Random Linear Coding Research Articles

Weighted partial network coding and its applications in wireless mesh networks

ABSTRACTNetwork coding (NC) has showed to be beneficial to improve transmission performance in wireless mesh networks. Random linear coding is usually applied as the default coding schema. However, random linear coding causes significant decoding delay and jitter at receiver. Further, current NC does not support weight assignment to original packets, which is however indispensable for popular applications such as quality of service control and multipath media streaming in wireless mesh networks. Partial network coding (PNC) can largely reduce decoding delay and receiving fluctuation while keeping the benefit of NC. However, PNC does not support weight‐based data replacement and weight assignment to original packets. In this work, we propose weighted partial network coding (WPNC), which is a generalized coding schema of PNC. WPNC inherits all merits of PNC and part of NC. With WPNC, both decoding delay and receiving fluctuation will be reduced as observed in PNC. Also, WPNC is quite suitable for those applications that require weight assignment to original packets. After providing the whole framework of WPNC and thorough theoretical analysis to its performance, we have demonstrated how WPNC can be integrated with quality of service control and multipath routing supported media streaming in wireless mesh networks. Performance of WPNC is inter‐validated by both theoretical analysis and numeric evaluations. Copyright © 2011 John; Wiley & Sons, Ltd.

ComboCoding: Combined intra-/inter-flow network coding for TCP over disruptive MANETs

TCP over wireless networks is challenging due to random losses and ACK interference. Although network coding schemes have been proposed to improve TCP robustness against extreme random losses, a critical problem still remains of DATA–ACK interference. To address this issue, we use inter-flow coding between DATA and ACK to reduce the number of transmissions among nodes. In addition, we also utilize a “pipeline” random linear coding scheme with adaptive redundancy to overcome high packet loss over unreliable links. The resulting coding scheme, ComboCoding, combines intra-flow and inter-flow coding to provide robust TCP transmission in disruptive wireless networks. The main contributions of our scheme are twofold; the efficient combination of random linear coding and XOR coding on bi-directional streams (DATA and ACK), and the novel redundancy control scheme that adapts to time-varying and space-varying link loss. The adaptive ComboCoding was tested on a variable hop string topology with unstable links and on a multipath MANET with dynamic topology. Simulation results show that TCP with ComboCoding delivers higher throughput than with other coding options in high loss and mobile scenarios, while introducing minimal overhead in normal operation.

Fountain Codes With PAPR Constraint for Multicast Communications

The Forward Link Only (FLO) merit of fountain codes is utilized to avoid excessive retransmission requests for multicast communications. Two important and basic classes of fountain codes, Random Linear (RL) codes and Luby Transform (LT) codes, are considered in this paper as application layer Forward Error Correction (FEC) codes. High Peak-to-Average Power Ratio (PAPR) is a notorious physical layer problem in OFDM systems. We take the PAPR constraint into account in the encoding of fountain codes for OFDM multicast communications. We propose a selective mapping (SLM) technique that selects the best P candidates among the QP candidates. This SLM technique performs significant better than the conventional SLM technique that selects the best candidate among the Q candidates. Mathematical analysis and computer simulation verify the excellence of our designs.

Multicast Queueing Delay: Performance Limits and Order-Optimality of Random Linear Coding

In this work we analyze the average queue backlog for transmission of a single multicast flow consisting of M destination nodes in a wireless network. In the model we consider, the channel between every pair of nodes is an independent identically distributed packet erasure channel. We first develop a lower bound on the average queue backlog achievable by any transmission strategy; for a single-hop multicast transmission, our bound indicates that the queue size must scale as at least Ω(ln(M)). Next, we generalize this result to a multihop network and obtain a lower bound on the queue backlog as it relates to the minimum-cut capacity of the network. We then analyze the queue backlog for a strategy in which random linear coding is performed over groups of packets in the queue at the source node of a single-hop multicast. We develop an upper bound on the average queue backlog for the packet-coding strategy to show that the queue size for this strategy scales as O(ln(M)). Our results demonstrate that in terms of the queue backlog for single-hop multicast, the packet coding strategy is order-optimal with respect to the number of receivers.

Diversity Embedded Streaming Erasure Codes (DE-SCo): Constructions and Optimality

Streaming erasure codes encode a source stream to guarantee that each source symbol is recovered within a fixed delay at the receiver over a burst-erasure channel. This paper introduces diversity embedded streaming erasure codes (DE-SCo), that provide a flexible tradeoff between the channel quality and receiver delay. When the channel conditions are good, the source stream is recovered with a low delay, whereas when the channel conditions are poor the source stream is still recovered, albeit with a larger delay. Information theoretic analysis of the underlying burst-erasure broadcast channel reveals that DE-SCo achieve the minimum possible delay for the weaker user, without sacrificing the performance of the stronger user. Our constructions are explicit, incur polynomial time encoding and decoding complexity and outperform random linear codes over bursty erasure channels.

Downloader-Initiated Random Linear Network Coding for Peer-to-Peer File Sharing

Network coding has recently drawn tremendous attention. Applying network coding to Peer-to-Peer (P2P) file sharing is still a relatively new area of study. Adopting network coding in a P2P file sharing network can make data distribution more effective and relieve the burden of the original seeder. In this paper, we first provide the mathematical analysis modeled from contemporary P2P file sharing protocols to analyze the efficiency of network coding in P2P file sharing networks. Then, we further investigate the “unlucky combination” problem encountered by the traditional random linear network coding algorithm when applied to a P2P file sharing network. Hence, we propose a “Downloader-Initiated Random Linear Network Coding” algorithm, which lets the downloader determine the random linear coding to eliminate the “unlucky combination” problem, and therefore further improves the performance of P2P file sharing networks.

Limits to List Decoding of Random Codes

It has been known since [Zyablov and Pinsker, 1982] that a random q-ary code of rate 1 - H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (ρ) - ε (where 0 <; ρ <; 1 - 1/q, ε >; 0 is small enough and -H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (·) is the g-ary entropy function) with high probability is a (ρ, 1/ε) -list decodable code (that is, every Hamming ball of radius at most pn has at most 1/ε codewords in it). In this paper, the "converse" result is proven. In particular, it is proven that for every 0 <; ρ <; 1 - 1/q, a random code of rate 1 - H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (ρ) - ε, with high probability, is not a (ρ, L)-list decodable code for any L ≤ c/ε, where c is some constant that depends only on ρ and q. A similar lower bound is also shown for random linear codes. Previously, such a tight lower bound on the list size was only known for the case when ρ ≥ 1 - 1/q O(√ε) for small enough ε >; 0 [Blinovsky, 1986, 2005, 2008; Guruswami and Vadhan, 2005]. A lower bound is known for all constant 0<;ρ<;1 - 1/q independent of ε, though the lower bound is asymptotically weaker than our bound [Blinovsky, 1986, 2005, 2008]. These results, however, are not subsumed by ours as these other results hold for arbitrary codes of rate 1 - H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (ρ) - ε.

On the List-Decodability of Random Linear Codes

On the List-Decodability of Random Linear Codes

Stable Throughput for Multicast With Random Linear Coding

This paper compares scheduling and coding strategies for a multicast version of a classic downlink problem. We consider scheduling strategies where, in each time slot, a scheduler observes the lengths of all queues and the connectivities of all links and can transmit the head-of-the-line packet from a single queue. We juxtapose this to a coding strategy that is simply a form of classical random linear coding. We show that there are configurations for which the stable throughput region of the scheduling strategy is a strict subset of the corresponding throughput region of the coding strategy. This analysis is performed for both time-invariant and time-varying channels. The analysis is also performed both with and without accounting for the impact on throughput of including coding overhead symbols in each encoded packet. Additionally, we compare coding strategies that only code within individual queues against a coding strategy that codes across separate queues. The strategy that codes across queues simply sends packets from all queues to all receivers. As a result, this strategy sends many packets to unnecessary recipients. We show, surprisingly, that there are cases where the strategy that codes across queues can achieve the same throughput region achievable by coding within individual queues.

Toward Increasing Packet Diversity for Relaying LT Fountain Codes in Wireless Sensor Networks

Diversity is a powerful means to increase the transmission performance of wireless communications. For the case of fountain codes relaying, it has been shown previously that introducing diversity is also beneficial since it counteracts transmission losses on the channel. Instead of simply hop-by-hop forwarding information, each sensor node diversifies the information flow using XOR combinations of stored packets. This approach has been shown to be efficient for random linear fountain codes. However, random linear codes exhibit high decoding complexity. In this paper, we propose diversity increased relaying strategies for the more realistic and lower complexity Luby Transform code in a linear network. Results are provided herein for a linear network assuming uniform imperfect channel states.

Buffer Asymptotics for Coding Over Networks

Traditionally, network buffer resources have been used at routers to queue transient packets to prevent packet drops. In contrast, we propose a scheme for large multihop networks where intermediate routers have no buffers for queueing transient packets. In the proposed scheme, network storage resources (memory) are used only at source and destination nodes to encode/decode packets using random linear coding over time. Our scheme capitalizes on the common empirical observation that for large networks with many flows through each router, if packet loss occurs in a flow path, it will very likely occur only at only a very few links on the path. Unfortunately, the location of this congested link varies with time, thereby preventing prevailing static buffer allocation strategies from exploiting this observation. We propose source coding over packets at the session layer as a means of “sharing” memory across links along a flow path. We call this spatial buffer multiplexing-where buffering and coding implemented at the source compensates for packet loss at any downstream bufferless link. First, we consider large spatial multihop networks with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> nodes (each with finite buffer space for coding/decoding) and Θ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) flows; the number of flows through each link scales as Ω( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> ) for some α ∈ (0,1). Using many-sources large deviations analysis, we show that to obtain comparable packet drop probabilities (QoS), spatial buffer multiplexing provides an order-wise buffer gain of Ω( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> ) per node over traditional static buffer allocation for queueing. Next we consider the complementary case of a network with a small number of flows through a buffer, but large source buffer. Here, we provide a sufficient condition under which the packet loss probability decreases exponentially in a function that is linear in the size of the input buffer, where the function is required to meet a predetermined negative slope -δ. We express the loss effective bandwidth for coding and compare it against that for queueing.

FTS: A Distributed Energy-Efficient Broadcasting Scheme Using Fountain Codes for Multihop Wireless Networks

We investigate the problem of reliable and energy-efficient one-to-all broadcasting in multihop wireless networks, and propose fractional transmission scheme (FTS) - a low-complexity and scalable broadcasting scheme. FTS exploits the broadcasting nature of wireless channels and random encoding of rateless codes to reduce energy consumption while ensuring reliable delivery of packets to all nodes in the network. In the proposed scheme, different neighbors of a node share the responsibility of transmitting packets by sending only a fraction of encoded packets required by the node to successfully receive the data sent by the source. A detailed analysis of the performance of FTS is presented for grid and random deployment networks. Further, extensive simulations compare our scheme with present energy-efficient methods such as random linear coding, multipoint relaying, dominant pruning, and broadcast incremental power scheme. Simulations reveal that FTS offers good performance and adaptability at a low computational cost.

Secrecy coding for the binary symmetric wiretap channel

AbstractIn this paper, we investigate the binary symmetric wiretap channel. We show that the secrecy capacitycan be achieved by using random linear codes. The random coding scheme gives insight into the structure of secrecy capacity achieving code but unfortunately involves a rather impractical decoder. We further explore the coset‐coding schemeconstructed by linear codes. As a result, we give an upper bound on the total information loss, which sheds light on the design of the applicable coset codes for the secure transmission with limited information leakage. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Practical entanglement distillation scheme using recurrence method and quantum low density parity check codes

Many entanglement distillation schemes use either universal random hashing or breeding as their final step to obtain almost perfect shared EPR pairs. In spite of a high yield, the hardness of decoding a random linear code makes the use of random hashing and breeding infeasible in practice. In this pilot study, we analyze the performance of the recurrence method, a well-known entanglement distillation scheme, with its final random hashing or breeding procedure being replaced by various efficiently decodable quantum codes. Among all the replacements investigated, the one using a certain adaptive quantum low density parity check (QLDPC) code is found to give the highest yield for Werner states over a wide range of noise level—the yield for using this QLDPC code is higher than the first runner up by more than 25% over a wide parameter range. In this respect, the effectiveness of using QLDPC codes in practical entanglement distillation is illustrated.

On Correctable Errors of Binary Linear Codes

The error correction capability of binary linear codes with minimum distance decoding, in particular the number of correctable/uncorrectable errors, is investigated for general linear codes and the first-order Reed-Muller codes. For linear codes, a lower bound on the number of uncorrectable errors is derived. The bound for uncorrectable errors with a weight of half the minimum distance asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. For the first-order Reed-Muller codes, the number of correctable/uncorrectable errors with a weight of half the minimum distance plus one is determined. This result is equivalent to deriving the number of Boolean functions of m variables with nonlinearity 2 m-2+1 . The monotone error structure and its related notions larger half and trial set, which were introduced by Helleseth, Klove, and Levenshtein, are mainly used to derive the results.

Parallel and Concurrent Security of the HB and HB+ Protocols

Hopper and Blum (Asiacrypt 2001) and Juels and Weis (Crypto 2005) recently proposed two shared-key authentication protocols—HB and HB+, respectively—whose extremely low computational cost makes them attractive for low-cost devices such as radio-frequency identification (RFID) tags. The security of these protocols is based on the conjectured hardness of the “learning parity with noise” (LPN) problem, which is equivalent to the problem of decoding random binary linear codes. The HB protocol is proven secure against a passive (eavesdropping) adversary, while the HB+ protocol is proven secure against active attacks.

Weight distribution moments of random linear/coset codes

A simple method is proposed for the calculation of moments of the weight function of a random linear code. Applying this method we compute the exact expression for the third-order covariance of the weight function. The results are also extended to random coset codes.

Minimum distance and convergence analysis of hamming-accumulate-accumulate codes

In this letter we consider the ensemble of codes formed by the serial concatenation of a Hamming code and two accumulate codes. We show that this ensemble is asymptotically good, in the sense that most codes in the ensemble have minimum distance growing linearly with the block length. Thus, the resulting codes achieve high minimum distances with high probability, about half or more of the minimum distance of a typical random linear code of the same rate and length in our examples. The proposed codes also show reasonably good iterative convergence thresholds, which makes them attractive for applications requiring high code rates and low error rates, such as optical communications and magnetic recording.

Priority Random Linear Codes in Distributed Storage Systems

Node churn and failures exist as fundamental characteristics in both peer-to-peer (P2P) and sensor networks. Peers in P2P networks are highly dynamic, whereas sensors are not dependable. As such, maintaining the persistence of periodically measured data in a scalable fashion has become a critical challenge in such systems, without the use of centralized servers. To better cope with node dynamics and failures, we propose priority random linear codes (RLCs), as well as their affiliated predistribution protocols, to maintain measurement data in different priorities, such that critical data have a higher opportunity to survive node failures than data of less importance. A salient feature of priority RLCs is the ability to partially recover more important subsets of the original data with higher priorities, when it is not feasible to recover all of them due to node dynamics. We present extensive analytical and experimental results to show the effectiveness of priority RLCs.

On lattices, learning with errors, random linear codes, and cryptography

Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size Õ( n 2 ) and encrypting a message increases its size by a factor of Õ( n ) (in previous cryptosystems these values are Õ( n 4 ) and Õ( n 2 ), respectively). In fact, under the assumption that all parties share a random bit string of length Õ( n 2 ), the size of the public key can be reduced to Õ( n ).