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Error Correcting Codes. To detect and correct errors Adding redundancy to the original message Crucial when it’s impossible to resend the message (interplanetary communications, storage..) and when the channel is very noisy (wireless communication). Simple example of coding.
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Error Correcting Codes • To detect and correct errors • Adding redundancy to the original message • Crucial when it’s impossible to resend the message (interplanetary communications, storage..) and when the channel is very noisy (wireless communication)
Simple example of coding The Hamming distance d(x,y) between two codewords x and y is the number of coordinate positions in which they differ d(A,B) =3 The Hamming distance d of a code is the minimum distance between two distinct codewords, over all pairs of codewords d=3 Hamming distance
Nearest neighbor decoding • a received vector is decoded to the codeword "closest" to it • sphere of radius e about the codeword • Perfect codes: the spheres cover the whole space
Nearest neighbor decoding is equivalent to finding the input message U that maximizes the probability P(U|Y) Decoding problem in terms of probability estimation. Graphical models. Maximum likelihood decoding X U Y noisy channel
Y X U Permuter U X Y Turbo Codes
