To provide reliable communication in data transmission, ability of correcting errors is of prime importance. This paper intends to suggest an easy algorithm to detect and correct errors in transmission codes using the well-known Karnaugh map. Referring to past research done and proving new theorems and also using a suggested simple technique taking advantage of the easy concept of Karnaugh map, we offer an algorithm to reduce the number of occupied squares in the map and therefore, reduce substantially the execution time for placing data bits in Karnaugh map. Based on earlier papers, we first propose an algorithm for correction of two simultaneous errors in a code. Then, defining specifications for empty squares of the map, we limit the choices for selection of new squares. In addition, burst errors in sending codes is discussed, and systematically code words for correcting them will be made.
In this work, we investigate the problem of constructing codes capable of correcting two deletions. In particular, we construct a code that requires redundancy approximately 8 log n + O(log log n) bits of redundancy, where n is the length of the code. To the best of the author's knowledge, this represents the best known construction in that it requires the lowest number of redundant bits for a code correcting two deletions.
When digital data are transmitted over a noisy channel, it is important to have a mechanism allowing recovery against a limited number of errors. Normally, a user string of 0's and 1's, called bits, is encoded by adding a number of redundant bits to it. When the receiver attempts to reconstruct the original message sent, it starts by examining a possibly corrupted version of the encoded message, and then makes a decision. This process is called the decoding. The purpose of this course is giving an introduction to the theory and practice of error-correcting codes.
In this paper, we construct systematic $q$-ary two-deletion correcting codes and burst-deletion correcting codes, where $q\geq 2$ is an even integer. For two-deletion codes, our construction has redundancy $5\log n+O(\log q\log\log n)$ and has encoding complexity near-linear in $n$, where $n$ is the length of the message sequences. For burst-deletion codes, we first present a construction of binary codes with redundancy $\log n+9\log\log n+γ_t+o(\log\log n)$ bits $(γ_t$ is a constant that depends only on $t)$ and capable of correcting a burst of at most $t$ deletions, which improves the Lenz-P…
This paper studies codes that correct bursts of deletions. Namely, a code will be called a $b$-burst-deletion-correcting code if it can correct a deletion of any $b$ consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically $\log(n)+b-1$, the redundancy of the best code construction by Cheng et al. is $b(\log (n/b+1))$. In this paper we close on this gap and provide codes with redundancy at most $\log(n) + (b-1)\log(\log(n)) +b -\log(b)$. We also derive a non-asymptotic upper bound on the size of $b$-burst-deletion-correcting codes an…
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 $\frac12$…
In this paper we describe a new error-correcting code (ECC) inspired by the Naccache-Stern cryptosystem. While by far less efficient than Turbo codes, the proposed ECC happens to be more efficient than some established ECCs for certain sets of parameters. The new ECC adds an appendix to the message. The appendix is the modular product of small primes representing the message bits. The receiver recomputes the product and detects transmission errors using modular division and lattice reduction.
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…
We consider the problem of constructing binary codes for correcting deletions that are localized within certain parts of the codeword that are unknown a priori. The model that we study is when $δ\leq w$ deletions are localized in a window of size $w$ bits. These $δ$ deletions do not necessarily occur in consecutive positions, but are restricted to the window of size $w$. The localized deletions model is a generalization of the bursty model, in which all the deleted bits are consecutive. In this paper, we construct new explicit codes for the localized model, based on the family of Guess & Check…
In this paper, two moment balancing schemes, namely a variable index scheme and a fixed index scheme, for either single insertion/deletion error correction or multiple substitution error correction are introduced for coded sequences originally developed for correcting substitution errors only. By judiciously flipping bits of the original substitution error correcting code word, the resulting word is able to correct either a reduced number of substitution errors or a single insertion/deletion error. The number of flips introduced by the two schemes can be kept small compared to the code length.…
Racetrack memory is a non-volatile memory engineered to provide both high density and low latency, that is subject to synchronization or shift errors. This paper describes a fast coding solution, in which delimiter bits assist in identifying the type of shift error, and easily implementable graph-based codes are used to correct the error, once identified. A code that is able to detect and correct double shift errors is described in detail.
This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n\times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $(t_r,t_c)$-criss-cross deletions if $t_r$ rows and $t_c$ columns are deleted, while a code correcting these deletion patterns is called a $(t_r,t_c)$-criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when $t_r=t_c$ the problems of correcting criss-cross deletions and criss-cross insertions are equival…
An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this paper, we investigate codes that combat either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with log n + O(log log n) redundancy bits, while the other corrects a single indel with log n + 2 redundant bits. These two encoders are order-optimal. The former encoder is the first known order-optimal e…
Assuming we are in a Word-RAM model with word size $w$, we show that we can construct in $o(w)$ time an error correcting code with a constant relative positive distance that maps numbers of $w$ bits into $Θ(w)$-bit numbers, and such that the application of the error-correcting code on any given number $x\in[0,2^w-1]$ takes constant time. Our result improves on a previously proposed error-correcting code with the same properties whose construction time was exponential in $w$.
Modern applications are driving demand for ultra-reliable low-latency communications, rekindling interest in the performance of short, high-rate error correcting codes. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can decode any short, high-rate block-code. For a code of $n$ bits, it avails of no more than $\lceil\log_2(n)\rceil$ bits of code-book-independent quantized soft detection information per received bit to determine an accurate decoding while retaining the original algorithm's suit…
Correcting insertions/deletions as well as substitution errors simultaneously plays an important role in DNA-based storage systems as well as in classical communications. This paper deals with the fundamental task of constructing codes that can correct a single insertion or deletion along with a single substitution. A non-asymptotic upper bound on the size of single-deletion single-substitution correcting codes is derived, showing that the redundancy of such a code of length $n$ has to be at least $2 \log n$. The bound is presented both for binary and non-binary codes while an extension to sin…
Construction of capacity achieving deletion correcting codes has been a baffling challenge for decades. A recent breakthrough by Brakensiek $et~al$., alongside novel applications in DNA storage, have reignited the interest in this longstanding open problem. In spite of recent advances, the amount of redundancy in existing codes is still orders of magnitude away from being optimal. In this paper, a novel approach for constructing binary two-deletion correcting codes is proposed. By this approach, parity symbols are computed from indicator vectors (i.e., vectors that indicate the positions of ce…
The design of block codes for short information blocks (e.g., a thousand or less information bits) is an open research problem that is gaining relevance thanks to emerging applications in wireless communication networks. In this paper, we review some of the most promising code constructions targeting the short block regime, and we compare them with both finite-length performance bounds and classical error-correction coding schemes. The work addresses the use of both binary and high-order modulations over the additive white Gaussian noise channel. We will illustrate how to effectively approach …
We compare the performance of short-length linear binary codes on the binary erasure channel and the binary-input Gaussian channel. We use a universal decoder that can decode any linear binary block code: Gaussian-elimination based Maximum-Likelihood decoder on the erasure channel and probabilistic Ordered Statistics Decoder on the Gaussian channel. As such we compare codes and not decoders. The word error rate versus the channel parameter is found for LDPC, Reed-Muller, Polar, and BCH codes at length 256 bits. BCH codes outperform other codes in absence of cyclic redundancy check. Under joint…
In diffusion based molecular communication, the intersymbol interference (ISI) is an important reason for system performance degradation, which is caused by the random movement, out-of-order arrival and indistinguishability of the moleclues. In this paper, a new metric called crossover distance is introduced to measure the distance between the received bit sequence and the probably tranmitted bit sequences. A new decoding scheme of conventional codes is proposed based on crossover distance, which can enhance the communication reliability significantly. The theoretic analysis indicates that the…
Due to its high data density and longevity, DNA is considered a promising medium for satisfying ever-increasing data storage needs. However, the diversity of errors that occur in DNA sequences makes efficient error-correction a challenging task. This paper aims to address simultaneously correcting two types of errors, namely, short tandem duplication and substitution errors. We focus on tandem repeats of length at most 3 and design codes for correcting an arbitrary number of duplication errors and one substitution error. Because a substituted symbol can be duplicated many times (as part of sub…
Order-Optimal Systematic Permutation Codes for Correcting t Deletions
Beijing University of Posts and Telecommunications（北京邮电大学）；VinUniversity（文林大学）；Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications（北京邮电大学通用无线通信教育部重点实验室）；National University of Singapore（新加坡国立大学）
In order to achieve fault tolerance, highly reliable system often require the ability to detect errors as soon as they occur and prevent the speared of erroneous information throughout the system. Thus, the need for codes capable of detecting and correcting byte errors are extremely important since many memory systems use b-bit-per-chip organization. Redundancy on the chip must be put to make fault-tolerant design available. This paper examined several methods of computer memory systems, and then a proposed technique is designed to choose a suitable method depending on the organization of memo…
Error correction techniques traditionally focus on the co-design of restricted code-structures in tandem with code-specific decoders that are computationally efficient when decoding long codes in hardware. Modern applications are, however, driving demand for ultra-reliable low-latency communications (URLLC), rekindling interest in the performance of shorter, higher-rate error correcting codes, and raising the possibility of revisiting universal, code-agnostic decoders. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reli…
The subject of this paper is transmission over a general class of binary-input memoryless symmetric channels using error correcting codes based on sparse graphs, namely low-density generator-matrix and low-density parity-check codes. The optimal (or ideal) decoder based on the posterior measure over the code bits, and its relationship to the sub-optimal belief propagation decoder, are investigated. We consider the correlation (or covariance) between two codebits, averaged over the noise realizations, as a function of the graph distance, for the optimal decoder. Our main result is that this cor…
In this paper, we consider quantized decoding of LDPC codes on the binary symmetric channel. The binary message passing algorithms, while allowing extremely fast hardware implementation, are not very attractive from the perspective of performance. More complex decoders such as the ones based on belief propagation exhibit superior performance but lead to slower decoders. The approach in this paper is to consider message passing decoders that have larger message alphabet (thereby providing performance improvement) as well as low complexity (thereby ensuring fast decoding). We propose a class of …
Substring Edit Correcting Codes and Optimal Single Burst-Deletion Correcting Codes
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与 interdisciplinary 科学研究院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
The energy used to transmit a single bit of data between the devices in wireless networks is equal to the energy for performing hundreds of instructions in those devices. Thus the reduction of the data necessary to transmit, while keeping the same functionality of the employed algorithms is a formidable and challenging scientific task. We describe an algorithm called Small Set of Allowed Coefficients (SSAC) that produces the shortest header overhead in random linear network coding schemes compared with all other approaches reported in the literature. The header overhead length is 2 to 7 times …
Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum informat…
Convolutional codes are error-correcting linear codes that utilize shift registers to encode. These codes have an arbitrary block size and they can incorporate both past and current information bits. DNA codes represent DNA sequences and are defined as sets of words comprised of the alphabet A, C, T, G satisfying certain mathematical bounds and constraints. The application of convolutional code models to DNA codes is a growing field of biocomputation. As opposed to block codes, convolutional codes factor in nearby information bits, which makes them an optimal model for representing biological …
The error pattern correcting code (EPCC) can be constructed to provide a syndrome decoding table targeting the dominant error events of an inter-symbol interference channel at the output of the Viterbi detector. For the size of the syndrome table to be manageable and the list of possible error events to be reasonable in size, the codeword length of EPCC needs to be short enough. However, the rate of such a short length code will be too low for hard drive applications. To accommodate the required large redundancy, it is possible to record only a highly compressed function of the parity bits of …
We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{Ω(1)}$. For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $…
A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of O(1)-query LDCs have super-polynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of ne…
Polar codes, as the first provable capacity-achieving error-correcting codes, have received much attention in recent years. However, the decoding performance of polar codes with traditional successive-cancellation (SC) algorithm cannot match that of the low-density parity-check (LDPC) or turbo codes. Because SC list (SCL) decoding algorithm can significantly improve the error-correcting performance of polar codes, design of SCL decoders is important for polar codes to be deployed in practical applications. However, because the prior latency reduction approaches for SC decoders are not applicab…
Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes and provide several CSS-type code constructions of quantum DS codes.
Staircase codes play an important role as error-correcting codes in optical communications. In this paper, a low-complexity method for resolving stall patterns when decoding staircase codes is described. Stall patterns are the dominating contributor to the error floor in the original decoding method. Our improvement is based on locating stall patterns by intersecting non-zero syndromes and flipping the corresponding bits. The approach effectively lowers the error floor and allows for a new range of block sizes to be considered for optical communications at a certain rate or, alternatively, a s…
The $m$-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). $m$-spotty byte error control codes are very suitable for burst correction. M. Özen and V. Siap [7] proved a MacWilliams identity for the $m$-spotty Rosenbloom-Tsfasman (shortly RT) weight enumerators of binary codes. The main purpose of this paper is to present the MacWilliams type identities for $m$-spotty RT weight enumerators of linear codes over finite commutative Frobenius rings.
Texas A&M University（德州农工大学）；University of California, San Diego（加州大学圣地亚哥分校）
When neural networks (NeuralNets) are implemented in hardware, their weights need to be stored in memory devices. As noise accumulates in the stored weights, the NeuralNet's performance will degrade. This paper studies how to use error correcting codes (ECCs) to protect the weights. Different from classic error correction in data storage, the optimization objective is to optimize the NeuralNet's performance after error correction, instead of minimizing the Uncorrectable Bit Error Rate in the protected bits. That is, by seeing the NeuralNet as a function of its input, the error correction schem…
Viterbi Algorithm Decoder Enhanced with Non-transmittable Codewords is one of the best decoding algorithm which effectively improves forward error correction performance. HoweverViterbi decoder enhanced with NTCs is not yet designed to work in storage media devices. Currently Reed Solomon (RS) Algorithm is almost the dominant algorithm used in correcting error in storage media. Conversely, recent studies show that there still exist low reliability of data in storage media while the demand for storage media increases drastically. This study proposes a design of the Soft Viterbi Algorithm decode…
