Luby Transform Research Articles

Enhancing Multi-Cloud Storage: A Hybrid Framework Using Interleaving and LT Codes

This article presents a novel framework for enhancing the integrity and availability of data stored on multi-cloud storage systems. The proposed technique leverages a combination of interleaving and Luby transform (LT) codes, specifically employing either ideal or robust soliton distribution during the LT encoding stage. The proposed technique is rigorously designed, incorporating data preparation facilitated by a cloud broker and culminating in the measurement of key performance indicators (KPIs). A central component of the framework is data processing, which involves interleaving followed by the LT encoding with soliton distribution selection. The performance of the proposed technique is evaluated through data integrity, data availability, and others. The results demonstrate that using the robust soliton distribution offers a pragmatic balance between computational efficiency and robust data integrity, particularly crucial in dynamic multi-cloud environments. Additionally, the proposed technique achieves a high level of data integrity of more than 0.97 and strong resilience with average data availability of 98%. However, further optimization is necessary to address storage overhead, retrieval time, and computational overhead. Despite these considerations, the framework remains significant, especially within the context of confidentiality, integrity, and availability for multi-cloud storage protection. The proposed technique underscores the potential of integrating interleaving and LT codes to revolutionize data storage and access within cloud environments, paving the way for future innovations in the field.

Raptor Codes with LT-LDPC Precoding Performance Analysis and FPGA Optimization

Raptor codes are designed to effectively handle packet loss and erasures as a type of fountain code, capable of generating an unlimited number of encoded symbols from source informations, which is particularly useful in scenarios where the number of transmissions is unknown or variable. Luby Transform (LT) codes, a kind of fountain code encoding method, exhibit linear complexity, and Low-Density Parity-Check (LDPC) codes can be constructed by the generator matrix and the parity-check matrix. The LT-LDPC pre-coding method combines the advantages of both, enhancing encoding and decoding efficiency. Deploying code on Field-Programmable Gate Arrays (FPGAs) involves several steps: synthesis, place and route, and bitstream generation, followed by loading the bitstream onto the FPGA to configure its internal logic according to the coding methods. This paper introduces a method of encoding and decoding Raptor codes, utilizing the intimal LT codes and the peripheral LDPC codes constructed by PEG. It employs a lop-BP decoding method over noise channels based on the Tanner graph for decoding and deploys the optimized algorithm on FPGA to enhance performance.

Augmented LT codes over binary extension fields with increased ratio of full‐degree columns

AbstractThis letter proposes an augmented scheme of LT codes to improve the decoding success rate. The method involves substituting the 1‐s in full‐degree columns of the generator matrix of conventional LT codes with binary extension field elements and simultaneously increasing the ratio of full‐degree columns in the ideal soliton distribution. For non‐full‐degree columns, we retain the nonzero elements as 1‐s in the conventional LT codes generator matrix to preserve computational efficiency. Compared to conventional LT codes, the proposed method enhances the linear independence of the generator matrix, leading to a higher decoding success rate with minimal data packets. Experimental results demonstrate the effectiveness of the method at improving the performance of LT codes, with close to 100% decoding success rate achieved with around 5% data redundancy.

Erasure Codes with a Hierarchical Bundle Structure

This paper presents a proof of the existence of computationally fast probabilistic erasure codes at distance ∈ from being MDS, namely the decoding algorithm is able with high probability to reconstruct the n letter message from any set of (1+∈)n letters. It can either be fixed rate or a rateless LT code [10] in that any number of code letters can be produced and each is produced independently of the others. We also decrease the minimum packet size from many to one letter. The key ingredient is a scheme Hierarchical Bun- dle/Bin (HB) which splits the message into a hierarchy of disjoint bundles and produces coded packets about each bundle. We show a correspondence of this to a particular game having to do with randomly throwing balls into a hierarchy of bins. The “information” that does not over flow from a smaller bin, contributes to the next larger bin that it is contained in. We prove matching upper and lower bounds on the cost of this game and provide the implementation details. This analysis is somewhat analogous to the evolution of the “ripple” in the LT decoding analysis [10]. The bundle size corresponds to the degree of the packet, therefore, smaller bundles tend to reduce encoding/decoding complexity, but packets coming from larger bundles ensure the approximately-MDS constraint, by ensuring more coverage. Our HB scheme with largest block size b requires encoding and decoding time O(∈b2) rather than the O(b2) needed for Reed-Solomon codes. This scheme HB (together with Spielman’s expanders) gives a probabilistic code with running time O(∈-1 ln(∈-1)n). Alon and Luby [5, 6] simultaneously developed a deterministic version but their running time is O(∈-4n). Both our and Alon’s results have since been completely subsumed by the latest generation of Shokrollahi and Luby’s Raptor codes [13, 14, 11].

Optimizing Distribution and Feedback for Short LT Codes with Reinforcement Learning

Optimizing Distribution and Feedback for Short LT Codes with Reinforcement Learning

CH-MAC: Achieving Low-latency Reliable Communication via Coding and Hopping in LPWAN

Wireless sensing has emerged as a powerful environmental sensing technology that is vulnerable to the impact of all kinds of ambient noises. LoRa is a novel interference-resilient technology of low-power wide-area networks (LPWAN), which has attracted wide attention from scientific and industrial communities. However, LoRa transmission suffers from serious latency in those complex wireless sensing environments requiring transmission reliability. In this article, we present CH-MAC, the first MAC-layer protocol based on the local corruption nature of packets and the time-varying nature of channels to reduce end-to-end transmission latency in LPWAN with reliable communication requirements. Specifically, CH-MAC employs Luby Transform code to divide and encode the payload into several blocks such that the receiver can retain part of the coded information in the corrupted packets. In addition, CH-MAC utilizes hopping to transmit different blocks of a packet with various channels to avoid sudden noise collision. Moreover, CH-MAC adopts a dynamic packet length adjustment mechanism to mitigate network congestion. Extensive evaluations on a real-world hardware testbed and a simulation platform show that CH-MAC can reduce end-to-end transmission latency by 2.63× with a communication success rate requirement of >95% compared with state-of-the-art methods.

Analysis and Research on Secondary LT Coding Anti-Eavesdropping Scheme Based on LT Code Degree-1

Fountain code can significantly increase eavesdroppers’ untranslated efficiency in the wireless communication eavesdropping channel. The secondary LT coding anti-eavesdropping scheme with fountain code degree-1 is the subject of a theoretical investigation in this paper. The fact that its channel security capacity is greater than that of traditional LT code is first deduced from an information-theoretic standpoint, and the impact of source symbol length on decoding complexity and decoding overhead is then examined. The experimental results show that, compared with the traditional anti-eavesdropping twice fountain code, selecting long source symbols for double LT coding, when the main channel is better than the eavesdropping channel, can ensure that the eavesdropper has a higher untranslated efficiency, and can effectively reduce the fountain code decoding complexity and the number of encoded symbols sent by the source to improve the efficiency of information transmission.

Towards long double-stranded chains and robust DNA-based data storage using the random code system.

DNA has become a popular choice for next-generation storage media due to its high storage density and stability. As the storage medium of life's information, DNA has significant storage capacity and low-cost, low-power replication and transcription capabilities. However, utilizing long double-stranded DNA for storage can introduce unstable factors that make it difficult to meet the constraints of biological systems. To address this challenge, we have designed a highly robust coding scheme called the "random code system," inspired by the idea of fountain codes. The random code system includes the establishment of a random matrix, Gaussian preprocessing, and random equilibrium. Compared to Luby transform codes (LT codes), random code (RC) has better robustness and recovery ability of lost information. In biological experiments, we successfully stored 29,390 bits of data in 25,700 bp chains, achieving a storage density of 1.78 bits per nucleotide. These results demonstrate the potential for using long double-stranded DNA and the random code system for robust DNA-based data storage.

Basis-Finding Algorithm for Decoding Fountain Codes for DNA-Based Data Storage

In this paper, we consider the decoding of fountain codes where the received symbols may have errors. It is motivated by the application of fountain codes in DNA-based data storage systems where the inner code decoding, which generally has undetectable errors, is performed before the outer fountain code decoding. We propose a novel and efficient decoding algorithm, namely basis-finding algorithm (BFA), followed by three implementations. The key idea of the BFA is to find a basis of the received symbols, and then use the most reliable basis elements to recover the source symbols with the inactivation decoding. Gaussian elimination is used to find the basis and to identify the most reliable basis elements. As a result, the BFA has polynomial time complexity. For random fountain codes, we are able to derive some theoretical bounds for the frame error rate (FER) of the BFA. Extensive simulations with Luby transform (LT) codes show that, the BFA has significantly lower FER than the belief propagation (BP) algorithm except for an extremely large amount of received symbols, and the FER of the BFA generally decreases as the average weight of basis elements increases.

Latency Optimization for Luby Transform Coded Computation in Wireless Networks

Luby transform (LT) code has attracted a significant attention due to its rateless property and low decoding complexity in distributed computing networks. However, for wireless networks, the frequent interaction in LT coded computation can give rise to the latency bottleneck considering the communication overhead of interaction. In this letter, we study a novel LT coded computation scheme for heterogeneous wireless networks. Considering nodes with the disparate computation and transmission capabilities, we propose a block-design based wireless LT coded computation (BD-WLTCC) scheme to balance both computation and transmission latency. First, we pre-allocate the maximum transmission rounds due to the limited bandwidth of wireless channel. Then, we design the sub-block size for each node by formulating a nonconvex optimization problem to avoid the frequent interaction. Finally, the sub-optimal sub-block size is obtained by solving the relaxed problem iteratively. And the optimized results provably reveal the generalization of existing works. Simulation results show the superiority of our scheme.

A Simulink model for modified fountain codes

This paper introduces a Simulink model design for a modified fountain code. The code is a new version of the traditional Luby transform (LT) codes. The design constructs the blocks required for generation of the generator matrix of a limited-degree-hopping-segment Luby transform (LDHS-LT) codes. This code is especially designed for short length data files which have assigned a great interest for wireless sensor networks. It generates the degrees in a predetermined sequence but random generation and partitioned the data file in segments. The data packets selection has been made serialy according to the integer generated from both degree and segment generators. The code is tested using Monte Carlo simulation approach with the conventional code generation using robust soliton distribution (RSD) for degree generation, and the simulation results approve better performance with all testing parameter.

Triangular code: Near-optimal linear time fountain code

In this paper, we propose Triangular Code (TC), a new class of fountain code with near-zero redundancy and linear encoding and decoding computational complexities of O(Lklogk), where k is the packet batch size and L is the packet data length. Different from previous works where the optimal performance of codes has been shown under asymptotic assumption, TC enjoys near-zero redundancy even under non-asymptotic settings for small-moderate number of packets. These features make TC suitable for practical implementation in battery-constrained devices in IoT, D2D and M2M network paradigms to achieve scalable reliability, and minimize latency due to its low decoding delay. TC is a non-linear code, which is encoded using the simple shift and XOR addition operations, and decoded using the simple back-substitution algorithm. Although it is nonlinear code at the packet level, it remains linear code when atomized at the bit level. We use this property to show that the back-substitution decoder of TC is equivalent to the Belief Propagation (BP) decoder of LT code. Therefore, TC can benefit from rich prolific literature published on LT code, to design efficient code for various applications. Despite the equivalency between the decoders of TC and LT code, we show that compared to state-of-the-art optimized LT code, TC reduces the redundancy of LT code by 68%–99% for k reaching 1024.© 2022 Published by Elsevier Ltd.

Shifted LT Code Security Scheme for Partial Information Encryption.

The existing physical layer security technology based on fountain codes needs to ensure that the legal channel is superior to the eavesdropping channel; when the quality of the legal channel and the eavesdropping channel are close, the information security cannot be guaranteed. Aiming at this problem, this paper proposes a shifted Luby transform (SLT) code security scheme for partial information encryption, which is mainly divided into two stages, partial information encryption transfer and degree distribution adjustment. The main idea is that the source randomly extracts part of the information symbols, and performs XOR encryption with the random sequence containing the main channel noise sent by the legitimate receiver. Afterward, the degree distribution is adjusted using the number of transfer information symbols received by the legitimate receiver to improve the average degree of the encoded codewords. Since the eavesdropper can only obtain fewer information symbols in the initial stage, it is difficult to decode the generated coded symbols after the degree distribution adjustment, thereby ensuring the safe transmission of information. The experimental results show that, compared with other LT anti-eavesdropping schemes, even if the legitimate channel is not dominant, the proposed scheme still has better security performance and less decoding overhead.

Performance and time improvement of LT code-based cloud storage

Outsourcing data on cloud storage services has already attracted great attention due to the prospect of rapid data growth and storing efficiencies for customers. The coding-based cloud storage approach can offer a more reliable and faster solution with less storage space in comparison with replication-based cloud storage. LT codes are the famous member of the rateless code family that can improve performance of storage systems utilizing good degree distributions. Since degree distribution plays a key role in LT codes performance, recently introduced Poisson robust soliton distribution (PRSD) and combined Poisson robust soliton distribution (CPRSD) motivate us to investigate LT code-based cloud storage system. So, we exploit LT codes with new degree distributions to provide lower average degree and higher decoding efficiency, specifically when receiving fewer encoding symbols, compared with popular degree distribution, robust soliton distribution (RSD). In this paper, we show that proposed cloud storage outperforms traditional ones in terms of storage space and robustness encountering unavailability of encoding symbols, due to compatible properties of PRSD and CPRSD with cloud storage essence. Furthermore, a modified decoding process is presented based on required encoding symbols behavior to reduce data retrieval time. Numerical results confirm improvement of cloud storage performance.

LT Codes with Double Encoding Matrix Reorder Physical Layer Secure Transmission

In traditional wireless sensor networks, information transmission usually uses data encryption methods to prevent information from being stolen illegally. However, once the encryption methods are leaked, eavesdropping nodes can easily obtain information. LT codes are rateless codes; if it is attacked by random channel noise, the decoding process will change and the decoding overhead will also randomly change. When it is used for physical layer communication of wireless sensor networks, it ensures that the destination node recovers all the information without adding the key, while the eavesdropping node can only obtain part of the information to achieve wireless information security transmission. To reduce the intercept efficiency of eavesdropping nodes, a physical layer security (PLS) method of LT codes with double encoding matrix reorder (DEMR-LT codes) is proposed. This method performs two consecutive LT code concatenated encoding on the source symbol, and part of the encoding matrix is reordered according to the degree value of each column from large to small, which reduces the probability of eavesdropping nodes recovering the source information. Experimental results show that compared with other LT code PLS schemes, DEMR-LT codes only increase the decoding overhead by a small amount. However, it can effectively reduce the intercept efficiency of eavesdropping nodes and improve information transmission security.

Enhanced Distributed Storage System Using Lower Triangular Matrix-Luby Transform Codes

In today’s digital environment, large volume of digital data is created daily and this data accumulates to unforeseen levels. Industries are finding it increasingly difficult to store data in an effective and trustworthy manner. Distributed storage appears to be the greatest approach for meeting current data storage demands at the moment. Furthermore, due to disc crashes or failures, efficient data recovery is becoming an issue. At present, new data storage techniques are required in order to restore data effectively even if some discs or servers are crashed. Hence, this proposed work aims to improve the storage efficiency and reliability in distributed storage systems (DSSs) to withstand disk failures and data erasures by modifying the degree distribution of Luby Transform (LT) codes. Recently, deriving an optimal degree distribution in LT codes becomes an upcoming research field. Hence, in this work, a novel approach called Lower Triangular Matrix (LTM) based degree distribution is presented for LT encoding. Unlike the traditional schemes such as Robust Soliton Distribution (RSD), there is no need for transmitting the neighbor symbols information along with each and every encoded symbol using LTM-LT codes which results in an efficient utilization of the bandwidth. Further, the overhead of LTM encoder is reduced by performing only one XOR operation for the generation of every new encoded symbol. The performance of LTM-LT codes is evaluated in terms of varying data erasure and disk failure conditions. Simulation results show that the proposed LTM-LT codes perform better compared to the traditional schemes such as RSD used for LT codes in Distributed Storage System.

Design of Anti-Eavesdropping Scheme for SLT-LT Codes Based on Random Symbol Sets

In recent years, physical layer security technology has provided a new way to solve the problem of secure transmissions in wireless communications from an information theory perspective by using physical layer security coding and the inherent characteristics of the channel. As a nonsystematic code, Luby transform (LT) code has the characteristics of random coding and rateless, therefore, eavesdroppers cannot directly obtain useful information from leaked code symbols. As a shifted LT (SLT) code, the SLT codes can efficiently recover information, with a smaller decoding overhead. Therefore, we propose an SLT-LT joint code anti-eavesdropping scheme based on random symbol sets. Based on the environment of the main channel, the sender selects a set of random symbols to send to legitimate receivers as known symbols and uses these random symbols and message symbols to form source symbols for the SLT-LT concatenated coding to increase the bit error rate (BER) of the eavesdroppers. The experimental results show that, compared with other anti-eavesdropping LT schemes, our proposed scheme only adds a small amount of decoding overhead but has a better security performance.

Memory-Based LT Codes for Efficient 5G Networks and Beyond

The next-generation networks (5G and beyond) require robust channel codes to support their high specifications, such as low latency, low complexity, significant coding gain, and flexibility. In this paper, we propose using a fountain code as a promising solution to 5G and 6G networks, and then we propose using a modified version of the fountain codes (Luby transform codes) over a network topology (Y-network) that is relevant in the context of the 5G networks. In such a network, the user can be connected to two different cells at the same time. In addition, the paper presents the necessary techniques for analyzing the system and shows that the proposed scheme enhances the system performance in terms of decoding success probability, error probability, and code rate (or overhead). Furthermore, the analyses in this paper allow us to quantify the trade-off between overhead, on the one hand, and the decoding success probability and error probability, on the other hand. Finally, based on the analytical approach and numerical results, our simulation results demonstrate that the proposed scheme achieves better performance than the regular LT codes and the other schemes in the literature.

Factored LT and Factored Raptor Codes for Large-Scale Distributed Matrix Multiplication

We propose two coding schemes for distributed matrix multiplication in the presence of stragglers. These coding schemes are adaptations of Luby Transform (LT) codes and Raptor codes to distributed matrix multiplication and are termed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Factored LT (FLT) codes</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Factored Raptor (FRT) codes</i> . We show that all nodes in the Tanner graph of a randomly sampled code have a tree-like neighborhood with high probability. This ensures that the density evolution analysis gives a reasonable estimate of the average recovery threshold of FLT codes. The recovery threshold of the proposed FLT codes is asymptotically optimal when the output degree distribution is Soliton. Empirically, we show that FRT codes have an excellent recovery threshold while the number of worker nodes is moderately large. In addition, using Azuma–Hoeffding inequality, we derive concentration results to show that the recovery threshold of a randomly chosen FLT code is close to the ensemble average. FLT and FRT codes have better recovery thresholds when compared to Product codes and they are expected to have better numerical stability when compared to Polynomial codes, while they can also be decoded with a low-complexity decoding algorithm. Finally, the proposed codes are better matched to the practically important case of sparse matrix-matrix multiplication as compared to many previous schemes.

NOREC4DNA: using near-optimal rateless erasure codes for DNA storage

BackgroundDNA is a promising storage medium for high-density long-term digital data storage. Since DNA synthesis and sequencing are still relatively expensive tasks, the coding methods used to store digital data in DNA should correct errors and avoid unstable or error-prone DNA sequences. Near-optimal rateless erasure codes, also called fountain codes, are particularly interesting codes to realize high-capacity and low-error DNA storage systems, as shown by Erlich and Zielinski in their approach based on the Luby transform (LT) code. Since LT is the most basic fountain code, there is a large untapped potential for improvement in using near-optimal erasure codes for DNA storage.ResultsWe present NOREC4DNA, a software framework to use, test, compare, and improve near-optimal rateless erasure codes (NORECs) for DNA storage systems. These codes can effectively be used to store digital information in DNA and cope with the restrictions of the DNA medium. Additionally, they can adapt to possible variable lengths of DNA strands and have nearly zero overhead. We describe the design and implementation of NOREC4DNA. Furthermore, we present experimental results demonstrating that NOREC4DNA can flexibly be used to evaluate the use of NORECs in DNA storage systems. In particular, we show that NORECs that apparently have not yet been used for DNA storage, such as Raptor and Online codes, can achieve significant improvements over LT codes that were used in previous work. NOREC4DNA is available on https://github.com/umr-ds/NOREC4DNA.ConclusionNOREC4DNA is a flexible and extensible software framework for using, evaluating, and comparing NORECs for DNA storage systems.