Linear Block Codes. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
|
|
- Hubert Sydney Griffin
- 6 years ago
- Views:
Transcription
1 1 / 26 Linear Block Codes Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay July 28, 2014
2 Binary Block Codes
3 3 / 26 Let F 2 be the set {0, 1}. Binary Block Code Definition An (n, k) binary block code is a subset of F n 2 containing 2k elements Example n = 3, k = 1, C = {000, 111} Example n 2, C = Set of vectors of even Hamming weight in F n 2, k = n 1 n = 3, k = 2, C = {000, 011, 101, 110} This code is called the single parity check code
4 4 / 26 Encoding Binary Block Codes The encoder maps k-bit information blocks to codewords. Definition An encoder for an (n, k) binary block code C is an injective function from F k 2 to C Example (3-Repetition Code) 0 000, or 1 000, 0 111
5 5 / 26 Decoding Binary Block Codes The decoder maps n-bit received blocks to codewords Definition A decoder for an (n, k) binary block code is a function from F n 2 to C Example (3-Repetition Code) n = 3, C = {000, 111} Since encoding is injective, information bits can be recovered as 000 0, 111 1
6 6 / 26 Optimal Decoder for Binary Block Codes Optimality criterion: Maximum probability of correct decision Let x C be the transmitted codeword Let y F n 2 be the received vector Maximum a posteriori (MAP) decoder is optimal ˆx MAP = argmax x C Pr(x y) If all codewords are equally likely to be transmitted, then maximum likelihood (ML) decoder is optimal ˆx ML = argmax x C Pr(y x) Over a BSC with p < 1 2, the minimum distance decoder is optimal if the codewords are equally likely ˆx = argmin x C d(x, y)
7 7 / 26 Error Correction Capability of Binary Block Codes Definition The minimum distance of a block code C is defined as d min = min d(x, y) x,y C,x y Example (3-Repetition Code) C = {000, 111}, d min = 3 Example (Single Parity Check Code) C = Set of vectors of even weight in F n 2, d min = 2 Theorem For a binary block code with minimum distance d min, the minimum distance decoder can correct upto d min 1 2 errors.
8 8 / 26 Complexity of Encoding and Decoding Encoder Map from F k 2 to C Worst case storage requirement = O(n2 k ) Decoder Map from F n 2 to C ˆx ML = argmax x C Pr(y x) Worst case storage requirement = O(n2 k ) Time complexity = O(n2 k ) Need more structure to reduce complexity
9 Binary Linear Block Codes
10 10 / 26 Vector Spaces over F 2 Define the following operations on F 2 Addition = = = = 0 Multiplication 0 0 = = = = 1 F 2 is also represented as GF(2) Fact The set F n 2 is a vector space over F 2
11 11 / 26 Binary Linear Block Code Definition An (n, k) binary linear block code is a k-dimensional subspace of F n 2 Theorem Let S be a nonempty subset of F n 2. Then S is a subspace of Fn 2 if u + v S for any two u and v in S. Example (3-Repetition Code) C = {000, 111} φ = 000, = 111, = 000 Example (Single Parity Check Code) C = Set of vectors of even weight in F n 2 wt(u + v) = wt(u) + wt(v) 2 wt(u v)
12 12 / 26 Encoding Binary Linear Block Codes Definition A generator matrix for a k-dimensional binary linear block code C is a k n matrix G whose rows form a basis for C. Linear Block Code Encoder Let u be a 1 k binary vector of information bits. The corresponding codeword is v = ug Example (3-Repetition Code) G = [ ] [ ] = [ 0 ] [ ] [ ] = [ 1 ] [ ]
13 13 / 26 Encoding Binary Linear Block Codes Example (Single Parity Check Code) n = 3, k = 2, C = {000, 011, 101, 110} [ ] G = [ 0 0 ] 0 [ 0 1 ] 1 [ 1 0 ] 1 [ 1 1 ] 0 = [ 0 0 ] [ ] = [ 0 1 ] [ ] = [ 1 0 ] [ ] = [ 1 1 ] [ ]
14 14 / 26 Encoding Complexity of Binary Linear Block Codes Need to store G Storage requirement = O(nk) O(n2 k ) Time complexity = O(nk) Complexity can be reduced further by imposing more structure in addition to linearity Decoding complexity? What is the optimal decoder?
15 15 / 26 Decoding Binary Linear Block Codes Codewords are equally likely ML decoder is optimal ˆx ML = argmax x C Pr(y x) Equally likely codewords and channel is BSC Minimum distance decoder is optimal ˆx ML = argmin x C d(x, y) To exploit linear structure to reduce decoding complexity, we need to study the dual code
16 16 / 26 Inner Product of Vectors in F n 2 Definition Let u = (u 1, u 2,..., u n ) and v = (v 1, v 2,..., v n ) belong to F n 2. The inner product of u and v is given by u v = n u i v i i=1 u v = 0 u and v are orthogonal. Examples ( ) (0 1 1 ) = = 0 ( ) (0 1 1 ) = = 1 ( ) (0 1 1 ) = = 0 ( ) (0 1 1 ) = = 0 Nonzero vectors can be self-orthogonal
17 Dual Code of a Linear Block Code Definition Let C be an (n, k) binary linear block code. Let C be the set of vectors in F n 2 which are orthogonal to all the codewords in C. { } C = u F n 2 u v = 0 for all v C C is a linear block code and is called the dual code of C. Example (3-Repetition Code) C = {000, 111}, C =? = = = = = = = = 0 C = {000, 011, 101, 110} = Single Parity Check Code 17 / 26
18 18 / 26 Dimension of the Dual Code Example (3-Repetition Code and SPC Code) C = {000, 111}, dim C = 1 C = {000, 011, 101, 110}, dim C = 2 dim C + dim C = = 3 Theorem dim C + dim C = n Corollary C is an (n, k) binary linear block code C is an (n, n k) binary linear block code
19 19 / 26 Parity Check Matrix of a Code Definition Let C be an (n, k) binary linear block code and let C be its dual code. A generator matrix H for C is called a parity check matrix for C. Example (3-Repetition Code) C = {000, 111} C = {000, 011, 101, 110} A generator matrix of C is H = H is a parity check matrix of C. [ 1 0 ]
20 20 / 26 Parity Check Matrix Completely Describes a Code Theorem Let C be a linear block code with parity check matrix H. Then v C v H T = 0 Example (3-Repetition [ Code) ] C = {000, 111}, H = Forward direction: v C v H T = 0 [ ] = [ 0 0 ], [ ] = [ 0 0 ]
21 21 / 26 Parity Check Matrix Completely Describes a Code Theorem Let C be a linear block code with parity check matrix H. Then v C v H T = 0 Example (3-Repetition [ Code) ] C = {000, 111}, H = Reverse direction: v C v H T = 0 v H T = [ ] 1 0 v 1 v 2 v = [ ] v 1 + v 3 v 2 + v v H T = 0 v 1 + v 3 = 0, v 2 + v 3 = 0 v 1 = v 3, v 2 = v 3 v 1 = v 2 = v 3
22 22 / 26 Decoding Binary Linear Block Codes Let a codeword x be sent through a BSC to get y, where e is the error vector y = x + e The probability of observing y given x was transmitted is given by Pr(y x) = p d(x,y) (1 p) n d(x,y) = p wt(e) (1 p) n wt(e) = (1 p) n ( p 1 p ) wt(e) If p < 1 2, lower weight error vectors are more likely
23 Decoding Binary Linear Block Codes Optimal decoder is given by ˆx ML = argmin x C d(x, y) = y + ê ML where ê ML = Most likely error vector such that y + e C. y + e C (y + e) H T = 0 e H T = y H T If s = y H T, the most likely error vector is ê ML = argmin wt(e) e F n 2,e HT =s Time complexity = O ( p(n)2 k) where p is a polynomial For each s, the ê ML can be precomputed and stored s is 1 n k binary vector Storage required is O(n2 n k ) 23 / 26
24 Summary
25 25 / 26 Complexity Comparison General Block Codes Encoding = O(n2 k ) Decoding = O(n2 k ) Linear Block Codes Encoding = O(nk) Decoding = O(p(n)2 min(k,n k) ) Observations Linear structure in codes reduces encoding complexity Decoding complexity is still exponential Need for codes with low complexity decoders
26 Questions? Takeaways? 26 / 26
EE512: 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 informationSIPCom8-1: Information Theory and Coding Linear Binary Codes Ingmar Land
SIPCom8-1: Information Theory and Coding Linear Binary Codes Ingmar Land Ingmar Land, SIPCom8-1: Information Theory and Coding (2005 Spring) p.1 Overview Basic Concepts of Channel Coding Block Codes I:
More informationCyclic Codes. Saravanan Vijayakumaran August 26, Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 25 Cyclic Codes Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 26, 2014 2 / 25 Cyclic Codes Definition A cyclic shift
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 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 informationFinite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together
More informationAlgebraic Geometry Codes. Shelly Manber. Linear Codes. Algebraic Geometry Codes. Example: Hermitian. Shelly Manber. Codes. Decoding.
Linear December 2, 2011 References Linear Main Source: Stichtenoth, Henning. Function Fields and. Springer, 2009. Other Sources: Høholdt, Lint and Pellikaan. geometry codes. Handbook of Coding Theory,
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 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 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 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 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 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 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 informationLDPC Codes. Slides originally from I. Land p.1
Slides originally from I. Land p.1 LDPC Codes Definition of LDPC Codes Factor Graphs to use in decoding Decoding for binary erasure channels EXIT charts Soft-Output Decoding Turbo principle applied to
More informationCode design: Computer search
Code design: Computer search Low rate codes Represent the code by its generator matrix Find one representative for each equivalence class of codes Permutation equivalences? Do NOT try several generator
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 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 informationMessage Passing Algorithm and Linear Programming Decoding for LDPC and Linear Block Codes
Message Passing Algorithm and Linear Programming Decoding for LDPC and Linear Block Codes Institute of Electronic Systems Signal and Information Processing in Communications Nana Traore Shashi Kant Tobias
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 informationCoding Theory and Applications. Linear Codes. Enes Pasalic University of Primorska Koper, 2013
Coding Theory and Applications Linear Codes Enes Pasalic University of Primorska Koper, 2013 2 Contents 1 Preface 5 2 Shannon theory and coding 7 3 Coding theory 31 4 Decoding of linear codes and MacWilliams
More informationLecture 4 : Introduction to Low-density Parity-check Codes
Lecture 4 : Introduction to Low-density Parity-check Codes LDPC codes are a class of linear block codes with implementable decoders, which provide near-capacity performance. History: 1. LDPC codes were
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 Fall 2011 Instructor : Stefaan Delcroix Contents 1 Introduction to Error-Correcting Codes 3 2 Basic Concepts and Properties 6 2.1 Definitions....................................
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 informationGraph-based codes for flash memory
1/28 Graph-based codes for flash memory Discrete Mathematics Seminar September 3, 2013 Katie Haymaker Joint work with Professor Christine Kelley University of Nebraska-Lincoln 2/28 Outline 1 Background
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 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 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 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 informationParameter Estimation
1 / 44 Parameter Estimation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay October 25, 2012 Motivation System Model used to Derive
More informationAn Introduction to (Network) Coding Theory
An Introduction to (Network) Coding Theory Anna-Lena Horlemann-Trautmann University of St. Gallen, Switzerland July 12th, 2018 1 Coding Theory Introduction Reed-Solomon codes 2 Introduction Coherent network
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 informationIntroduction to Convolutional Codes, Part 1
Introduction to Convolutional Codes, Part 1 Frans M.J. Willems, Eindhoven University of Technology September 29, 2009 Elias, Father of Coding Theory Textbook Encoder Encoder Properties Systematic Codes
More informationLecture 14: Hamming and Hadamard Codes
CSCI-B69: A Theorist s Toolkit, Fall 6 Oct 6 Lecture 4: Hamming and Hadamard Codes Lecturer: Yuan Zhou Scribe: Kaiyuan Zhu Recap Recall from the last lecture that error-correcting codes are in fact injective
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 informationBelief-Propagation Decoding of LDPC Codes
LDPC Codes: Motivation Belief-Propagation Decoding of LDPC Codes Amir Bennatan, Princeton University Revolution in coding theory Reliable transmission, rates approaching capacity. BIAWGN, Rate =.5, Threshold.45
More informationBinary Linear Codes G = = [ I 3 B ] , G 4 = None of these matrices are in standard form. Note that the matrix 1 0 0
Coding Theory Massoud Malek Binary Linear Codes Generator and Parity-Check Matrices. A subset C of IK n is called a linear code, if C is a subspace of IK n (i.e., C is closed under addition). A linear
More informationChapter 2. Error Correcting Codes. 2.1 Basic Notions
Chapter 2 Error Correcting Codes The identification number schemes we discussed in the previous chapter give us the ability to determine if an error has been made in recording or transmitting information.
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 informationMT5821 Advanced Combinatorics
MT5821 Advanced Combinatorics 1 Error-correcting codes In this section of the notes, we have a quick look at coding theory. After a motivating introduction, we discuss the weight enumerator of a code,
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 informationLecture Notes on Channel Coding
Lecture Notes on Channel Coding arxiv:1607.00974v1 [cs.it] 4 Jul 2016 Georg Böcherer Institute for Communications Engineering Technical University of Munich, Germany georg.boecherer@tum.de July 5, 2016
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 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 informationAn Introduction to (Network) Coding Theory
An to (Network) Anna-Lena Horlemann-Trautmann University of St. Gallen, Switzerland April 24th, 2018 Outline 1 Reed-Solomon Codes 2 Network Gabidulin Codes 3 Summary and Outlook A little bit of history
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 informationECEN 655: Advanced Channel Coding
ECEN 655: Advanced Channel Coding Course Introduction Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 655: Advanced Channel Coding 1 / 19 Outline 1 History
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 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 informationQuasi-cyclic Low Density Parity Check codes with high girth
Quasi-cyclic Low Density Parity Check codes with high girth, a work with Marta Rossi, Richard Bresnan, Massimilliano Sala Summer Doctoral School 2009 Groebner bases, Geometric codes and Order Domains Dept
More information4 An Introduction to Channel Coding and Decoding over BSC
4 An Introduction to Channel Coding and Decoding over BSC 4.1. Recall that channel coding introduces, in a controlled manner, some redundancy in the (binary information sequence that can be used at the
More informationVector Spaces. distributive law u,v. Associative Law. 1 v v. Let 1 be the unit element in F, then
1 Def: V be a set of elements with a binary operation + is defined. F be a field. A multiplication operator between a F and v V is also defined. The V is called a vector space over the field F if: V is
More informationOptimum Soft Decision Decoding of Linear Block Codes
Optimum Soft Decision Decoding of Linear Block Codes {m i } Channel encoder C=(C n-1,,c 0 ) BPSK S(t) (n,k,d) linear modulator block code Optimal receiver AWGN Assume that [n,k,d] linear block code C is
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 informationChannel Coding for Secure Transmissions
Channel Coding for Secure Transmissions March 27, 2017 1 / 51 McEliece Cryptosystem Coding Approach: Noiseless Main Channel Coding Approach: Noisy Main Channel 2 / 51 Outline We present an overiew of linear
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 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 informationLecture 4: Linear Codes. Copyright G. Caire 88
Lecture 4: Linear Codes Copyright G. Caire 88 Linear codes over F q We let X = F q for some prime power q. Most important case: q =2(binary codes). Without loss of generality, we may represent the information
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 informationU Logo Use Guidelines
COMP2610/6261 - Information Theory Lecture 22: Hamming Codes U Logo Use Guidelines Mark Reid and Aditya Menon logo is a contemporary n of our heritage. presents our name, d and our motto: arn the nature
More informationLow-Density Parity-Check Codes
Department of Computer Sciences Applied Algorithms Lab. July 24, 2011 Outline 1 Introduction 2 Algorithms for LDPC 3 Properties 4 Iterative Learning in Crowds 5 Algorithm 6 Results 7 Conclusion PART I
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 information6.1.1 What is channel coding and why do we use it?
Chapter 6 Channel Coding 6.1 Introduction 6.1.1 What is channel coding and why do we use it? Channel coding is the art of adding redundancy to a message in order to make it more robust against noise. It
More information9 THEORY OF CODES. 9.0 Introduction. 9.1 Noise
9 THEORY OF CODES Chapter 9 Theory of Codes After studying this chapter you should understand what is meant by noise, error detection and correction; be able to find and use the Hamming distance for a
More informationOn Extremal Codes With Automorphisms
On Extremal Codes With Automorphisms Anton Malevich Magdeburg, 20 April 2010 joint work with S. Bouyuklieva and W. Willems 1/ 33 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes
More informationCoding Techniques for Data Storage Systems
Coding Techniques for Data Storage Systems Thomas Mittelholzer IBM Zurich Research Laboratory /8 Göttingen Agenda. Channel Coding and Practical Coding Constraints. Linear Codes 3. Weight Enumerators and
More informationOn Two Probabilistic Decoding Algorithms for Binary Linear Codes
On Two Probabilistic Decoding Algorithms for Binary Linear Codes Miodrag Živković Abstract A generalization of Sullivan inequality on the ratio of the probability of a linear code to that of any of its
More informationchannel of communication noise Each codeword has length 2, and all digits are either 0 or 1. Such codes are called Binary Codes.
5 Binary Codes You have already seen how check digits for bar codes (in Unit 3) and ISBN numbers (Unit 4) are used to detect errors. Here you will look at codes relevant for data transmission, for example,
More informationLinear Codes, Target Function Classes, and Network Computing Capacity
Linear Codes, Target Function Classes, and Network Computing Capacity Rathinakumar Appuswamy, Massimo Franceschetti, Nikhil Karamchandani, and Kenneth Zeger IEEE Transactions on Information Theory Submitted:
More informationIntroduction to Low-Density Parity Check Codes. Brian Kurkoski
Introduction to Low-Density Parity Check Codes Brian Kurkoski kurkoski@ice.uec.ac.jp Outline: Low Density Parity Check Codes Review block codes History Low Density Parity Check Codes Gallager s LDPC code
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 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 informationBinary Convolutional Codes
Binary Convolutional Codes A convolutional code has memory over a short block length. This memory results in encoded output symbols that depend not only on the present input, but also on past inputs. An
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A real-valued
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 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 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 informationECEN 604: Channel Coding for Communications
ECEN 604: Channel Coding for Communications Lecture: Introduction to Cyclic Codes Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 604: Channel Coding for Communications
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 informationELEC 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 informationIntroduction into Quantum Computations Alexei Ashikhmin Bell Labs
Introduction into Quantum Computations Alexei Ashikhmin Bell Labs Workshop on Quantum Computing and its Application March 16, 2017 Qubits Unitary transformations Quantum Circuits Quantum Measurements Quantum
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 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 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 informationMTAT : Introduction to Coding Theory. Lecture 1
MTAT05082: Introduction to Coding Theory Instructor: Dr Vitaly Skachek Lecture 1 University of Tartu Scribe: Saad Usman Khan Introduction Information theory studies reliable information transmission over
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 informationAnother view of the division property
Another view of the division property Christina Boura and Anne Canteaut Université de Versailles-St Quentin, France Inria Paris, France Dagstuhl seminar, January 2016 Motivation E K : block cipher with
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 informationFactor Graphs and Message Passing Algorithms Part 1: Introduction
Factor Graphs and Message Passing Algorithms Part 1: Introduction Hans-Andrea Loeliger December 2007 1 The Two Basic Problems 1. Marginalization: Compute f k (x k ) f(x 1,..., x n ) x 1,..., x n except
More informationLinear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x f(x) = q(x)h(x) + r(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 + + a n 1 x n 1 + a n x n, where the coefficients a 1, a 2,, a n are
More informationNetwork Coding and Schubert Varieties over Finite Fields
Network Coding and Schubert Varieties over Finite Fields Anna-Lena Horlemann-Trautmann Algorithmics Laboratory, EPFL, Schweiz October 12th, 2016 University of Kentucky What is this talk about? 1 / 31 Overview
More informationLecture 8: Channel and source-channel coding theorems; BEC & linear codes. 1 Intuitive justification for upper bound on channel capacity
5-859: Information Theory and Applications in TCS CMU: Spring 23 Lecture 8: Channel and source-channel coding theorems; BEC & linear codes February 7, 23 Lecturer: Venkatesan Guruswami Scribe: Dan Stahlke
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 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 informationCombinatória e Teoria de Códigos Exercises from the notes. Chapter 1
Combinatória e Teoria de Códigos Exercises from the notes Chapter 1 1.1. The following binary word 01111000000?001110000?00110011001010111000000000?01110 encodes a date. The encoding method used consisted
More informationFlip-N-Write: A Simple Deterministic Technique to Improve PRAM Write Performance, Energy and Endurance. Presenter: Brian Wongchaowart March 17, 2010
Flip-N-Write: A Simple Deterministic Technique to Improve PRAM Write Performance, Energy and Endurance Sangyeun Cho Hyunjin Lee Presenter: Brian Wongchaowart March 17, 2010 Motivation Suppose that you
More informationEE512: Error Control Coding
EE51: Error Control Coding Solution for Assignment on BCH and RS Codes March, 007 1. To determine the dimension and generator polynomial of all narrow sense binary BCH codes of length n = 31, we have to
More informationThe idea is that if we restrict our attention to any k positions in x, no matter how many times we
k-wise Independence and -biased k-wise Indepedence February 0, 999 Scribe: Felix Wu Denitions Consider a distribution D on n bits x x x n. D is k-wise independent i for all sets of k indices S fi ;:::;i
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 information