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)

Hamming distance 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 A -> 00 -> B -> 10 -> C -> 01 -> D -> 11 -> 11101

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

Maximum likelihood decoding 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. U Y X noisy channel

Convolution codes as graphical models

Turbo Codes