Hamming Codes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The One-Way Communication Complexity of Hamming Distance
THEORY OF COMPUTING, Volume 4 (2008), pp. 129–135 http://theoryofcomputing.org The One-Way Communication Complexity of Hamming Distance T. S. Jayram Ravi Kumar∗ D. Sivakumar∗ Received: January 23, 2008; revised: September 12, 2008; published: October 11, 2008. Abstract: Consider the following version of the Hamming distance problem for ±1 vec- n √ n √ tors of length n: the promise is that the distance is either at least 2 + n or at most 2 − n, and the goal is to find out which of these two cases occurs. Woodruff (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2004) gave a linear lower bound for the randomized one-way communication complexity of this problem. In this note we give a simple proof of this result. Our proof uses a simple reduction from the indexing problem and avoids the VC-dimension arguments used in the previous paper. As shown by Woodruff (loc. cit.), this implies an Ω(1/ε2)-space lower bound for approximating frequency moments within a factor 1 + ε in the data stream model. ACM Classification: F.2.2 AMS Classification: 68Q25 Key words and phrases: One-way communication complexity, indexing, Hamming distance 1 Introduction The Hamming distance H(x,y) between two vectors x and y is defined to be the number of positions i such that xi 6= yi. Let GapHD denote the following promise problem: the input consists of ±1 vectors n √ n √ x and y of length n, together with the promise that either H(x,y) ≤ 2 − n or H(x,y) ≥ 2 + n, and the goal is to find out which of these two cases occurs. -
A [49 16 6 ]-Linear Code As Product of the [7 4 2 ] Code Due to Aunu and the Hamming [7 4 3 ] Code
General Letters in Mathematics, Vol. 4, No.3 , June 2018, pp.114 -119 Available online at http:// www.refaad.com https://doi.org/10.31559/glm2018.4.3.4 A [49 16 6 ]-Linear Code as Product Of The [7 4 2 ] Code Due to Aunu and the Hamming [7 4 3 ] Code CHUN, Pamson Bentse1, IBRAHIM Alhaji Aminu2, and MAGAMI, Mohe'd Sani3 1 Department Of Mathematics, Plateau State University Bokkos, Jos Nigeria [email protected] 2 Department Of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto Nigeria [email protected] 3 Department Of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto Nigeria [email protected] Abstract: The enumeration of the construction due to "Audu and Aminu"(AUNU) Permutation patterns, of a [ 7 4 2 ]- linear code which is an extended code of the [ 6 4 1 ] code and is in one-one correspondence with the known [ 7 4 3 ] - Hamming code has been reported by the Authors. The [ 7 4 2 ] linear code, so constructed was combined with the known Hamming [ 7 4 3 ] code using the ( u|u+v)-construction method to obtain a new hybrid and more practical single [14 8 3 ] error- correcting code. In this paper, we provide an improvement by obtaining a much more practical and applicable double error correcting code whose extended version is a triple error correcting code, by combining the same codes as in [1]. Our goal is achieved through using the product code construction approach with the aid of some proven theorems. Keywords: Cayley tables, AUNU Scheme, Hamming codes, Generator matrix,Tensor product 2010 MSC No: 97H60, 94A05 and 94B05 1. -
Coding Theory: Linear Error-Correcting Codes Coding Theory Basic Definitions Error Detection and Correction Finite Fields Anna Dovzhik
Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Outline Coding Theory: Linear Error-Correcting Codes Coding Theory Basic Definitions Error Detection and Correction Finite Fields Anna Dovzhik Linear Codes Hamming Codes Finite Fields Revisited BCH Codes April 23, 2014 Reed-Solomon Codes Conclusion Coding Theory: Linear Error- Correcting 1 Coding Theory Codes Basic Definitions Anna Dovzhik Error Detection and Correction Outline Coding Theory 2 Finite Fields Basic Definitions Error Detection and Correction 3 Finite Fields Linear Codes Linear Codes Hamming Codes Hamming Codes Finite Fields Finite Fields Revisited Revisited BCH Codes BCH Codes Reed-Solomon Codes Reed-Solomon Codes Conclusion 4 Conclusion Definition A q-ary word w = w1w2w3 ::: wn is a vector where wi 2 A. Definition A q-ary block code is a set C over an alphabet A, where each element, or codeword, is a q-ary word of length n. Basic Definitions Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Definition If A = a ; a ;:::; a , then A is a code alphabet of size q. Outline 1 2 q Coding Theory Basic Definitions Error Detection and Correction Finite Fields Linear Codes Hamming Codes Finite Fields Revisited BCH Codes Reed-Solomon Codes Conclusion Definition A q-ary block code is a set C over an alphabet A, where each element, or codeword, is a q-ary word of length n. Basic Definitions Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Definition If A = a ; a ;:::; a , then A is a code alphabet of size q. Outline 1 2 q Coding Theory Basic Definitions Error Detection Definition and Correction Finite Fields A q-ary word w = w1w2w3 ::: wn is a vector where wi 2 A. -
Fault Tolerant Computing CS 530 Information Redundancy: Coding Theory
Fault Tolerant Computing CS 530 Information redundancy: Coding theory Yashwant K. Malaiya Colorado State University April 3, 2018 1 Information redundancy: Outline • Using a parity bit • Codes & code words • Hamming distance . Error detection capability . Error correction capability • Parity check codes and ECC systems • Cyclic codes . Polynomial division and LFSRs 4/3/2018 Fault Tolerant Computing ©YKM 2 Redundancy at the Bit level • Errors can bits to be flipped during transmission or storage. • An extra parity bit can detect if a bit in the word has flipped. • Some errors an be corrected if there is enough redundancy such that the correct word can be guessed. • Tukey: “bit” 1948 • Hamming codes: 1950s • Teletype, ASCII: 1960: 7+1 Parity bit • Codes are widely used. 304,805 letters in the Torah Redundancy at the Bit level Even/odd parity (1) • Errors can bits to be flipped during transmission/storage. • Even/odd parity: . is basic method for detecting if one bit (or an odd number of bits) has been switched by accident. • Odd parity: . The number of 1-bit must add up to an odd number • Even parity: . The number of 1-bit must add up to an even number Even/odd parity (2) • The it is known which parity it is being used. • If it uses an even parity: . If the number of of 1-bit add up to an odd number then it knows there was an error: • If it uses an odd: . If the number of of 1-bit add up to an even number then it knows there was an error: • However, If an even number of 1-bit is flipped the parity will still be the same. -
Sequence Distance Embeddings
Sequence Distance Embeddings by Graham Cormode Thesis Submitted to the University of Warwick for the degree of Doctor of Philosophy Computer Science January 2003 Contents List of Figures vi Acknowledgments viii Declarations ix Abstract xi Abbreviations xii Chapter 1 Starting 1 1.1 Sequence Distances . ....................................... 2 1.1.1 Metrics ............................................ 3 1.1.2 Editing Distances ...................................... 3 1.1.3 Embeddings . ....................................... 3 1.2 Sets and Vectors ........................................... 4 1.2.1 Set Difference and Set Union . .............................. 4 1.2.2 Symmetric Difference . .................................. 5 1.2.3 Intersection Size ...................................... 5 1.2.4 Vector Norms . ....................................... 5 1.3 Permutations ............................................ 6 1.3.1 Reversal Distance ...................................... 7 1.3.2 Transposition Distance . .................................. 7 1.3.3 Swap Distance ....................................... 8 1.3.4 Permutation Edit Distance . .............................. 8 1.3.5 Reversals, Indels, Transpositions, Edits (RITE) . .................... 9 1.4 Strings ................................................ 9 1.4.1 Hamming Distance . .................................. 10 1.4.2 Edit Distance . ....................................... 10 1.4.3 Block Edit Distances . .................................. 11 1.5 Sequence Distance Problems . ................................. -
Dictionary Look-Up Within Small Edit Distance
Dictionary Lo okUp Within Small Edit Distance Ab dullah N Arslan and Omer Egeciog lu Department of Computer Science University of California Santa Barbara Santa Barbara CA USA farslanomergcsucsbedu Abstract Let W b e a dictionary consisting of n binary strings of length m each represented as a trie The usual dquery asks if there exists a string in W within Hamming distance d of a given binary query string q We present an algorithm to determine if there is a memb er in W within edit distance d of a given query string q of length m The metho d d+1 takes time O dm in the RAM mo del indep endent of n and requires O dm additional space Intro duction Let W b e a dictionary consisting of n binary strings of length m each A dquery asks if there exists a string in W within Hamming distance d of a given binary query string q Algorithms for answering dqueries eciently has b een a topic of interest for some time and have also b een studied as the approximate query and the approximate query retrieval problems in the literature The problem was originally p osed by Minsky and Pap ert in in which they asked if there is a data structure that supp orts fast dqueries The cases of small d and large d for this problem seem to require dierent techniques for their solutions The case when d is small was studied by Yao and Yao Dolev et al and Greene et al have made some progress when d is relatively large There are ecient algorithms only when d prop osed by Bro dal and Venkadesh Yao and Yao and Bro dal and Gasieniec The small d case has applications -
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). -
Hamming Code - Wikipedia, the Free Encyclopedia Hamming Code Hamming Code
Hamming code - Wikipedia, the free encyclopedia Hamming code Hamming code From Wikipedia, the free encyclopedia In telecommunication, a Hamming code is a linear error-correcting code named after its inventor, Richard Hamming. Hamming codes can detect and correct single-bit errors, and can detect (but not correct) double-bit errors. In contrast, the simple parity code cannot detect errors where two bits are transposed, nor can it correct the errors it can find. Contents [hide] • 1 History • 2 Codes predating Hamming ♦ 2.1 Parity ♦ 2.2 Two-out-of-five code ♦ 2.3 Repetition • 3 Hamming codes • 4 Example using the (11,7) Hamming code • 5 Hamming code (7,4) ♦ 5.1 Hamming matrices ♦ 5.2 Channel coding ♦ 5.3 Parity check ♦ 5.4 Error correction • 6 Hamming codes with additional parity • 7 See also • 8 References • 9 External links History Hamming worked at Bell Labs in the 1940s on the Bell Model V computer, an electromechanical relay-based machine with cycle times in seconds. Input was fed in on punch cards, which would invariably have read errors. During weekdays, special code would find errors and flash lights so the operators could correct the problem. During after-hours periods and on weekends, when there were no operators, the machine simply moved on to the next job. Hamming worked on weekends, and grew increasingly frustrated with having to restart his programs from scratch due to the unreliability of the card reader. Over the next few years he worked on the problem of error-correction, developing an increasingly powerful array of algorithms. -
Searching All Seeds of Strings with Hamming Distance Using Finite Automata
Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol I IMECS 2009, March 18 - 20, 2009, Hong Kong Searching All Seeds of Strings with Hamming Distance using Finite Automata Ondřej Guth, Bořivoj Melichar ∗ Abstract—Seed is a type of a regularity of strings. Finite automata provide common formalism for many al- A restricted approximate seed w of string T is a fac- gorithms in the area of text processing (stringology), in- tor of T such that w covers a superstring of T under volving forward exact and approximate pattern match- some distance rule. In this paper, the problem of all ing and searching for borders, periods, and repetitions restricted seeds with the smallest Hamming distance [7], backward pattern matching [8], pattern matching in is studied and a polynomial time and space algorithm a compressed text [9], the longest common subsequence for solving the problem is presented. It searches for [10], exact and approximate 2D pattern matching [11], all restricted approximate seeds of a string with given limited approximation using Hamming distance and and already mentioned computing approximate covers it computes the smallest distance for each found seed. [5, 6] and exact covers [12] and seeds [2] in generalized The solution is based on a finite (suffix) automata ap- strings. Therefore, we would like to further extend the set proach that provides a straightforward way to design of problems solved using finite automata. Such a problem algorithms to many problems in stringology. There- is studied in this paper. fore, it is shown that the set of problems solvable using finite automata includes the one studied in this Finite automaton as a data structure may be easily imple- paper. -
Coded Computing Via Binary Linear Codes: Designs and Performance Limits Mahdi Soleymani∗, Mohammad Vahid Jamali∗, and Hessam Mahdavifar, Member, IEEE
1 Coded Computing via Binary Linear Codes: Designs and Performance Limits Mahdi Soleymani∗, Mohammad Vahid Jamali∗, and Hessam Mahdavifar, Member, IEEE Abstract—We consider the problem of coded distributed processing capabilities. A coded computing scheme aims to computing where a large linear computational job, such as a divide the job to k < n tasks and then to introduce n − k matrix multiplication, is divided into k smaller tasks, encoded redundant tasks using an (n; k) code, in order to alleviate the using an (n; k) linear code, and performed over n distributed nodes. The goal is to reduce the average execution time of effect of slower nodes, also referred to as stragglers. In such a the computational job. We provide a connection between the setup, it is often assumed that each node is assigned one task problem of characterizing the average execution time of a coded and hence, the total number of encoded tasks is n equal to the distributed computing system and the problem of analyzing the number of nodes. error probability of codes of length n used over erasure channels. Recently, there has been extensive research activities to Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution leverage coding schemes in order to boost the performance time any linear-coded distributed computing system can achieve. of distributed computing systems [6]–[18]. However, most It is also shown that there exist good binary linear codes that not of the work in the literature focus on the application of only attain (asymptotically) the best performance that any linear maximum distance separable (MDS) codes. -
ERROR CORRECTING CODES Definition (Alphabet)
ERROR CORRECTING CODES Definition (Alphabet) An alphabet is a finite set of symbols. Example: A simple example of an alphabet is the set A := fB; #; 17; P; $; 0; ug. Definition (Codeword) A codeword is a string of symbols in an alphabet. Example: In the alphabet A := fB; #; 17; P; $; 0; ug, some examples of codewords of length 4 are: P 17#u; 0B$#; uBuB; 0000: Definition (Code) A code is a set of codewords in an alphabet. Example: In the alphabet A := fB; #; 17; P; $; 0; ug, an example of a code is the set of 5 words: fP 17#u; 0B$#; uBuB; 0000;PP #$g: If all the codewords are known and recorded in some dictionary then it is possible to transmit messages (strings of codewords) to another entity over a noisy channel and detect and possibly correct errors. For example in the code C := fP 17#u; 0B$#; uBuB; 0000;PP #$g if the word 0B$# is sent and received as PB$#, then if we knew that only one error was made it can be determined that the letter P was an error which can be corrected by changing P to 0. Definition (Linear Code) A linear code is a code where the codewords form a finite abelian group under addition. Main Problem of Coding Theory: To construct an alphabet and a code in that alphabet so it is as easy as possible to detect and correct errors in transmissions of codewords. A key idea for solving this problem is the notion of Hamming distance. Definition (Hamming distance dH ) Let C be a code and let 0 0 0 0 w = w1w2 : : : wn; w = w1w2 : : : wn; 0 0 be two codewords in C where wi; wj are elements of the alphabet of C. -
B-Bit Sketch Trie: Scalable Similarity Search on Integer Sketches
b-Bit Sketch Trie: Scalable Similarity Search on Integer Sketches Shunsuke Kanda Yasuo Tabei RIKEN Center for Advanced Intelligence Project RIKEN Center for Advanced Intelligence Project Tokyo, Japan Tokyo, Japan [email protected] [email protected] Abstract—Recently, randomly mapping vectorial data to algorithms intending to build sketches of non-negative inte- strings of discrete symbols (i.e., sketches) for fast and space- gers (i.e., b-bit sketches) have been proposed for efficiently efficient similarity searches has become popular. Such random approximating various similarity measures. Examples are b-bit mapping is called similarity-preserving hashing and approximates a similarity metric by using the Hamming distance. Although minwise hashing (minhash) [12]–[14] for Jaccard similarity, many efficient similarity searches have been proposed, most of 0-bit consistent weighted sampling (CWS) for min-max ker- them are designed for binary sketches. Similarity searches on nel [15], and 0-bit CWS for generalized min-max kernel [16]. integer sketches are in their infancy. In this paper, we present Thus, developing scalable similarity search methods for b-bit a novel space-efficient trie named b-bit sketch trie on integer sketches is a key issue in large-scale applications of similarity sketches for scalable similarity searches by leveraging the idea behind succinct data structures (i.e., space-efficient data structures search. while supporting various data operations in the compressed Similarity searches on binary sketches are classified