Information Theory Introduction to Channel Coding Jalal Al Roumy

2 Introduction Error control for data integrity may be exercised by means of either forward error correction (FEC) or automatic request for re- transmission (ARQ). For FEC, redundant data is added to the code for the purposes of error detection and correction whereas ARQ utilises redundancy for the sole purpose or error detection. Upon detection the receiver requests a repeat transmission. Channel coding is shown in the following diagram – modulation and channel coding may be combined. Information to be transmitted Source coding Channel coding Modulation Transmitter Channel Information received Source decoding Channel decoding Demodulation Receiver Channel coding Channel decoding Crypto source

3 Coding - basic concepts Without coding theory and error-correcting codes there would be no deep-space travel and pictures, no satellite TV, no mobile communications no e-business, no e-commerce, no control systems, etc. Error-correcting codes are used to correct messages when they are transmitted through noisy channels. Error correcting framework Example A code C over an alphabet S is a subset of S* - (C ε S*). A q -nary code is a code over an alphabet of q -symbols. A binary code is a code over the alphabet {0,1}. Examples of codesC 1 = {00, 01, 10, 11} C 2 = {000, 010, 101, 100} C 3 = {00000, 01101, 10111, 11011}

4 Background Historically FECs are divided into linear block codes (eg Hamming, cyclic, Reed-Solomon etc) and codes such as convolution codes. From a purist viewpoint the division is drawn between linear block codes and convolution codes. The next sessions will examine particular examples of FECs covering: An overview to linear codes Hamming codes Cyclic codes Convolutional codes The session will conclude with a brief excursion into ARQs. Firstly a reminder of the most basic form or error detection – the parity check. As usual we need some basic theory to begin!

5 Parity Codes Single Bit Parity: Detect single bit errors Two Dimensional Bit Parity: Detect and correct single bit errors 0 0 1

6 Example code

7 The ISBN-code; an example Each book has International Standard Book Number has a 10-digit codeword produced by the publisher with the following structure: l p m w= x 1 … x 10 language publisher number weighted check sum such that The publisher has to put X (in this case 4 ) into the 10-th position. The ISBN code is designed to detect: (a) any single error (b) any double error created by a transposition Single error detection Let X = x 1 … x 10 be a correct code and let Y = x 1 … x J-1 y J x J+1 … x 10 with y J = x J + a, a  0 In such a case: ≡ 0 (mod 11)

8 Binary symmetric channel Consider a transmission of binary symbols such that each symbol has probability of error p < 1/2. Binary symmetric channel If n symbols are transmitted, then the probability of t errors is In the case of binary symmetric channels the ”nearest neighbour decoding strategy” is also “maximum likelihood decoding strategy''. Example Consider C = {000, 111} and the nearest neighbour decoding strategy. Probability that the received word is decoded correctly as 000 is (1 - p) 3 + 3p(1 - p) 2, as 111 is (1 - p) 3 + 3p(1 - p) 2. ThereforeP err (C) = 1 - ((1 - p) 3 + 3p(1 - p) 2 ) is the so-called word error probability. Example If p = 0.01, then P err (C) = and only one word in 3555 will reach the user with an error.

9 Hamming distance The intuitive concept of “closeness'' of two words is formalized through Hamming distance d (x, y) of words x, y. For two words (or vectors) x, y; d (x, y) = the number of symbols x and y differ. Example:d (10101, 01100) = 3,d (first, second, fifth) = 3 Properties of Hamming distance (1) d (x, y) = 0; iff x = y (2) d (x, y) = d (y, x) (3) d (x, z) ≤ d (x, y) + d (y, z) triangle inequality An important parameter of codes C is their minimal distance. d (C) = min {d (x, y) | x, y ε C, x ≠ y}, because it gives the smallest number of errors needed to change one codeword into another. Theorem Basic error correcting theorem (1) A code C can detect up to s errors if d (C) ≥ s + 1. (2) A code C can correct up to t errors if d (C) ≥ 2t + 1. Note – for binary linear codes d (C) = smallest weight W (C) of non-zero codeword,

10 Some notation Notation: Notation: An (n,M,d) - code C is a code such that n - is the length of codewords. M - is the number of codewords. d - is the minimum distance in C. Example: C 1 = {00, 01, 10, 11} is a (2,4,1)-code. C 2 = {000, 011, 101, 110} is a (3,4,2)-code. C 3 = {00000, 01101, 10110, 11011} is a (5,4,3)-code. Comment: A good (n,M,d) code has small n and large M and d.

11 Code Rate For q-nary (n,M,d)-code we define code rate, or information rate, R, by The code rate represents the ratio of the number of input data symbols to the number of transmitted code symbols. For a Hadamard code eg, this is an important parameter for real implementations, because it shows what fraction of the bandwidth is being used to transmit actual data. log2(n) = ln(n)/ln(2) Recall that

12 The main coding theory problem A good (n,M,d) -code has small n, large M and large d. The main coding theory problem is to optimize one of the parameters n, M, d for given values of the other two. Notation: Notation: A q (n,d) is the largest M such that there is an q -nary (n,M,d) -code.

13 A general upper bound on A q (n,d) Notation F q n – is a set of all words of length n over alphabet {0,1,2,…,q -1} Definition For any codeword u  F q n and any integer r  0 the sphere of radius r and centre u is denoted by S (u,r) = {v  F q n | d (u,v)  r }. Theorem Theorem A sphere of radius r in F q n, 0  r  n contains Theorem Theorem (The sphere-packing or Hamming bound) If C is a q -nary (n,M,2t +1) -code, then A code which achieves the sphere-packing bound as above, i.e. such that equality holds, is called a perfect code. words The Singleton result given for upper bound is

14 A general upper bound on A q (n,d) For Binary: The Singleton result becomes

15 A general upper bound on A q (n,d) Example Example An (7,M,3) -code is perfect if i.e. M = 16 An example of such a code: C 4 = { , , , , , , , , , , , , , , , }

16 Lower Bound For A q (N,d) The following lower bound for A q (n,d) is known as Gilbert- Varshanov bound: Theorem Theorem Given d  n, there exists a q -ary (n,M,d) -code with For binary

17 Examples using bounds Does there exist a linear code of length n=9, dimension k=2, and distance d=5? Yes, because What is a lower and an upper bound on the size or the dimension, k, of a linear code with n=9 and d=5? Hamming upper bound:

18 Does there exists a (15, 7, 5) linear code? Check G-V condition G-V condition does not hold, so G-V bound does not tell us whether or not such a code exists. Further example

19 Another example