Error Detection and Correction Using the BCH Code 1
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10. on Extending Bch Codes
University of Plymouth PEARL https://pearl.plymouth.ac.uk 04 University of Plymouth Research Theses 01 Research Theses Main Collection 2011 Algebraic Codes For Error Correction In Digital Communication Systems Jibril, Mubarak http://hdl.handle.net/10026.1/1188 University of Plymouth All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. Copyright Statement This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and information derived from it may be published without the author’s prior consent. Copyright © Mubarak Jibril, December 2011 Algebraic Codes For Error Correction In Digital Communication Systems by Mubarak Jibril A thesis submitted to the University of Plymouth in partial fulfilment for the degree of Doctor of Philosophy School of Computing and Mathematics Faculty of Technology Under the supervision of Dr. M. Z. Ahmed, Prof. M. Tomlinson and Dr. C. Tjhai December 2011 ABSTRACT Algebraic Codes For Error Correction In Digital Communication Systems Mubarak Jibril C. Shannon presented theoretical conditions under which communication was pos- sible error-free in the presence of noise. Subsequently the notion of using error correcting codes to mitigate the effects of noise in digital transmission was intro- duced by R. -
Decoding Beyond the Designed Distance for Certain Algebraic Codes
INFORMATION AND CONTROL 35, 209--228 (1977) Decoding beyond the Designed Distance for Certain Algebraic Codes DAVID MANDELBAUM P.O. Box 645, Eatontown, New Jersey 07724 It is shown how decoding beyond the designed distance can be accomplished for a certain decoding algorithm. This method can be used for subcodes of generalized Goppa codes and generalized Reed-Solomon codes. 1. INTRODUCTION A decoding algorithm for generalized Goppa codes is presented here. This method directly uses the Euclidean algorithm, it is shown how subcodes of these generalized codes can be decoded for errors beyond the designed distance. It is proved that all coset leaders can be decoded by this method. Decoding of errors beyond the BCH bound may involve solution of nonlinear equations. The method used is similar to the use of continued fractions for decoding within the BCH bound (Mandelbaum, 1976). The Euclidean algorithm is also used by Sugiyama et aL (1975) in a different manner for decoding Goppa codes within the BCH bound. Section 2 defines the generalized codes and shows the relationship with the Mattson-Solomon polynomial. Section 3 defines the decoding algorithm and gives, as a detailed example, the decoding of the (15, 5) BCH code for a quadruple error. Application to the Golay code is pointed out. This method can also be used for decoding the Generalized Reed-Solomon codes constructed by Delsarte (1975). Section 4 presents a method of obtaining the original information and syndrome in an iterated manner, which results in reduced computation for codewords with no errors. Other methods for decoding beyond the BCH bound for BCH codes are given by Berlekamp (1968) and Harmann (1972). -
Chapter 6 Modeling and Simulations of Reed-Solomon Encoder and Decoder
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE The Implementation of a Reed Solomon Code Encoder /Decoder A graduate project submitted in partial fulfillment of the requirements For the degree of Master of Science in Electrical Engineering By Qiang Zhang May 2014 Copyright by Qiang Zhang 2014 ii The graduate project of Qiang Zhang is approved: Dr. Ali Amini Date Dr. Sharlene Katz Date Dr. Nagi El Naga, Chair Date California State University, Northridge iii Acknowledgements I would like to thank my project professor and committee: Dr. Nagi El Naga, Dr. Ali Amini, and Dr. Sharlene Katz for their time and help. I would also like to thank my parents for their love and care. I could not have finished this project without them, because they give me the huge encouragement during last two years. iv Dedication To my parents v Table of Contents Copyright Page.................................................................................................................... ii Signature Page ................................................................................................................... iii Acknowledgement ............................................................................................................. iv Dedication ............................................................................................................................v List of Figures .................................................................................................................. viii Abstract ................................................................................................................................x -
Mceliece Cryptosystem Based on Extended Golay Code
McEliece Cryptosystem Based On Extended Golay Code Amandeep Singh Bhatia∗ and Ajay Kumar Department of Computer Science, Thapar university, India E-mail: ∗[email protected] (Dated: November 16, 2018) With increasing advancements in technology, it is expected that the emergence of a quantum computer will potentially break many of the public-key cryptosystems currently in use. It will negotiate the confidentiality and integrity of communications. In this regard, we have privacy protectors (i.e. Post-Quantum Cryptography), which resists attacks by quantum computers, deals with cryptosystems that run on conventional computers and are secure against attacks by quantum computers. The practice of code-based cryptography is a trade-off between security and efficiency. In this chapter, we have explored The most successful McEliece cryptosystem, based on extended Golay code [24, 12, 8]. We have examined the implications of using an extended Golay code in place of usual Goppa code in McEliece cryptosystem. Further, we have implemented a McEliece cryptosystem based on extended Golay code using MATLAB. The extended Golay code has lots of practical applications. The main advantage of using extended Golay code is that it has codeword of length 24, a minimum Hamming distance of 8 allows us to detect 7-bit errors while correcting for 3 or fewer errors simultaneously and can be transmitted at high data rate. I. INTRODUCTION Over the last three decades, public key cryptosystems (Diffie-Hellman key exchange, the RSA cryptosystem, digital signature algorithm (DSA), and Elliptic curve cryptosystems) has become a crucial component of cyber security. In this regard, security depends on the difficulty of a definite number of theoretic problems (integer factorization or the discrete log problem). -
Chapter 9: BCH, Reed-Solomon, and Related Codes
Draft of February 23, 2001 Chapter 9: BCH, Reed-Solomon, and Related Codes 9.1 Introduction. In Chapter 7 we gave one useful generalization of the (7, 4) Hamming code of the Introduction: the family of (2m − 1, 2m − m − 1) single error-correcting Hamming Codes. In Chapter 8 we gave a further generalization, to a class of codes capable of correcting a single burst of errors. In this Chapter, however, we will give a far more important and extensive generalization, the multiple-error correcting BCH and Reed-Solomon Codes. To motivate the general definition, recall that the parity-check matrix of a Hamming Code of length n =2m − 1 is given by (see Section 7.4) H =[v0 v1 ··· vn−1 ] , (9.1) m m where (v0, v1,...,vn−1) is some ordering of the 2 − 1 nonzero (column) vectors from Vm = GF (2) . The matrix H has dimensions m × n, which means that it takes m parity-check bits to correct one error. If we wish to correct two errors, it stands to reason that m more parity checks will be required. Thus we might guess that a matrix of the general form v0 v1 ··· vn−1 H2 = , w0 w1 ··· wn−1 where w0, w1,...,wn−1 ∈ Vm, will serve as the parity-check matrix for a two-error-correcting code of length n. Since however the vi’s are distinct, we may view the correspondence vi → wi as a function from Vm into itself, and write H2 as v0 v1 ··· vn−1 H2 = . (9.3) f(v0) f(v1) ··· f(vn−1) But how should the function f be chosen? According to the results of Section 7.3, H2 will define a n two-error-correcting code iff the syndromes of the 1 + n + 2 error pattern of weights 0, 1 and 2 are all distinct. -
The Theory of Error-Correcting Codes the Ternary Golay Codes and Related Structures
Math 28: The Theory of Error-Correcting Codes The ternary Golay codes and related structures We have seen already that if C is an extremal [12; 6; 6] Type III code then C attains equality in the LP bounds (and conversely the LP bounds show that any ternary (12; 36; 6) code is a translate of such a C); ear- lier we observed that removing any coordinate from C yields a perfect 2-error-correcting [11; 6; 5] ternary code. Since C is extremal, we also know that its Hamming weight enumerator is uniquely determined, and compute 3 3 3 3 3 3 12 6 6 3 9 12 WC (X; Y ) = (X(X + 8Y )) − 24(Y (X − Y )) = X + 264X Y + 440X Y + 24Y : Here are some further remarkable properties that quickly follow: A (5,6,12) Steiner system. The minimal words form 132 pairs fc; −cg with the same support. Let S be the family of these 132 supports. We claim S is a (5; 6; 12) Steiner system; that is, that any pentad (5-element 12 6 set of coordinates) is a subset of a unique hexad in S. Because S has the right size 5 5 , it is enough to show that no two hexads intersect in exactly 5 points. If they did, then we would have some c; c0 2 C, both of weight 6, whose supports have exactly 5 points in common. But then c0 would agree with either c or −c on at least 3 of these points (and thus on exactly 4, because (c; c0) = 0), making c0 ∓ c a nonzero codeword of weight less than 5 (and thus exactly 3), which is a contradiction. -
Chapter 9: BCH, Reed-Solomon, and Related Codes
Draft of February 21, 1999 Chapter 9: BCH, Reed-Solomon, and Related Codes 9.1 Introduction. In Chapter 7 we gave one useful generalization of the (7, 4)Hamming code of the Introduction: the family of (2m − 1, 2m − m − 1)single error-correcting Hamming Codes. In Chapter 8 we gave a further generalization, to a class of codes capable of correcting a single burst of errors. In this Chapter, however, we will give a far more important and extensive generalization, the multiple-error correcting BCH and Reed-Solomon Codes. To motivate the general definition, recall that the parity-check matrix of a Hamming Code of length n =2m − 1 is given by (see Section 7.4) H =[v0 v1 ··· vn−1 ] , (9.1) m m where (v0, v1,...,vn−1)is some ordering of the 2 − 1 nonzero (column)vectors from Vm = GF (2) . The matrix H has dimensions m × n, which means that it takes m parity-check bits to correct one error. If we wish to correct two errors, it stands to reason that m more parity checks will be required. Thus we might guess that a matrix of the general form v0 v1 ··· vn−1 H2 = , w0 w1 ··· wn−1 where w0, w1,...,wn−1 ∈ Vm, will serve as the parity-check matrix for a two-error-correcting code of length n. Since however the vi’s are distinct, we may view the correspondence vi → wi as a function from Vm into itself, and write H2 as v0 v1 ··· vn−1 H2 = . (9.3) f(v0) f(v1) ··· f(vn−1) But how should the function f be chosen? According to the results of Section 7.3, H2 will define a n two-error-correcting code iff the syndromes of the 1 + n + 2 error pattern of weights 0, 1 and 2 are all distinct. -
Review of Binary Codes for Error Detection and Correction
` ISSN(Online): 2319-8753 ISSN (Print): 2347-6710 International Journal of Innovative Research in Science, Engineering and Technology (A High Impact Factor, Monthly, Peer Reviewed Journal) Visit: www.ijirset.com Vol. 7, Issue 4, April 2018 Review of Binary Codes for Error Detection and Correction Eisha Khan, Nishi Pandey and Neelesh Gupta M. Tech. Scholar, VLSI Design and Embedded System, TCST, Bhopal, India Assistant Professor, VLSI Design and Embedded System, TCST, Bhopal, India Head of Dept., VLSI Design and Embedded System, TCST, Bhopal, India ABSTRACT: Golay Code is a type of Error Correction code discovered and performance very close to Shanon‟s limit . Good error correcting performance enables reliable communication. Since its discovery by Marcel J.E. Golay there is more research going on for its efficient construction and implementation. The binary Golay code (G23) is represented as (23, 12, 7), while the extended binary Golay code (G24) is as (24, 12, 8). High speed with low-latency architecture has been designed and implemented in Virtex-4 FPGA for Golay encoder without incorporating linear feedback shift register. This brief also presents an optimized and low-complexity decoding architecture for extended binary Golay code (24, 12, 8) based on an incomplete maximum likelihood decoding scheme. KEYWORDS: Golay Code, Decoder, Encoder, Field Programmable Gate Array (FPGA) I. INTRODUCTION Communication system transmits data from source to transmitter through a channel or medium such as wired or wireless. The reliability of received data depends on the channel medium and external noise and this noise creates interference to the signal and introduces errors in transmitted data. -
A Parallel Algorithm for Query Adaptive, Locality Sensitive Hash
A CUDA Based Parallel Decoding Algorithm for the Leech Lattice Locality Sensitive Hash Family A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the School of Electric and Computing Systems of the College of Engineering and Applied Sciences November 03, 2011 by Lee A Carraher BSCE, University of Cincinnati, 2008 Thesis Advisor and Committee Chair: Dr. Fred Annexstein Abstract Nearest neighbor search is a fundamental requirement of many machine learning algorithms and is essential to fuzzy information retrieval. The utility of efficient database search and construction has broad utility in a variety of computing fields. Applications such as coding theory and compression for electronic commu- nication systems as well as use in artificial intelligence for pattern and object recognition. In this thesis, a particular subset of nearest neighbors is consider, referred to as c-approximate k-nearest neighbors search. This particular variation relaxes the constraints of exact nearest neighbors by introducing a probability of finding the correct nearest neighbor c, which offers considerable advantages to the computational complexity of the search algorithm and the database overhead requirements. Furthermore, it extends the original nearest neighbors algorithm by returning a set of k candidate nearest neighbors, from which expert or exact distance calculations can be considered. Furthermore thesis extends the implementation of c-approximate k-nearest neighbors search so that it is able to utilize the burgeoning GPGPU computing field. The specific form of c-approximate k-nearest neighbors search implemented is based on the locality sensitive hash search from the E2LSH package of Indyk and Andoni [1]. -
On the Algebraic Structure of Quasi Group Codes
ON THE ALGEBRAIC STRUCTURE OF QUASI GROUP CODES MARTINO BORELLO AND WOLFGANG WILLEMS Abstract. In this note, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi group codes, that is as linear codes with a free group of permutation automorphisms. An algebraic description, including the concatenated structure, of such codes is presented. Finally, self-duality of quasi group codes is investigated. 1. Introduction In the theory of error correcting codes, the linear ones play a central role for their algebraic properties which allow, for example, their easy description and storage. Quite early in their study, it appeared convenient to add additional algebraic structure in order to get more information about the parameters and to facilitate the decoding process. In 1957, E. Prange introduced the now well- known class of cyclic codes [18], which are the forefathers of many other families of codes with symmetries introduced later. In particular, abelian codes [2], group codes [15], quasi-cyclic codes [9] and quasi-abelian codes [19] are distinguished descendants of cyclic codes: they have a nice algebraic structure and many optimal codes belong to these families. The aim of the current note is to give an intrinsic description of these classes in terms of their permutation automorphism groups: we will show that all of them have a free group of permutation automorphisms (which is not necessarily the full permutation automorphism group of the code). We will define a code with this property as a quasi group code. So all families cited above can be described in this way. -
Error-Correction and the Binary Golay Code
London Mathematical Society Impact150 Stories 1 (2016) 51{58 C 2016 Author(s) doi:10.1112/i150lms/t.0003 e Error-correction and the binary Golay code R.T.Curtis Abstract Linear algebra and, in particular, the theory of vector spaces over finite fields has provided a rich source of error-correcting codes which enable the receiver of a corrupted message to deduce what was originally sent. Although more efficient codes have been devised in recent years, the 12- dimensional binary Golay code remains one of the most mathematically fascinating such codes: not only has it been used to send messages back to earth from the Voyager space probes, but it is also involved in many of the most extraordinary and beautiful algebraic structures. This article describes how the code can be obtained in two elementary ways, and explains how it can be used to correct errors which arise during transmission 1. Introduction Whenever a message is sent - along a telephone wire, over the internet, or simply by shouting across a crowded room - there is always the possibly that the message will be damaged during transmission. This may be caused by electronic noise or static, by electromagnetic radiation, or simply by the hubbub of people in the room chattering to one another. An error-correcting code is a means by which the original message can be recovered by the recipient even though it has been corrupted en route. Most commonly a message is sent as a sequence of 0s and 1s. Usually when a 0 is sent a 0 is received, and when a 1 is sent a 1 is received; however occasionally, and hopefully rarely, a 0 is sent but a 1 is received, or vice versa. -
Lecture 14: BCH Codes Overview 1 General Codes
CS681 Computational Number Theory Lecture 14: BCH Codes Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview We shall look a a special form of linear codes called cyclic codes. These have very nice structures underlying them and we shall study BCH codes. 1 General Codes n Recall that a linear code C is just a subspace of Fq . We saw last time that by picking a basis of C we can construct what is known as a parity check matrix H that is zero precisely at C. Let us understand how the procedure works. Alice has a message, of length k and she wishes to transmit across the channel. The channel is unreliable and therefore both Alice and Bob first agree on some code C. Now how does Alice convert her message into a code word in C? If Alice’s message could be written as (x1, x2, ··· , xk) where each xi ∈ Fq, then Alice Pk simply sends i=1 xibi which is a codeword. Bob receives some y and he checks if Hy = 0. Assuming that they choose a good distance code (the channel cannot alter one code into an- other), if Bob finds that Hy = 0, then he knows that the message he received was untampered with. But what sort of errors can the channel give? Let us say that the channel can change atmost t positions of the codeword. If x was sent and x0 was received with atmost t changes between them, then the vector e = x0−x can be thought of as the error vector.