In this paper, a simple, general-purpose and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists in starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance on the memory-less erasure channel, and suitable also for the iterative correction of single bursts of erasures. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm, which executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed for achieving good performance on the memory-less erasure channel, then the code obtained after the application of the PSS algorithm provides good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the effectiveness of the PSS algorithm when applied to different categories of LDPC codes.
In this paper we study array-based codes over graphs for correcting multiple node failures. These codes have applications to neural networks, associative memories, and distributed storage systems. We assume that the information is stored on the edges of a complete undirected graph and a node failure is the event where all the edges in the neighborhood of a given node have been erased. A code over graphs is called $ρ$-node-erasure-correcting if it allows to reconstruct the erased edges upon the failure of any $ρ$ nodes or less. We present a binary optimal construction for double-node-erasure co…
In this paper, we show that the single deletion correcting Varshamov-Tenengolts code, with minor modifications, can also correct an ordered deletion-erasure pattern where one deletion and at most one erasure occur and the deletion always occurs before the erasure. For large code lengths, the constructed code has the same logarithmic redundancy as optimal codes.
This paper proposes an erasure correcting code and its systematic form for the distributed storage system. The proposed codes are encoded by exclusive OR and bit-level shift operation. By the shift operation, the encoded packets are slightly longer than the source packets. This paper evaluates the extra length of encoded packets, called overhead, and shows that the proposed codes have smaller overheads than the zigzag decodable code, which is an existing code using exclusive OR and bit-level shift operation.
Motivated by modern network communication applications which require low latency, we study codes that correct erasures with low decoding delay. We provide a simple explicit construction that yields convolutional codes that can correct both burst and arbitrary erasures under a maximum decoding delay constraint $T$. Our proposed code has efficient encoding/decoding algorithms and requires a field size that is linear in $T$. We study the performance of our code over the Gilbert-Elliot channel; our simulation results show significant performance gains over low-delay codes existing in the literatur…
An interactive error correcting code ($\mathsf{iECC}$) is an interactive protocol with the guarantee that the receiver can correctly determine the sender's message, even in the presence of noise. This generalizes the concept of an error correcting code ($\mathsf{ECC}$), which is a non-interactive $\mathsf{iECC}$ that is known to have erasure resilience capped at $\frac12$. The work of \cite{GuptaTZ21} constructed the first $\mathsf{iECC}$ resilient to $> \frac12$ adversarial erasures. However, their $\mathsf{iECC}$ has communication complexity quadratic in the message size. In our work, we con…
An error correcting code ($\mathsf{ECC}$) allows a sender to send a message to a receiver such that even if a constant fraction of the communicated bits are corrupted, the receiver can still learn the message correctly. Due to their importance and fundamental nature, $\mathsf{ECC}$s have been extensively studied, one of the main goals being to maximize the fraction of errors that the $\mathsf{ECC}$ is resilient to. For adversarial erasure errors (over a binary channel) the maximal error resilience of an $\mathsf{ECC}$ is $\frac12$ of the communicated bits. In this work, we break this $\frac12$…
Optimal codes for correcting a single (wrap-around) burst of errors
2007年，Martinian和Trott提出了一种以最小解码延迟纠正擦除突发的代码。它们的构造采用[n，k]码，可以纠正长度为n-k的任何擦除突发（包括环绕突发）。他们提出了这样一个问题，即这样的[n，k]码是否存在于所有整数k和n（1<= k <= n）以及所有域（特别是二进制域）。本文给出了两个递归构造和一个直接构造，肯定地回答了这个问题。
An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the require…
Despite their exceptional error-correcting properties, Reed-Solomon codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth: A naive repair approach would require the whole file to be reconstructed in order to recover a single erased codeword symbol. In a recent work, Guruswami and Wootters (STOC'16) proposed a single-erasure repair method for Reed-Solomon codes that achieves the optimal repair bandwidth amongst all linear encoding schemes. Their key idea is to recover the erased symbol by collecting a sufficiently large numb…
We derive the probability distribution of the link outage duration at a typical receiver in a wireless network with Poisson distributed interferers sending messages with slotted random access over a Rayleigh fading channel. This result is used to analyze the performance of random linear network coding, showing that there is an optimum code rate and that interference correlation affects the decoding probability and throughput.
This paper presents a new construction of error correcting codes which achieves optimal recovery of a streaming source over a packet erasure channel. The channel model considered is the sliding window erasure model, with burst and arbitrary losses, introduced by Badr et al. . Recently, two independents works by Fong et al. and Krishnan and Kumar have identified optimal streaming codes within this framework. In this paper, we introduce streaming code when the rate of the code is at least 1/2. Our proposed construction is explicit and systematic, uses off-the-shelf maximum distance separable (MD…
In the setting of error-correcting codes with feedback, Alice wishes to communicate a $k$-bit message $x$ to Bob by sending a sequence of bits over a channel while noiselessly receiving feedback from Bob. It has been long known (Berlekamp, 1964) that in this model, Bob can still correctly determine $x$ even if $\approx \frac13$ of Alice's bits are flipped adversarially. This improves upon the classical setting without feedback, where recovery is not possible for error fractions exceeding $\frac14$. The original feedback setting assumes that after transmitting each bit, Alice knows (via feedbac…
A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.
In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$. One needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which gro…
This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t \log n - \log(t!)$. Then, we present an existential construction of $t$-criss…
We describe a replacement for RAID 6, based on a new linear, systematic code, which detects and corrects any combination of $E$ errors (unknown location) and $Z$ erasures (known location) provided that $Z+2E \leq 4$. We investigate some scenarios for error correction beyond the code's minimum distance, using list decoding. We describe a decoding algorithm with quasi-logarithmic time complexity, when parallel processing is used: $\approx O(\log N)$ where $N$ is the number of disks in the array (similar to RAID 6). By comparison, the error correcting code implemented by RAID 6 allows error detec…
Random jammers that overpower transmitted signals are a practical concern for many wireless communication protocols. As such, wireless receivers must be able to cope with standard channel noise and jamming (intentional or unintentional). To address this challenge, we propose a novel method to augment the resilience of the recent family of universal error-correcting GRAND algorithms. This method, called Erasure Decoding by Gaussian Elimination (EDGE), impacts the syndrome check block and is applicable to any variant of GRAND. We show that the proposed EDGE method naturally reverts to the origin…
Partial MDS (PMDS) and sector-disk (SD) codes are classes of erasure correcting codes that combine locality with strong erasure correction capabilities. We construct PMDS and SD codes with local regeneration where each local code is a bandwidth-optimal regenerating MDS code. In the event of a node failure, these codes reduce both, the number of servers that have to be contacted as well as the amount of network traffic required for the repair process. The constructions require significantly smaller field size than the only other construction known in literature. Further, we present a constructi…
