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70 CS Spring 2008 Discrete Mathematics for CS David Wagner Note 11 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 fixed-length packets, and the unreliability is manifest in the fact that some of the packets are lost during...
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70 CS Spring 2008
Discrete Mathematics for CS David Wagner
Note 11
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 fixed-length packets, and the unreliability is manifest in the fact that some of the packets are 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 practice 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 labelled 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. 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 m1 , . . . , mn and make the following crucial observations: 1) There is a unique polynomial...
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Berkeley - CS - 170
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Berkeley - CS - 70
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Berkeley - CS - 174
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Berkeley - CS - 174
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Berkeley - CS - 24
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Berkeley - CS - 12
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Berkeley - CS - 174
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Berkeley - CS - 174
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Berkeley - CS - 174
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Berkeley - CS - 174
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Berkeley - CS - 174
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Berkeley - CS - 29
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Berkeley - CS - 174
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Berkeley - HISTORY - 90
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