lec13rev09

# lec13rev09 - CS 70 Fall 2006 Discrete Mathematics for CS...

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Discrete Mathematics for CS Fall 2006 Lecture 13 Error Correcting Codes Erasure Errors We will consider two situations in which we wish to transmit information on an unreliable channel. The first is exemplified by the internet, where the information (say a file) is broken up into packets, and the unreliability is manifest in the fact that some of the packets are lost during transmission, as shown below: Suppose that the message consists of n packets and suppose that k packets are lost during transmission. We will show how to encode the initial message consisting of n packets into a redundant encoding consisting of n + k packets such that the recipient can reconstruct the message from any n received packets. Note that in this setting the packets are labelled and thus the recipient knows exactly which packets were dropped during transmission. We can assume without loss of generality that the contents of each packet is a number modulo q , where q is a prime. This is because the contents of the packet might be a 32-bit string and can therefore be regarded as a number between 0 and 2 32 - 1. Thus we can choose q to be any prime larger than 2 32 . The properties of polynomials over GF ( q ) (i.e. with coefficients and values reduced modulo q ) are perfectly suited to solve this problem and are the backbone to this error-correcting scheme. To see this, let us denote the message to be sent by m 1 , . . . , m n and make the following crucial observations: 1) There is a unique polynomial P ( x ) of degree n - 1 such that P ( i ) = m i for 1 i n (i.e. P ( x ) contains all of the information about the message, and evaluating P ( i ) gives the contents of the i - th packet). 2) The message to be sent is now m 1 = P ( 1 ) , . . . , m n = P ( n ) . We can generate additional packets by eval- uating P ( x ) at points n + j (remember, our transmitted message must be redundant, i.e. it must contain more packets than the original message to account for the lost packets). Thus the transmitted message is c 1 = P ( 1 ) , c 2 = P ( 2 ) , . . . , c n + k = P ( n + k ) . Since we are working modulo q , we must make sure that n + k q , but this condition does not impose a serious constraint since q is very large. 3) We can uniquely reconstruct P ( x ) from its values at any n distinct points, since it has degree n - 1. This means that P ( x ) can be reconstructed from any n of the transmitted packets. Evaluating this reconstructed polynomial P ( x ) at x = 1 , . . . , n yields the original message m 1 , . . . , m n . Example Suppose we are working over GF ( 7 ) (i.e. all coefficients and numbers can take on values between 0 and 6), and the number of packets in the message is n = 4. Suppose the message that Alice wants to send to Bob is m 1 = 3, m 2 = 1, m 3 = 5, and m 4 = 0. The unique degree n - 1 = 3 polynomial described by these 4 points is P ( x ) = x 3 + 4 x 2 + 5 (verify that P ( i ) = m i for 1 i 4). Now, suppose that Alice wishes to guard against

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## This note was uploaded on 11/03/2008 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Penn State.

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lec13rev09 - CS 70 Fall 2006 Discrete Mathematics for CS...

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