scholarly journals Hamming Codes: Error Reducing Techniques

2021 ◽  
Vol 9 (11) ◽  
pp. 1972-1974
Author(s):  
Rohitkumar R Upadhyay
Keyword(s):  
Lower Bound ◽  
Error Correction ◽  
Coding Theory ◽  
Hamming Distance ◽  
Error Reduction ◽  
Error Correcting Codes ◽  
Significant Other ◽  
Hamming Codes ◽  
Decoding Algorithms ◽  
Reducing Properties

Abstract: Hamming codes for all intents and purposes are the first nontrivial family of error-correcting codes that can actually correct one error in a block of binary symbols, which literally is fairly significant. In this paper we definitely extend the notion of error correction to error-reduction and particularly present particularly several decoding methods with the particularly goal of improving the error-reducing capabilities of Hamming codes, which is quite significant. First, the error-reducing properties of Hamming codes with pretty standard decoding definitely are demonstrated and explored. We show a sort of lower bound on the definitely average number of errors present in a decoded message when two errors for the most part are introduced by the channel for for all intents and purposes general Hamming codes, which actually is quite significant. Other decoding algorithms are investigated experimentally, and it generally is definitely found that these algorithms for the most part improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of for all intents and purposes standard decoding. Keywords: coding theory, hamming codes, hamming distance

scholarly journals Iterative Soft-Decision Decoding of Binary Cyclic Codes

2008 ◽  
Vol 4 (2) ◽  
pp. 142 ◽  
Author(s):  
Marco Baldi ◽  
Giovanni Cancellieri ◽  
Franco Chiaraluce
Keyword(s):  
Error Correction ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Parity Check ◽  
Soft Decision ◽  
Coding Gain ◽  
New Approach ◽  
Decoding Algorithms ◽  
Soft Decision Decoding ◽  
Hard Decision

Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.

scholarly journals Deep Neural Networks Classification via Binary Error-Detecting Output Codes

2021 ◽  
Vol 11 (8) ◽  
pp. 3563
Author(s):  
Martin Klimo ◽  
Peter Lukáč ◽  
Peter Tarábek
Keyword(s):  
Neural Networks ◽  
Categorical Data ◽  
Coding Theory ◽  
Hamming Distance ◽  
Activation Function ◽  
Block Codes ◽  
Ease Of Use ◽  
Linear Block Codes ◽  
Binary Coding ◽  
The One

One-hot encoding is the prevalent method used in neural networks to represent multi-class categorical data. Its success stems from its ease of use and interpretability as a probability distribution when accompanied by a softmax activation function. However, one-hot encoding leads to very high dimensional vector representations when the categorical data’s cardinality is high. The Hamming distance in one-hot encoding is equal to two from the coding theory perspective, which does not allow detection or error-correcting capabilities. Binary coding provides more possibilities for encoding categorical data into the output codes, which mitigates the limitations of the one-hot encoding mentioned above. We propose a novel method based on Zadeh fuzzy logic to train binary output codes holistically. We study linear block codes for their possibility of separating class information from the checksum part of the codeword, showing their ability not only to detect recognition errors by calculating non-zero syndrome, but also to evaluate the truth-value of the decision. Experimental results show that the proposed approach achieves similar results as one-hot encoding with a softmax function in terms of accuracy, reliability, and out-of-distribution performance. It suggests a good foundation for future applications, mainly classification tasks with a high number of classes.

scholarly journals Semantically Secure Symmetric Encryption with Error Correction for Distributed Storage

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Juha Partala
Keyword(s):  
Error Correction ◽  
Network Coding ◽  
Linear Combination ◽  
Distributed Storage ◽  
Homomorphic Encryption ◽  
Error Correcting Codes ◽  
Symmetric Encryption ◽  
Goppa Codes ◽  
Linear Network ◽  
Linear Network Coding

A distributed storage system (DSS) is a fundamental building block in many distributed applications. It applies linear network coding to achieve an optimal tradeoff between storage and repair bandwidth when node failures occur. Additively homomorphic encryption is compatible with linear network coding. The homomorphic property ensures that a linear combination of ciphertext messages decrypts to the same linear combination of the corresponding plaintext messages. In this paper, we construct a linearly homomorphic symmetric encryption scheme that is designed for a DSS. Our proposal provides simultaneous encryption and error correction by applying linear error correcting codes. We show its IND-CPA security for a limited number of messages based on binary Goppa codes and the following assumption: when dividing a scrambled generator matrix G^ into two parts G1^ and G2^, it is infeasible to distinguish G2^ from random and to find a statistical connection between G1^ and G2^. Our infeasibility assumptions are closely related to those underlying the McEliece public key cryptosystem but are considerably weaker. We believe that the proposed problem has independent cryptographic interest.

Cooperative Sequence Clustering and Decoding for DNA Storage System with Fountain Codes

2021 ◽  
Author(s):  
Jaeho Jeong ◽  
Seong-Joon Park ◽  
Jae-Won Kim ◽  
Jong-Seon No ◽  
Ha Hyeon Jeon ◽  
...  
Keyword(s):  
Hamming Distance ◽  
Storage Systems ◽  
Storage System ◽  
Data Retrieval ◽  
Illumina Miseq ◽  
Error Correcting Codes ◽  
Read Length ◽  
Sequence Coverage ◽  
Source Codes ◽  
Dna Storage

Abstract Motivation In DNA storage systems, there are tradeoffs between writing and reading costs. Increasing the code rate of error-correcting codes may save writing cost, but it will need more sequence reads for data retrieval. There is potentially a way to improve sequencing and decoding processes in such a way that the reading cost induced by this tradeoff is reduced without increasing the writing cost. In past researches, clustering, alignment, and decoding processes were considered as separate stages but we believe that using the information from all these processes together may improve decoding performance. Actual experiments of DNA synthesis and sequencing should be performed because simulations cannot be relied on to cover all error possibilities in practical circumstances. Results For DNA storage systems using fountain code and Reed-Solomon (RS) code, we introduce several techniques to improve the decoding performance. We designed the decoding process focusing on the cooperation of key components: Hamming-distance based clustering, discarding of abnormal sequence reads, RS error correction as well as detection, and quality score-based ordering of sequences. We synthesized 513.6KB data into DNA oligo pools and sequenced this data successfully with Illumina MiSeq instrument. Compared to Erlich’s research, the proposed decoding method additionally incorporates sequence reads with minor errors which had been discarded before, and thuswas able to make use of 10.6–11.9% more sequence reads from the same sequencing environment, this resulted in 6.5–8.9% reduction in the reading cost. Channel characteristics including sequence coverage and read-length distributions are provided as well. Availability The raw data files and the source codes of our experiments are available at: https://github.com/jhjeong0702/dna-storage.

scholarly journals A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

2019 ◽  
Vol 9 (2) ◽  
pp. 1232
Author(s):  
Issam Abderrahman Joundan ◽  
Said Nouh ◽  
Mohamed Azouazi ◽  
Abdelwahed Namir
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Search Problem ◽  
Public Key Cryptosystems ◽  
True Value ◽  
Efficient Scheme ◽  
Minimum Distances

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity.  ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1107-1116
Author(s):  
Ruihu Li ◽  
Luobin Guo ◽  
Zongben Xu
Keyword(s):  
Error Correction ◽  
Infinite Family ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Minimum Distances ◽  
Quantum Error

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

scholarly journals Opportunities and Challenges for Error Control Schemes for Wireless Sensor Networks: A Review

Electronics ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 504
Author(s):  
Rajan Kadel ◽  
Krishna Paudel ◽  
Deepani B. Guruge ◽  
Sharly J. Halder
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Error Correction ◽  
Communication Systems ◽  
Research Work ◽  
Error Correcting Codes ◽  
Wireless Sensor ◽  
Future Research ◽  
Asymmetric Structure ◽  
Design And Implementation

Error Correction Schemes (ECSs) significantly contribute to enhancing reliability and energy efficiency of Wireless Sensor Networks (WSNs). This review paper offers an overview of the different types of ECS used in communication systems and a synopsis of the standards for WSN. We also discuss channels and network models for WSN as they are crucial for efficient ECS design and implementation. The literature review conducted on the proposed energy consumption and efficiency models for WSN indicates that existing research work has not considered Single Hop Asymmetric Structure (SHAS) with high performing Error Correcting Codes (ECCs). We present a review on proposed ECS for WSN based on three criteria: Forward Error Correction (FEC), adaptive error correction techniques, and other techniques. Based on our review work, we found that there are limited works on ECS design on a realistic network model i.e., a modified multi-hop WSN model. Finally, we offer future research challenges and opportunities on ECS design and implementation for WSN.

scholarly journals Palindromic Decompositions with Gaps and Errors

2018 ◽  
Vol 29 (08) ◽  
pp. 1311-1329
Author(s):  
Michał Adamczyk ◽  
Mai Alzamel ◽  
Panagiotis Charalampopoulos ◽  
Jakub Radoszewski
Keyword(s):  
Lower Bound ◽  
Computational Biology ◽  
Dna Sequence ◽  
Hamming Distance ◽  
Efficient Algorithms ◽  
Combinatorics On Words ◽  
Hiv Virus ◽  
On Line

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an on-line algorithm for obtaining a palindromic decomposition of a string of length [Formula: see text] with the minimal total gap length in time [Formula: see text] and space [Formula: see text], where [Formula: see text] is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal [Formula: see text]-palindromes (i.e. palindromes with [Formula: see text] errors under the edit or Hamming distance) and [Formula: see text] allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time [Formula: see text] and space [Formula: see text]. Finally, we provide an implementation of our algorithms.

scholarly journals A Single Error Correcting Code with One-Step Group Partitioned Decoding Based on Shared Majority-Vote

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 709
Author(s):  
Abhishek Das ◽  
Nur A. Touba
Keyword(s):  
Optimization Technique ◽  
Error Correcting Code ◽  
Latin Square ◽  
Error Correcting Codes ◽  
Trade Off ◽  
Cache Memories ◽  
Hamming Codes ◽  
Single Error ◽  
Memory Overhead ◽  
On Chip

Technology scaling has led to an increase in density and capacity of on-chip caches. This has enabled higher throughput by enabling more low latency memory transfers. With the reduction in size of SRAMs and development of emerging technologies, e.g., STT-MRAM, for on-chip cache memories, reliability of such memories becomes a major concern. Traditional error correcting codes, e.g., Hamming codes and orthogonal Latin square codes, either suffer from high decoding latency, which leads to lower overall throughput, or high memory overhead. In this paper, a new single error correcting code based on a shared majority voting logic is presented. The proposed codes trade off decoding latency in order to improve the memory overhead posed by orthogonal Latin square codes. A latency optimization technique is also proposed which lowers the decoding latency by incurring a slight memory overhead. It is shown that the proposed codes achieve better redundancy compared to orthogonal Latin square codes. The proposed codes are also shown to achieve lower decoding latency compared to Hamming codes. Thus, the proposed codes achieve a balanced trade-off between memory overhead and decoding latency, which makes them highly suitable for on-chip cache memories which have stringent throughput and memory overhead constraints.

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