Hamming Codes 11/17/04
|
|
- Clara Hubbard
- 5 years ago
- Views:
Transcription
1 Hamming Codes 11/17/04
2 History In the late 1940 s Richard Hamming recognized that the further evolution of computers required greater reliability, in particular the ability to not only detect errors, but correct them. His search for error-correcting codes led to the Hamming Codes, perfect 1-error correcting codes, and the extended Hamming Codes, 1-error correcting and 2-error detecting codes.
3 Uses Hamming Codes are still widely used in computing, telecommunication, and other applications. Hamming Codes also applied in Data compression Some solutions to the popular puzzle The Hat Game Block Turbo Codes
4 A [7,4] binary Hamming Code Let our codeword be (x 1 x 2 x 7 ) ε F 2 7 x 3, x 5, x 6, x 7 are chosen according to the message (perhaps the message itself is (x 3 x 5 x 6 x 7 )). x 4 := x 5 + x 6 + x 7 (mod 2) x 2 := x 3 + x 6 + x 7 x 1 := x 3 + x 5 + x 7
5 [7,4] binary Hamming codewords
6 A [7,4] binary Hamming Code Let a = x 4 + x 5 + x 6 + x 7 (=1 iff one of these bits is in error) Let b = x 2 + x 3 + x 6 + x 7 Let c = x 1 + x 3 + x 5 + x 7 If there is an error (assuming at most one) then abc will be binary representation of the subscript of the offending bit.
7 A [7,4] binary Hamming Code If (y 1 y 2 y 7 ) is received and abc 000, then we assume the bit abc is in error and switch it. If abc=000, we assume there were no errors (so if there are three or more errors we may recover the wrong codeword).
8 Definition: Generator and Check Matrices For an [n, k] linear code, the generator matrix is a k n matrix for which the row space is the given code. A check matrix for an [n, k] is a generator matrix for the dual code. In other words, an (n-k) k matrix M for which Mx = 0 for all x in the code.
9 A Construction for binary Hamming Codes For a given r, form an r 2 r -1 matrix M, the columns of which are the binary representations (r bits long) of 1,, 2 r -1. The linear code for which this is the check matrix is a [2 r -1, 2 r -1 r] binary Hamming Code = {x=(x 1 x 2 x n ) : Mx T = 0}.
10 Example Check Matrix A check matrix for a [7,4] binary Hamming Code:
11 Syndrome Decoding Let y = (y 1 y 2 y n ) be a received codeword. The syndrome of y is S:=L r y T. If S=0 then there was no error. If S 0 then S is the binary representation of some integer 1 t n=2 r -1 and the intended codeword is x = (y 1 y r +1 y n ).
12 Example Using L 3 Suppose ( ) is received. 100 is 4 in binary, so the intended codeword was ( ).
13 Extended [8,4] binary Hamm. Code As with the [7,4] binary Hamming Code: x 3, x 5, x 6, x 7 are chosen according to the message. x 4 := x 5 + x 6 + x 7 x 2 := x 3 + x 6 + x 7 x 1 := x 3 + x 5 + x 7 Add a new bit x 0 such that x 0 = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7. i.e., the new bit makes the sum of all the bits zero. x 0 is called a parity check.
14 Extended binary Hamming Code The minimum distance between any two codewords is now 4, so an extended Hamming Code is a 1-error correcting and 2-error detecting code. The general construction of a [2r, 2r-1 - r] extended code from a [2r 1, 2r 1 r] binary Hamming Code is the same: add a parity check bit.
15 Check Matrix Construction of Extended Hamming Code The check matrix of an extended Hamming Code can be constructed from the check matrix of a Hamming code by adding a zero column on the left and a row of 1 s to the bottom.
16 Decoding Extended Hamming Code
17 q-ary Hamming Codes The binary construction generalizes to Hamming Codes over an alphabet A={0,, q}, q 2. For a given r, form an r (q r -1)/(q-1) matrix M over A, any two columns of which are linearly independent. M determines a [(q r -1)/(q-1), (q r -1)/(q-1) r] (= [n,k]) q-ary Hamming Code for which M is the check matrix.
18 Example: ternary [4, 2] Hamming Two check matrices for the some [4, 2] ternary Hamming Codes:
19 Syndrome decoding: the q-ary case The syndrome of received word y, S:=My T, will be a multiple of one of the columns of M, say S=αm i, α scalar, m i the i th column of M. Assume an error vector of weight 1 was introduced y = x + (0 α 0), α in the i th spot.
20 Example: q-ary Syndrome [4,2] ternary with check matrix, word ( ) received. So decode ( ) as ( ) ( ) = ( ).
21 Perfect 1-error correcting Hamming Codes are perfect 1-error correcting codes. That is, any received word with at most one error will be decoded correctly and the code has the smallest possible size of any code that does this. For a given r, any perfect 1-error correcting linear code of length n=2 r -1 and dimension n-r is a Hamming Code.
22 Proof: 1-error correcting A code will be 1-error correcting if spheres of radius 1 centered at codewords cover the codespace, and if the minimum distance between any two codewords 3, since then spheres of radius 1 centered at codewords will be disjoint.
23 Proof: 1-error correcting Suppose codewords x, y differ by 1 bit. Then x-y is a codeword of weight 1, and M(x-y) 0. Contradiction. If x, y differ by 2 bits, then M(x-y) is the difference of two multiples of columns of M. No two columns of M are linearly dependent, so M(x-y) 0, another contradiction. Thus the minimum distance is at least 3.
24 Perfect A sphere of radius δ centered at x is S δ (x)={y in A n : d H (x,y) δ }. Where A is the alphabet, F q, and d H is the Hamming distance. A sphere of radius e contains words. If C is an e-error correcting code then, so.
25 Perfect This last inequality is called the sphere packing bound for an e-error correcting code C of length n over F m : where n is the length of the code and in this case e=1. A code for which equality holds is called perfect.
26 Proof: Perfect The right side of this, for e=1 is q n /(1+n(q-1)). The left side is q n-r where n= (q r -1)/(q-1). q n-r (1+n(q-1)) = q n-r (1+(q r -1)) = q n.
27 Applications Data compression. Turbo Codes The Hat Game
28 Data Compression Hamming Codes can be used for a form of lossy compression. If n=2 r -1 for some r, then any n-tuple of bits x is within distance at most 1 from a Hamming codeword c. Let G be a generator matrix for the Hamming Code, and mg=c. For compression, store x as m. For decompression, decode m as c. This saves r bits of space but corrupts (at most) 1 bit.
29 The Hat Game A group of n players enter a room whereupon they each receive a hat. Each player can see everyone else s hat but not his own. The players must each simultaneously guess a hat color, or pass. The group loses if any player guesses the wrong hat color or if every player passes. Players are not necessarily anonymous, they can be numbered.
30 The Hat Game Assignment of hats is assumed to be random. The players can meet beforehand to devise a strategy. The goal is to devise the strategy that gives the highest probability of winning.
31 Source Notes on Coding Theory by J.I. Hall otes/coding-notes.html
ELEC 519A Selected Topics in Digital Communications: Information Theory. Hamming Codes and Bounds on Codes
ELEC 519A Selected Topics in Digital Communications: Information Theory Hamming Codes and Bounds on Codes Single Error Correcting Codes 2 Hamming Codes (7,4,3) Hamming code 1 0 0 0 0 1 1 0 1 0 0 1 0 1
More informationELEC 405/ELEC 511 Error Control Coding and Sequences. Hamming Codes and the Hamming Bound
ELEC 45/ELEC 5 Error Control Coding and Sequences Hamming Codes and the Hamming Bound Single Error Correcting Codes ELEC 45 2 Hamming Codes One form of the (7,4,3) Hamming code is generated by This is
More informationOrthogonal Arrays & Codes
Orthogonal Arrays & Codes Orthogonal Arrays - Redux An orthogonal array of strength t, a t-(v,k,λ)-oa, is a λv t x k array of v symbols, such that in any t columns of the array every one of the possible
More informationELEC 405/ELEC 511 Error Control Coding. Hamming Codes and Bounds on Codes
ELEC 405/ELEC 511 Error Control Coding Hamming Codes and Bounds on Codes Single Error Correcting Codes (3,1,3) code (5,2,3) code (6,3,3) code G = rate R=1/3 n-k=2 [ 1 1 1] rate R=2/5 n-k=3 1 0 1 1 0 G
More informationCommunications II Lecture 9: Error Correction Coding. Professor Kin K. Leung EEE and Computing Departments Imperial College London Copyright reserved
Communications II Lecture 9: Error Correction Coding Professor Kin K. Leung EEE and Computing Departments Imperial College London Copyright reserved Outline Introduction Linear block codes Decoding Hamming
More informationAnd for polynomials with coefficients in F 2 = Z/2 Euclidean algorithm for gcd s Concept of equality mod M(x) Extended Euclid for inverses mod M(x)
Outline Recall: For integers Euclidean algorithm for finding gcd s Extended Euclid for finding multiplicative inverses Extended Euclid for computing Sun-Ze Test for primitive roots And for polynomials
More informationMATH Examination for the Module MATH-3152 (May 2009) Coding Theory. Time allowed: 2 hours. S = q
MATH-315201 This question paper consists of 6 printed pages, each of which is identified by the reference MATH-3152 Only approved basic scientific calculators may be used. c UNIVERSITY OF LEEDS Examination
More informationHamming codes and simplex codes ( )
Chapter 6 Hamming codes and simplex codes (2018-03-17) Synopsis. Hamming codes are essentially the first non-trivial family of codes that we shall meet. We start by proving the Distance Theorem for linear
More informationCyclic Redundancy Check Codes
Cyclic Redundancy Check Codes Lectures No. 17 and 18 Dr. Aoife Moloney School of Electronics and Communications Dublin Institute of Technology Overview These lectures will look at the following: Cyclic
More informationarxiv:cs/ v1 [cs.it] 15 Sep 2005
On Hats and other Covers (Extended Summary) arxiv:cs/0509045v1 [cs.it] 15 Sep 2005 Hendrik W. Lenstra Gadiel Seroussi Abstract. We study a game puzzle that has enjoyed recent popularity among mathematicians,
More informationSolutions of Exam Coding Theory (2MMC30), 23 June (1.a) Consider the 4 4 matrices as words in F 16
Solutions of Exam Coding Theory (2MMC30), 23 June 2016 (1.a) Consider the 4 4 matrices as words in F 16 2, the binary vector space of dimension 16. C is the code of all binary 4 4 matrices such that the
More informationCan You Hear Me Now?
Can You Hear Me Now? An Introduction to Coding Theory William J. Turner Department of Mathematics & Computer Science Wabash College Crawfordsville, IN 47933 19 October 2004 W. J. Turner (Wabash College)
More informationThe Hamming Codes and Delsarte s Linear Programming Bound
The Hamming Codes and Delsarte s Linear Programming Bound by Sky McKinley Under the Astute Tutelage of Professor John S. Caughman, IV A thesis submitted in partial fulfillment of the requirements for the
More informationexercise in the previous class (1)
exercise in the previous class () Consider an odd parity check code C whose codewords are (x,, x k, p) with p = x + +x k +. Is C a linear code? No. x =, x 2 =x =...=x k = p =, and... is a codeword x 2
More informationSolutions to problems from Chapter 3
Solutions to problems from Chapter 3 Manjunatha. P manjup.jnnce@gmail.com Professor Dept. of ECE J.N.N. College of Engineering, Shimoga February 28, 2016 For a systematic (7,4) linear block code, the parity
More informationMATH 291T CODING THEORY
California State University, Fresno MATH 291T CODING THEORY Spring 2009 Instructor : Stefaan Delcroix Chapter 1 Introduction to Error-Correcting Codes It happens quite often that a message becomes corrupt
More informationChapter 3 Linear Block Codes
Wireless Information Transmission System Lab. Chapter 3 Linear Block Codes Institute of Communications Engineering National Sun Yat-sen University Outlines Introduction to linear block codes Syndrome and
More informationEE 229B ERROR CONTROL CODING Spring 2005
EE 229B ERROR CONTROL CODING Spring 2005 Solutions for Homework 1 1. Is there room? Prove or disprove : There is a (12,7) binary linear code with d min = 5. If there were a (12,7) binary linear code with
More informationAnswers and Solutions to (Even Numbered) Suggested Exercises in Sections of Grimaldi s Discrete and Combinatorial Mathematics
Answers and Solutions to (Even Numbered) Suggested Exercises in Sections 6.5-6.9 of Grimaldi s Discrete and Combinatorial Mathematics Section 6.5 6.5.2. a. r = = + = c + e. So the error pattern is e =.
More informationMATH 291T CODING THEORY
California State University, Fresno MATH 291T CODING THEORY Fall 2011 Instructor : Stefaan Delcroix Contents 1 Introduction to Error-Correcting Codes 3 2 Basic Concepts and Properties 6 2.1 Definitions....................................
More informationCodes for Partially Stuck-at Memory Cells
1 Codes for Partially Stuck-at Memory Cells Antonia Wachter-Zeh and Eitan Yaakobi Department of Computer Science Technion Israel Institute of Technology, Haifa, Israel Email: {antonia, yaakobi@cs.technion.ac.il
More informationWe saw in the last chapter that the linear Hamming codes are nontrivial perfect codes.
Chapter 5 Golay Codes Lecture 16, March 10, 2011 We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes. Question. Are there any other nontrivial perfect codes? Answer. Yes,
More informationLecture 8: Shannon s Noise Models
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 8: Shannon s Noise Models September 14, 2007 Lecturer: Atri Rudra Scribe: Sandipan Kundu& Atri Rudra Till now we have
More informationChapter 7. Error Control Coding. 7.1 Historical background. Mikael Olofsson 2005
Chapter 7 Error Control Coding Mikael Olofsson 2005 We have seen in Chapters 4 through 6 how digital modulation can be used to control error probabilities. This gives us a digital channel that in each
More informationMATH32031: Coding Theory Part 15: Summary
MATH32031: Coding Theory Part 15: Summary 1 The initial problem The main goal of coding theory is to develop techniques which permit the detection of errors in the transmission of information and, if necessary,
More informationERROR CORRECTING CODES
ERROR CORRECTING CODES To send a message of 0 s and 1 s from my computer on Earth to Mr. Spock s computer on the planet Vulcan we use codes which include redundancy to correct errors. n q Definition. A
More informationGreedy Codes. Theodore Rice
Greedy Codes Theodore Rice 1 Key ideas These greedy codes come from certain positions of combinatorial games. These codes will be identical to other well known codes. Little sophisticated mathematics is
More informationMATH 433 Applied Algebra Lecture 22: Review for Exam 2.
MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric
More informationMATH3302. Coding and Cryptography. Coding Theory
MATH3302 Coding and Cryptography Coding Theory 2010 Contents 1 Introduction to coding theory 2 1.1 Introduction.......................................... 2 1.2 Basic definitions and assumptions..............................
More informationKnow the meaning of the basic concepts: ring, field, characteristic of a ring, the ring of polynomials R[x].
The second exam will be on Friday, October 28, 2. It will cover Sections.7,.8, 3., 3.2, 3.4 (except 3.4.), 4. and 4.2 plus the handout on calculation of high powers of an integer modulo n via successive
More informationERROR-CORRECTING CODES AND LATIN SQUARES
ERROR-CORRECTING CODES AND LATIN SQUARES Ritu Ahuja Department of Mathematics, Khalsa College for Women, Civil Lines, Ludhiana 141001, Punjab, (India) ABSTRACT Data stored and transmitted in digital form
More informationx n k m(x) ) Codewords can be characterized by (and errors detected by): c(x) mod g(x) = 0 c(x)h(x) = 0 mod (x n 1)
Cyclic codes: review EE 387, Notes 15, Handout #26 A cyclic code is a LBC such that every cyclic shift of a codeword is a codeword. A cyclic code has generator polynomial g(x) that is a divisor of every
More informationArrangements, matroids and codes
Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011 References 2/43 1. Codes, arrangements and matroids by Relinde
More informationMATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups.
MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups. Binary codes Let us assume that a message to be transmitted is in binary form. That is, it is a word in the alphabet
More informationPhysical Layer and Coding
Physical Layer and Coding Muriel Médard Professor EECS Overview A variety of physical media: copper, free space, optical fiber Unified way of addressing signals at the input and the output of these media:
More information1. How many errors may be detected (not necessarily corrected) if a code has a Hamming Distance of 6?
Answers to Practice Problems Practice Problems - Hamming distance 1. How many errors may be detected (not necessarily corrected) if a code has a Hamming Distance of 6? 2n = 6; n=3 2. How many errors may
More informationB. Cyclic Codes. Primitive polynomials are the generator polynomials of cyclic codes.
B. Cyclic Codes A cyclic code is a linear block code with the further property that a shift of a codeword results in another codeword. These are based on polynomials whose elements are coefficients from
More informationMathematics Department
Mathematics Department Matthew Pressland Room 7.355 V57 WT 27/8 Advanced Higher Mathematics for INFOTECH Exercise Sheet 2. Let C F 6 3 be the linear code defined by the generator matrix G = 2 2 (a) Find
More informationError Detection and Correction: Hamming Code; Reed-Muller Code
Error Detection and Correction: Hamming Code; Reed-Muller Code Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Hamming Code: Motivation
More information: Error Correcting Codes. October 2017 Lecture 1
03683072: Error Correcting Codes. October 2017 Lecture 1 First Definitions and Basic Codes Amnon Ta-Shma and Dean Doron 1 Error Correcting Codes Basics Definition 1. An (n, K, d) q code is a subset of
More information3. Coding theory 3.1. Basic concepts
3. CODING THEORY 1 3. Coding theory 3.1. Basic concepts In this chapter we will discuss briefly some aspects of error correcting codes. The main problem is that if information is sent via a noisy channel,
More information7.1 Definitions and Generator Polynomials
Chapter 7 Cyclic Codes Lecture 21, March 29, 2011 7.1 Definitions and Generator Polynomials Cyclic codes are an important class of linear codes for which the encoding and decoding can be efficiently implemented
More informationMATH3302 Coding Theory Problem Set The following ISBN was received with a smudge. What is the missing digit? x9139 9
Problem Set 1 These questions are based on the material in Section 1: Introduction to coding theory. You do not need to submit your answers to any of these questions. 1. The following ISBN was received
More informationMATH/MTHE 406 Homework Assignment 2 due date: October 17, 2016
MATH/MTHE 406 Homework Assignment 2 due date: October 17, 2016 Notation: We will use the notations x 1 x 2 x n and also (x 1, x 2,, x n ) to denote a vector x F n where F is a finite field. 1. [20=6+5+9]
More informationELEC-E7240 Coding Methods L (5 cr)
Introduction ELEC-E7240 Coding Methods L (5 cr) Patric Östergård Department of Communications and Networking Aalto University School of Electrical Engineering Spring 2017 Patric Östergård (Aalto) ELEC-E7240
More informationMa/CS 6b Class 24: Error Correcting Codes
Ma/CS 6b Class 24: Error Correcting Codes By Adam Sheffer Communicating Over a Noisy Channel Problem. We wish to transmit a message which is composed of 0 s and 1 s, but noise might accidentally flip some
More informationLower Bounds for q-ary Codes with Large Covering Radius
Lower Bounds for q-ary Codes with Large Covering Radius Wolfgang Haas Immanuel Halupczok Jan-Christoph Schlage-Puchta Albert-Ludwigs-Universität Mathematisches Institut Eckerstr. 1 79104 Freiburg, Germany
More informationMa/CS 6b Class 25: Error Correcting Codes 2
Ma/CS 6b Class 25: Error Correcting Codes 2 By Adam Sheffer Recall: Codes V n the set of binary sequences of length n. For example, V 3 = 000,001,010,011,100,101,110,111. Codes of length n are subsets
More informationCoding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014
Anna Dovzhik 1 Coding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014 Sharing data across channels, such as satellite, television, or compact disc, often
More informationLecture 3: Error Correcting Codes
CS 880: Pseudorandomness and Derandomization 1/30/2013 Lecture 3: Error Correcting Codes Instructors: Holger Dell and Dieter van Melkebeek Scribe: Xi Wu In this lecture we review some background on error
More informationChapter 2 Date Compression: Source Coding. 2.1 An Introduction to Source Coding 2.2 Optimal Source Codes 2.3 Huffman Code
Chapter 2 Date Compression: Source Coding 2.1 An Introduction to Source Coding 2.2 Optimal Source Codes 2.3 Huffman Code 2.1 An Introduction to Source Coding Source coding can be seen as an efficient way
More information: Coding Theory. Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, upattane
2301532 : Coding Theory Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, 2006 http://pioneer.chula.ac.th/ upattane Chapter 1 Error detection, correction and decoding 1.1 Basic definitions and
More informationLecture 12: November 6, 2017
Information and Coding Theory Autumn 017 Lecturer: Madhur Tulsiani Lecture 1: November 6, 017 Recall: We were looking at codes of the form C : F k p F n p, where p is prime, k is the message length, and
More informationInformation redundancy
Information redundancy Information redundancy add information to date to tolerate faults error detecting codes error correcting codes data applications communication memory p. 2 - Design of Fault Tolerant
More information11 Minimal Distance and the Parity Check Matrix
MATH32031: Coding Theory Part 12: Hamming Codes 11 Minimal Distance and the Parity Check Matrix Theorem 23 (Distance Theorem for Linear Codes) Let C be an [n, k] F q -code with parity check matrix H. Then
More informationLecture 17: Perfect Codes and Gilbert-Varshamov Bound
Lecture 17: Perfect Codes and Gilbert-Varshamov Bound Maximality of Hamming code Lemma Let C be a code with distance 3, then: C 2n n + 1 Codes that meet this bound: Perfect codes Hamming code is a perfect
More informationMT361/461/5461 Error Correcting Codes: Preliminary Sheet
MT361/461/5461 Error Correcting Codes: Preliminary Sheet Solutions to this sheet will be posted on Moodle on 14th January so you can check your answers. Please do Question 2 by 14th January. You are welcome
More informationError Correcting Codes Prof. Dr. P. Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore
(Refer Slide Time: 00:15) Error Correcting Codes Prof. Dr. P. Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore Lecture No. # 03 Mathematical Preliminaries:
More informationG Solution (10 points) Using elementary row operations, we transform the original generator matrix as follows.
EE 387 October 28, 2015 Algebraic Error-Control Codes Homework #4 Solutions Handout #24 1. LBC over GF(5). Let G be a nonsystematic generator matrix for a linear block code over GF(5). 2 4 2 2 4 4 G =
More informationLecture Introduction. 2 Linear codes. CS CTT Current Topics in Theoretical CS Oct 4, 2012
CS 59000 CTT Current Topics in Theoretical CS Oct 4, 01 Lecturer: Elena Grigorescu Lecture 14 Scribe: Selvakumaran Vadivelmurugan 1 Introduction We introduced error-correcting codes and linear codes in
More informationMultimedia Communications. Mathematical Preliminaries for Lossless Compression
Multimedia Communications Mathematical Preliminaries for Lossless Compression What we will see in this chapter Definition of information and entropy Modeling a data source Definition of coding and when
More informationCOMM901 Source Coding and Compression. Quiz 1
German University in Cairo - GUC Faculty of Information Engineering & Technology - IET Department of Communication Engineering Winter Semester 2013/2014 Students Name: Students ID: COMM901 Source Coding
More informationMath 512 Syllabus Spring 2017, LIU Post
Week Class Date Material Math 512 Syllabus Spring 2017, LIU Post 1 1/23 ISBN, error-detecting codes HW: Exercises 1.1, 1.3, 1.5, 1.8, 1.14, 1.15 If x, y satisfy ISBN-10 check, then so does x + y. 2 1/30
More informationEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Linear Block Codes February 14, 2007 1. Code 1: n = 4, n k = 2 Parity Check Equations: x 1 + x 3 = 0, x 1 + x 2 + x 4 = 0 Parity Bits: x 3 = x 1,
More informationA Brief Encounter with Linear Codes
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 8-2014 A Brief Encounter with Linear Codes Brent El-Bakri Boise State University, brentelbakri@boisestate.edu
More informationWeek 3: January 22-26, 2018
EE564/CSE554: Error Correcting Codes Spring 2018 Lecturer: Viveck R. Cadambe Week 3: January 22-26, 2018 Scribe: Yu-Tse Lin Disclaimer: These notes have not been subjected to the usual scrutiny reserved
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is
More informationMathematics 222 (A1) Midterm Examination May 24, 2002
Mathematics (A) Midterm Examination May 4, 00 Department of Mathematical and Statistical Sciences University of Alberta Instructor: I. E. Leonard Time: 70 Minutes. There were 5 children in the Emergency
More informationSection 3 Error Correcting Codes (ECC): Fundamentals
Section 3 Error Correcting Codes (ECC): Fundamentals Communication systems and channel models Definition and examples of ECCs Distance For the contents relevant to distance, Lin & Xing s book, Chapter
More informationCS6304 / Analog and Digital Communication UNIT IV - SOURCE AND ERROR CONTROL CODING PART A 1. What is the use of error control coding? The main use of error control coding is to reduce the overall probability
More informationReed-Muller Codes. These codes were discovered by Muller and the decoding by Reed in Code length: n = 2 m, Dimension: Minimum Distance
Reed-Muller Codes Ammar Abh-Hhdrohss Islamic University -Gaza ١ Reed-Muller Codes These codes were discovered by Muller and the decoding by Reed in 954. Code length: n = 2 m, Dimension: Minimum Distance
More informationTHIS paper is aimed at designing efficient decoding algorithms
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 7, NOVEMBER 1999 2333 Sort-and-Match Algorithm for Soft-Decision Decoding Ilya Dumer, Member, IEEE Abstract Let a q-ary linear (n; k)-code C be used
More informationLinear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x
Coding Theory Massoud Malek Linear Cyclic Codes Polynomial and Words A polynomial of degree n over IK is a polynomial p(x) = a 0 + a 1 x + + a n 1 x n 1 + a n x n, where the coefficients a 0, a 1, a 2,,
More informationThe extended coset leader weight enumerator
The extended coset leader weight enumerator Relinde Jurrius Ruud Pellikaan Eindhoven University of Technology, The Netherlands Symposium on Information Theory in the Benelux, 2009 1/14 Outline Codes, weights
More informationNotes on Alekhnovich s cryptosystems
Notes on Alekhnovich s cryptosystems Gilles Zémor November 2016 Decisional Decoding Hypothesis with parameter t. Let 0 < R 1 < R 2 < 1. There is no polynomial-time decoding algorithm A such that: Given
More informationMAS309 Coding theory
MAS309 Coding theory Matthew Fayers January March 2008 This is a set of notes which is supposed to augment your own notes for the Coding Theory course They were written by Matthew Fayers, and very lightly
More informationCSCI 2570 Introduction to Nanocomputing
CSCI 2570 Introduction to Nanocomputing Information Theory John E Savage What is Information Theory Introduced by Claude Shannon. See Wikipedia Two foci: a) data compression and b) reliable communication
More informationCodes over Subfields. Chapter Basics
Chapter 7 Codes over Subfields In Chapter 6 we looked at various general methods for constructing new codes from old codes. Here we concentrate on two more specialized techniques that result from writing
More informationThe BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes
S-723410 BCH and Reed-Solomon Codes 1 S-723410 BCH and Reed-Solomon Codes 3 Background The algebraic structure of linear codes and, in particular, cyclic linear codes, enables efficient encoding and decoding
More informationGenerator Matrix. Theorem 6: If the generator polynomial g(x) of C has degree n-k then C is an [n,k]-cyclic code. If g(x) = a 0. a 1 a n k 1.
Cyclic Codes II Generator Matrix We would now like to consider how the ideas we have previously discussed for linear codes are interpreted in this polynomial version of cyclic codes. Theorem 6: If the
More informationError Correcting Codes: Combinatorics, Algorithms and Applications Spring Homework Due Monday March 23, 2009 in class
Error Correcting Codes: Combinatorics, Algorithms and Applications Spring 2009 Homework Due Monday March 23, 2009 in class You can collaborate in groups of up to 3. However, the write-ups must be done
More information3F1 Information Theory, Lecture 3
3F1 Information Theory, Lecture 3 Jossy Sayir Department of Engineering Michaelmas 2011, 28 November 2011 Memoryless Sources Arithmetic Coding Sources with Memory 2 / 19 Summary of last lecture Prefix-free
More informationCyclic codes: overview
Cyclic codes: overview EE 387, Notes 14, Handout #22 A linear block code is cyclic if the cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c(x)
More informationError Correction Review
Error Correction Review A single overall parity-check equation detects single errors. Hamming codes used m equations to correct one error in 2 m 1 bits. We can use nonbinary equations if we create symbols
More informationModular numbers and Error Correcting Codes. Introduction. Modular Arithmetic.
Modular numbers and Error Correcting Codes Introduction Modular Arithmetic Finite fields n-space over a finite field Error correcting codes Exercises Introduction. Data transmission is not normally perfect;
More informationChapter 4. Data Transmission and Channel Capacity. Po-Ning Chen, Professor. Department of Communications Engineering. National Chiao Tung University
Chapter 4 Data Transmission and Channel Capacity Po-Ning Chen, Professor Department of Communications Engineering National Chiao Tung University Hsin Chu, Taiwan 30050, R.O.C. Principle of Data Transmission
More informationLecture B04 : Linear codes and singleton bound
IITM-CS6845: Theory Toolkit February 1, 2012 Lecture B04 : Linear codes and singleton bound Lecturer: Jayalal Sarma Scribe: T Devanathan We start by proving a generalization of Hamming Bound, which we
More informationA 2-error Correcting Code
A 2-error Correcting Code Basic Idea We will now try to generalize the idea used in Hamming decoding to obtain a linear code that is 2-error correcting. In the Hamming decoding scheme, the parity check
More informationLinear Codes and Syndrome Decoding
Linear Codes and Syndrome Decoding These notes are intended to be used as supplementary reading to Sections 6.7 9 of Grimaldi s Discrete and Combinatorial Mathematics. The proofs of the theorems are left
More informationPerfect Two-Fault Tolerant Search with Minimum Adaptiveness 1
Advances in Applied Mathematics 25, 65 101 (2000) doi:10.1006/aama.2000.0688, available online at http://www.idealibrary.com on Perfect Two-Fault Tolerant Search with Minimum Adaptiveness 1 Ferdinando
More informationLecture 12. Block Diagram
Lecture 12 Goals Be able to encode using a linear block code Be able to decode a linear block code received over a binary symmetric channel or an additive white Gaussian channel XII-1 Block Diagram Data
More informationCyclic codes. Vahid Meghdadi Reference: Error Correction Coding by Todd K. Moon. February 2008
Cyclic codes Vahid Meghdadi Reference: Error Correction Coding by Todd K. Moon February 2008 1 Definitions Definition 1. A ring < R, +,. > is a set R with two binary operation + (addition) and. (multiplication)
More informationApril 25 May 6, 2016, Verona, Italy. GAME THEORY and APPLICATIONS Mikhail Ivanov Krastanov
April 25 May 6, 2016, Verona, Italy GAME THEORY and APPLICATIONS Mikhail Ivanov Krastanov Games in normal form There are given n-players. The set of all strategies (possible actions) of the i-th player
More informationError-Correcting Codes
Error-Correcting Codes HMC Algebraic Geometry Final Project Dmitri Skjorshammer December 14, 2010 1 Introduction Transmission of information takes place over noisy signals. This is the case in satellite
More informationLecture 2 Linear Codes
Lecture 2 Linear Codes 2.1. Linear Codes From now on we want to identify the alphabet Σ with a finite field F q. For general codes, introduced in the last section, the description is hard. For a code of
More informationCoding for a Non-symmetric Ternary Channel
Coding for a Non-symmetric Ternary Channel Nicolas Bitouzé and Alexandre Graell i Amat Department of Electronics, Institut TELECOM-TELECOM Bretagne, 938 Brest, France Email: nicolas.bitouze@telecom-bretagne.eu,
More informationFinite Mathematics. Nik Ruškuc and Colva M. Roney-Dougal
Finite Mathematics Nik Ruškuc and Colva M. Roney-Dougal September 19, 2011 Contents 1 Introduction 3 1 About the course............................. 3 2 A review of some algebraic structures.................
More informationPUTNAM PROBLEM SOLVING SEMINAR WEEK 7 This is the last meeting before the Putnam. The Rules. These are way too many problems to consider. Just pick a
PUTNAM PROBLEM SOLVING SEMINAR WEEK 7 This is the last meeting before the Putnam The Rules These are way too many problems to consider Just pick a few problems in one of the sections and play around with
More information16.36 Communication Systems Engineering
MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.36: Communication
More informationCross-Error Correcting Integer Codes over Z 2
Cross-Error Correcting Integer Codes over Z 2 m Anna-Lena Trautmann Department of Electrical & Computer Systems Eng., Monash University Department of Electrical & Electronic Engineering, University of
More information