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Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Note 8 Error Correcting Codes Erasure Errors We will consider the situation where we wish to transmit information over an unreliable communication channel. This is exemplified by the internet, where the information (say a file) is broken up into fixed-length packets, and the unreliability is manifest in the fact that some of the packets may be lost during transmission, as shown below: Suppose that, in the absence of packet loss, it would take n packets to send the entire message—but in reality up to k packets may be 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. We will assume that the packets are labeled and thus the recipient knows exactly which packets were dropped during transmission. In our scheme, the contents of each packet is a number modulo q , where q is a prime. For example, a 32-bit string can be regarded as a number between 0 and 2 32- 1; then we could choose q to be any prime larger than 2 32 and view it as a number modulo q . 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 of this error- correcting scheme. To see this, let us denote the message to be sent by m 1 , . . . , m n . Then we can form a....
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