms.

Section 6: Joint Network-Channel Decoding
Description 6: Provide an overview of the joint network-channel decoding approach to improve the reliability of network-coded wireless architectures, emphasizing the cross-layer optimization of network and channel coding.

Section 7: Understanding Joint Network-Channel Decoding
Description 7: Offer a simple example to explain the rationale and benefits of joint network-channel decoding, illustrating its potential gains over separate decoding methods.

Section 8: Recent Results
Description 8: Highlight recent research findings and studies that demonstrate the effectiveness and advantages of joint network-channel code design and decoding in robust and reliable wireless networks.

Section 9: Concluding Remarks
Description 9: Conclude the paper by summarizing the key points discussed, emphasizing the current research interests, and identifying open issues and future directions in the field of robust network-coded wireless architectures.