MATH32031: Coding Theory Part 15: Summary

Size: px
Start display at page:

Download "MATH32031: Coding Theory Part 15: Summary"

Transcription

1 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, allow the original message to be reconstructed. For our purposes, a p-ary code C of length n is a subset of F (n p = (F p,..., F p, A p-ary codeword of length n is simply a vector x = (x 1,..., x n with the x i F p (that is, a member of the n-dimensional vector space over F p and the property that xis a member of the code, that is x C F (n p. 2 Hamming distance Definition The Hamming distance d(x, y between two vectors x, y F (n is the number of coefficients in which they differ. Symmetric channels. Suppose that a codeword x is transmitted and y is received. If the nearest neighbour of y in C is x C then it seems reasonable to suppose that x was the original message. To justify this we suppose: Errors in different positions in a word are independent; the occurrence of an error in one position in the word does not affect the probability of error in another position. Each symbol f F has the same probability r of being erroneously transmitted. We also assume that this probability of error is small, r 1/2. If f F is mistransmitted, then we suppose all q 1 remaining symbols are equally likely to be received. Definition. The minimum distance of a code C, denoted d(c, is d(c = min (d(x, y x, y C, x y Theorem 2 (a If, for a code C, then C can detect up to s errors. d(c s + 1 (b If d(c 2t + 1 then the code C can correct up to t errors. i

2 3 Equivalence of codes Definition. Two q-ary codes are called equivalent if one can be obtained from the other by means of the following 2 types of operation: (a a permutation of the positions of the code; (b a permutation of the symbols appearing in a fixed position. Given a code of length n with M elements we can write the elements of C in an M n matrix. For this representation: operation (a corresponds to a permutations of columns; operation (b is just a relabelling of the symbols in a given column. Notice that a permutation of the rows does not alter the code, since the codewords are just permuted. Proposition 5 The distance function is invariant under these operations (and hence under equivalence. The sphere of radius r about u is Lemma 7 S r (u = {x F (n d(x, u r}. (a The number of vectors in F (n 2 of weight i is ( n n! = i i!(n i! (b For r n, and F = F q, S r (u contains exactly S r (u = ( n 0 + ( n 1 ( n (q r (q 1 r = r i=0 ( n i (q 1 i points of F (n. Theorem 8 (Hamming Bound or Sphere Packing Bound If there is a q-ary (n, M, 2t + 1-code, then the following inequality is satisfied: { t ( } n M S t (u q n, or equivalently M (q 1 i q n. i For a linear code of dimension k this becomes S t (u q n k. Perfect codes. A code whose parameters give equality in the sphere packing bound is called a perfect code. (This amounts to saying F (n q is completely covered by the nonintersecting spheres S t (c. ii i=0

3 Linear codes. A code C F n is called an F -linear code if C is an F -vector subspace. Weight. in v. Let v F n. Then the weight of v, w(v, is the number of non-zero co-ordinates Lemma For x, y F n d(x, y = w(x y. Weight of a code. If C is a code then the weight w(c of C is defined as w(c = min{ w(x x C, x 0 }. Theorem If C is a linear code then d(c = w(c. 4 Generator matrices Let r 1,..., r k denote a basis for an F -linear code C F (n. Recall every c C can be written uniquely as c = λ i r i for λ i F. Thus the code C is completely specified by the k n matrix This is called a generator matrix for C, and is usually denoted by G = G(C. r 1. r k. 5 Equivalence of linear codes Definition. Two F -linear codes C, C are (F -linearly equivalent if one can be obtained from the other by means of a combination of transpositions of the following types: (a a permutation of the positions of the code; (b multiplication of symbols in a fixed position by a non-zero scalar in F. Theorem 9 Two k n matrices generate (F -linearly equivalent codes if one can be obtained from the other by a sequence of operations of the following types: (R1 Permutation of rows. (R2 Multiplication of a row by a non-zero scalar. iii

4 (R3 Adding a scalar multiple of one row to another row. (C1 Permutation of columns. (C2 Multiplication of any fixed column by a non-zero scalar. Theorem 10 Let G be the generator matrix of an [n, k] code over F. Then by using operations R1, R2, R3, C1, C2, G can be transformed to a standard form matrix [I k A] where I k = is the (k k identity matrix and A is a (k (n k matrix. 6 Encoding From the point of view of applications, C encodes an arbitrary vector from F (k as a codeword in F (n. Let u = (u 1,..., u k F (k be the message. Produce a codeword x by the (linear transformation u k u i r i = ug C F n. i=1 7 Decoding Consider the cosets F (n /C. A coset leader is a choice of vector of minimal weight in its coset. Write the coset leaders as a i, codewords as c i. Standard array decoding 0 c 1... c q k 1 a 1 a 1 + c 1... a 1 + c q k 1... a i a i + c 1... a i + c q k 1... a q n k 1 a q n k 1 + c 1... a q n k 1 + c q k 1 In terms of the standard array, y = a i + c j for some c j and c j is the decoded vector; hence once y is located, we need only read off the code vector of the top of the column. Proposition 13 (a The received vectors that are decoded as 0 by the algorithm are the coset leaders. (b The algorithm fails to detect an error if and only if y C. (c This is a nearest neighbour decoding. iv

5 Theorem 14 Let C be a [n, k] F 2 -code transmitted by a binary symmetric channel with symbol error rate r. Let α i be the number of coset leaders of weight i and let A i be the number of codewords of weight i. Then P corr (C = n α i r i (1 r n i. i=0 P undetect (C = n A i r i (1 r n i. i=1 8 The dual code Given a linear code C V we define the dual code (or orthogonal vector space C as Proposition 15 C is a linear code. (C = C. C = { v V v c = 0 for every c C }. Lemma 16 Let C be an [n, k] F -code in V with generator matrix G; then x C if and only if xg T = 0. Theorem 17 Suppose C is an [n, k]-code, then C is an [n, n k]-code. Definition. A parity check matrix H for C is a generator matrix for the dual code C. Proposition 20 C = {x V xh T = 0}. Theorem 21 Let C be [n, k] F -code, and suppose C has a generator matrix G = [I k A] in standard form; then H = [ A T I n k ] is a parity check matrix for C. 9 Syndrome decoding Let C V be an F linear code with parity check matrix H. For y V the syndrome of y, denoted S(y, is S(y = yh T. Lemma 22 S(y = S(z if and only if y z C. For a given received vector y: Step 1. Determine S(y. Step 2. Locate the a i with S(a i = S(y; by the above it follows that y a i + C. Step 3. (Recall that a i is taken as the error. Decode y as y a i. v

6 10 Incomplete decoding Step 1. Given a received vector y, calculate the syndrome S(y. Step 2. Find the unique coset leader a i with S(a i = S(y. (Recall that a i is taken as the error. So if w(a i t, then we take x = y a i as the (unique nearest neighbour decode of y. (Note: this decode of y will be correct if y contains t errors, since x is the (unique nearest neighbour. Step 3. If w(a i > t, then y may well have several nearest neighbours, so we seek re-transmission. 11 Minimal distance and the parity check matrix Theorem 23 (Distance Theorem for Linear Codes Let C be a linear code with parity check matrix H. Then d(c = d if and only if every set of d 1 columns of H is linearly independent and some set of d columns of H is linearly dependent. 12 Hamming codes We introduce a relation on K (n \ {0} by saying This is an equivalence relation. We define v u if and only if v = λu for some λ K. P n 1 (K = (K (n \ {0}/ (the set of equivalence classes. Geometrically P n 1 (K should be thought of as the set of (undirected lines in K (n. Lemma 24 P n 1 (F q = qn 1 q 1. Let s 1 be given. We let Ham(s, q denote the F q -linear code whose parity check matrix has the (representative vectors of P s 1 (F q as its columns. Theorem 25 (a Ham(s, q is an F q -code with parameters [n, n s, 3] where n = qs 1 q 1. (b Ham(s, q is a perfect, single error correcting code. vi

7 13 Decoding a Hamming code For a Hamming code we may proceed as follows: 1. Calculate S(y. 2. Check S(y against the columns of H and find S(y = λh j for some j, some λ F. We then know by ( that S(y = S(a for a the coset leader a = (0,..., 0, j λ, 0,..., 0, and so to decode just subtract λ from the j-th co-ordinate of y. 14 Construction of codes with large distance Theorem 27 Let n > k, d n k and suppose d 2 ( n 1 (q 1 i < q n k. i i=0 Then there exists an [n, k, d] F q -code. 15 Rings of polynomials and factor rings Proposition 28 Let f be a polynomial of degree deg f = m. Then R f has { 1, x,..., x m 1 } as an F p -basis. In particular R f has p m elements. 16 Cyclic codes With the previous notation we set f n = x n 1, and we write R n for Definition of cyclic codes. then belongs to C. R fn = F p [x] /(f n. An F p -linear code in F (n p (a 0, a 1,..., a n 1 C (a n 1, a 0, a 1,..., a n 2 is called cyclic if whenever Theorem 29 A code C R n is a cyclic code if and only if C is an R n -ideal. Theorem 30 Let C be an ideal in R n. Then there exists a unique monic polynomial g C such that: (a C = gr n, (b g (x n 1 (divisibility in F p [x]. We call such a g the generator of the code C. vii

8 17 Generator matrices Theorem 31 Let g(x = g g r 1 x r 1 + x r be the generator of a cyclic code C in R n. Then dim Fp (C = n r and if g f n then C has an (n r n generator matrix given by g 0 g 1... g r g G = g r g g r Given a cyclic code C = gr n as above, we write gh = f n (in F p [x]. Definition. h is called the check polynomial of the code C. Proposition 32 An element c R n is a codeword in gr n if and only if ch = 0 (in R n. Theorem 33 With the above notation, set k = n r and write h = h h k 1 x k 1 + x k. (a If h f n then a parity check matrix for the code gr n is 1 h k 1... h h H = k 1... h h 0. (b C is the cyclic code generated by ĥ(x = 1 + h k 1 x + + h 0 x k. 18 The u u + v construction If C 1 is an (n, M 1, d 1 F -code and C 2 is an (n, M 2, d 2 F -code, let C 1 C 2 = { u u + v F (2n u C 1, v C 2 }. Theorem 34 C 1 C 2 is a (2n, M 1 M 2, min{2d 1, d 2 } F -code. 19 Boolean Functions Theorem 35 The square-free monomials in the v i s form a basis for the Boolean functions on V m. viii

9 20 Reed-Muller Codes Definition For 0 r m, the rth order binary Reed-Muller code R(r, m is the set of all vectors f that correspond to Boolean functions f(v 1,..., v m in m variables that have degree at most r. It has length 2 m. Lemma The dimension of R(r, m is k = 1 + ( m 1 + ( m ( m r. Theorem 36 R(r + 1, m + 1 = R(r + 1, m R(r, m. Theorem 37 d(r(r, m = 2 m r. Lemma R(m 1, m is the even weight code. Theorem 38 R(m r 1, m = R(r, m for 0 r m 1. ix

11 Minimal Distance and the Parity Check Matrix

11 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 information

MATH Examination for the Module MATH-3152 (May 2009) Coding Theory. Time allowed: 2 hours. S = q

MATH 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 information

3. Coding theory 3.1. Basic concepts

3. 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 information

MATH 291T CODING THEORY

MATH 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 information

MATH 291T CODING THEORY

MATH 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 information

: Coding Theory. Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, upattane

: 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 information

Math 512 Syllabus Spring 2017, LIU Post

Math 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 information

Hamming codes and simplex codes ( )

Hamming 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

Know the meaning of the basic concepts: ring, field, characteristic of a ring, the ring of polynomials R[x].

Know 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 information

Mathematics Department

Mathematics 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 information

Coding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014

Coding 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 information

MATH3302. Coding and Cryptography. Coding Theory

MATH3302. 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 information

EE 229B ERROR CONTROL CODING Spring 2005

EE 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 information

MATH3302 Coding Theory Problem Set The following ISBN was received with a smudge. What is the missing digit? x9139 9

MATH3302 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 information

Binary Linear Codes G = = [ I 3 B ] , G 4 = None of these matrices are in standard form. Note that the matrix 1 0 0

Binary 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 information

7.1 Definitions and Generator Polynomials

7.1 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 information

Section 3 Error Correcting Codes (ECC): Fundamentals

Section 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 information

Generator 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.

Generator 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 information

B. Cyclic Codes. Primitive polynomials are the generator polynomials of cyclic codes.

B. 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 information

: Error Correcting Codes. October 2017 Lecture 1

: 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 information

Error-Correcting Codes

Error-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 information

EE512: Error Control Coding

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 information

The extended coset leader weight enumerator

The 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 information

Combinatória e Teoria de Códigos Exercises from the notes. Chapter 1

Combinatória 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 information

Lecture B04 : Linear codes and singleton bound

Lecture 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 information

MATH/MTHE 406 Homework Assignment 2 due date: October 17, 2016

MATH/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

Lecture 4: Linear Codes. Copyright G. Caire 88

Lecture 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 information

MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups.

MATH 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 information

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x

Linear 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 information

The extended Golay code

The extended Golay code The extended Golay code N. E. Straathof July 6, 2014 Master thesis Mathematics Supervisor: Dr R. R. J. Bocklandt Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en

More information

MTH6108 Coding theory

MTH6108 Coding theory MTH6108 Coding theory Contents 1 Introduction and definitions 2 2 Good codes 6 2.1 The main coding theory problem............................ 6 2.2 The Singleton bound...................................

More information

Coding 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 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 information

Vector spaces. EE 387, Notes 8, Handout #12

Vector 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 information

Chapter 3 Linear Block Codes

Chapter 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 information

Orthogonal Arrays & Codes

Orthogonal 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 information

ECEN 604: Channel Coding for Communications

ECEN 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 information

Arrangements, matroids and codes

Arrangements, 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 information

Solutions 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 (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 information

We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes.

We 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 information

ELEC 405/ELEC 511 Error Control Coding and Sequences. Hamming Codes and the Hamming Bound

ELEC 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 information

Codes over Subfields. Chapter Basics

Codes 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 information

Solutions to problems from Chapter 3

Solutions 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 information

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 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 information

Cyclic codes: overview

Cyclic 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 information

Ma/CS 6b Class 25: Error Correcting Codes 2

Ma/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 information

Chapter 2. Error Correcting Codes. 2.1 Basic Notions

Chapter 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 information

Linear Block Codes. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay

Linear Block Codes. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay 1 / 26 Linear Block Codes Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay July 28, 2014 Binary Block Codes 3 / 26 Let F 2 be the set

More information

Lecture 14: Hamming and Hadamard Codes

Lecture 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 information

The Hamming Codes and Delsarte s Linear Programming Bound

The 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 information

Some error-correcting codes and their applications

Some error-correcting codes and their applications Chapter 14 Some error-correcting codes and their applications J. D. Key 1 14.1 Introduction In this chapter we describe three types of error-correcting linear codes that have been used in major applications,

More information

Cyclic Redundancy Check Codes

Cyclic 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 information

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x f(x) = q(x)h(x) + r(x),

Linear 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 information

Reed Muller Error Correcting Codes

Reed Muller Error Correcting Codes Reed Muller Error Correcting Codes Ben Cooke Abstract. This paper starts by defining key terms and operations used with Reed Muller codes and binary numbers. Reed Muller codes are then defined and encoding

More information

ELEC 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 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 information

Finite Mathematics. Nik Ruškuc and Colva M. Roney-Dougal

Finite 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 information

MATH 433 Applied Algebra Lecture 22: Review for Exam 2.

MATH 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 information

FUNDAMENTALS OF ERROR-CORRECTING CODES - NOTES. Presenting: Wednesday, June 8. Section 1.6 Problem Set: 35, 40, 41, 43

FUNDAMENTALS OF ERROR-CORRECTING CODES - NOTES. Presenting: Wednesday, June 8. Section 1.6 Problem Set: 35, 40, 41, 43 FUNDAMENTALS OF ERROR-CORRECTING CODES - NOTES BRIAN BOCKELMAN Monday, June 6 2005 Presenter: J. Walker. Material: 1.1-1.4 For Lecture: 1,4,5,8 Problem Set: 6, 10, 14, 17, 19. 1. Basic concepts of linear

More information

CS6304 / 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 information

Chapter 7. Error Control Coding. 7.1 Historical background. Mikael Olofsson 2005

Chapter 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

} has dimension = k rank A > 0 over F. For any vector b!

} has dimension = k rank A > 0 over F. For any vector b! FINAL EXAM Math 115B, UCSB, Winter 2009 - SOLUTIONS Due in SH6518 or as an email attachment at 12:00pm, March 16, 2009. You are to work on your own, and may only consult your notes, text and the class

More information

Reed-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. 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 information

Error Detection and Correction: Hamming Code; Reed-Muller Code

Error 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

Channel Coding for Secure Transmissions

Channel 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 information

Chapter 5. Cyclic Codes

Chapter 5. Cyclic Codes Wireless Information Transmission System Lab. Chapter 5 Cyclic Codes Institute of Communications Engineering National Sun Yat-sen University Outlines Description of Cyclic Codes Generator and Parity-Check

More information

Roll No. :... Invigilator's Signature :.. CS/B.TECH(ECE)/SEM-7/EC-703/ CODING & INFORMATION THEORY. Time Allotted : 3 Hours Full Marks : 70

Roll No. :... Invigilator's Signature :.. CS/B.TECH(ECE)/SEM-7/EC-703/ CODING & INFORMATION THEORY. Time Allotted : 3 Hours Full Marks : 70 Name : Roll No. :.... Invigilator's Signature :.. CS/B.TECH(ECE)/SEM-7/EC-703/2011-12 2011 CODING & INFORMATION THEORY Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks

More information

Lecture Introduction. 2 Linear codes. CS CTT Current Topics in Theoretical CS Oct 4, 2012

Lecture 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 information

Answers 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 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 information

4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER /$ IEEE

4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER /$ IEEE 4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER 2008 List Decoding of Biorthogonal Codes the Hadamard Transform With Linear Complexity Ilya Dumer, Fellow, IEEE, Grigory Kabatiansky,

More information

Error Correcting Codes Questions Pool

Error Correcting Codes Questions Pool Error Correcting Codes Questions Pool Amnon Ta-Shma and Dean Doron January 3, 018 General guidelines The questions fall into several categories: (Know). (Mandatory). (Bonus). Make sure you know how to

More information

MAS309 Coding theory

MAS309 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 information

Support weight enumerators and coset weight distributions of isodual codes

Support weight enumerators and coset weight distributions of isodual codes Support weight enumerators and coset weight distributions of isodual codes Olgica Milenkovic Department of Electrical and Computer Engineering University of Colorado, Boulder March 31, 2003 Abstract In

More information

Optimum Soft Decision Decoding of Linear Block Codes

Optimum 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 information

MT361/461/5461 Error Correcting Codes: Preliminary Sheet

MT361/461/5461 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 information

Communications 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 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 information

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Cyclic Codes March 22, 2007 1. A cyclic code, C, is an ideal genarated by its minimal degree polynomial, g(x). C = < g(x) >, = {m(x)g(x) : m(x) is

More information

Lecture 12. Block Diagram

Lecture 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 information

A Brief Encounter with Linear Codes

A 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 information

Linear Codes and Syndrome Decoding

Linear 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 information

ERROR CORRECTING CODES

ERROR 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 information

ELEC-E7240 Coding Methods L (5 cr)

ELEC-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 information

NEW CLASSES OF CODES FOR CRYPTOLOGISTS AND COMPUTER SCIENTISTS

NEW CLASSES OF CODES FOR CRYPTOLOGISTS AND COMPUTER SCIENTISTS NEW CLASSES OF CODES FOR CRYPTOLOGISTS AND COMPUTER SCIENTISTS W. B. Vasantha Kandasamy e-mail: vasanthakandasamy@gmail.com web: http://mat.iitm.ac.in/~wbv www.vasantha.net Florentin Smarandache e-mail:

More information

16.36 Communication Systems Engineering

16.36 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 information

Hamming Codes 11/17/04

Hamming Codes 11/17/04 Hamming Codes 11/17/04 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

More information

Decomposing Bent Functions

Decomposing Bent Functions 2004 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 8, AUGUST 2003 Decomposing Bent Functions Anne Canteaut and Pascale Charpin Abstract In a recent paper [1], it is shown that the restrictions

More information

Information redundancy

Information 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 information

x 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)

x 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 information

Reed-Muller Codes. Sebastian Raaphorst Carleton University

Reed-Muller Codes. Sebastian Raaphorst Carleton University Reed-Muller Codes Sebastian Raaphorst Carleton University May 9, 2003 Abstract This paper examines the family of codes known as Reed-Muller codes. We begin by briefly introducing the codes and their history

More information

G Solution (10 points) Using elementary row operations, we transform the original generator matrix as follows.

G 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 information

On Two Probabilistic Decoding Algorithms for Binary Linear Codes

On 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 information

The BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes

The 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 information

MT5821 Advanced Combinatorics

MT5821 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 information

Codes for Partially Stuck-at Memory Cells

Codes 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 information

exercise in the previous class (1)

exercise 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 information

ELEC3227/4247 Mid term Quiz2 Solution with explanation

ELEC3227/4247 Mid term Quiz2 Solution with explanation ELEC7/447 Mid term Quiz Solution with explanation Ang Man Shun Department of Electrical and Electronic Engineering, University of Hong Kong Document creation date : 015 1 05 This document explain the solution

More information

Lecture 19 : Reed-Muller, Concatenation Codes & Decoding problem

Lecture 19 : Reed-Muller, Concatenation Codes & Decoding problem IITM-CS6845: Theory Toolkit February 08, 2012 Lecture 19 : Reed-Muller, Concatenation Codes & Decoding problem Lecturer: Jayalal Sarma Scribe: Dinesh K Theme: Error correcting codes In the previous lecture,

More information

: Error Correcting Codes. November 2017 Lecture 2

: Error Correcting Codes. November 2017 Lecture 2 03683072: Error Correcting Codes. November 2017 Lecture 2 Polynomial Codes and Cyclic Codes Amnon Ta-Shma and Dean Doron 1 Polynomial Codes Fix a finite field F q. For the purpose of constructing polynomial

More information

The Golay codes. Mario de Boer and Ruud Pellikaan

The Golay codes. Mario de Boer and Ruud Pellikaan The Golay codes Mario de Boer and Ruud Pellikaan Appeared in Some tapas of computer algebra (A.M. Cohen, H. Cuypers and H. Sterk eds.), Project 7, The Golay codes, pp. 338-347, Springer, Berlin 1999, after

More information

MTAT : Introduction to Coding Theory. Lecture 1

MTAT : 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 information

Error Correcting Codes: Combinatorics, Algorithms and Applications Spring Homework Due Monday March 23, 2009 in class

Error 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 information

POLYNOMIAL CODES AND FINITE GEOMETRIES

POLYNOMIAL CODES AND FINITE GEOMETRIES POLYNOMIAL CODES AND FINITE GEOMETRIES E. F. Assmus, Jr and J. D. Key Contents 1 Introduction 2 2 Projective and affine geometries 3 2.1 Projective geometry....................... 3 2.2 Affine geometry..........................

More information