What are topics in error-correcting coding which are related to interesting math. ? I am primarely interested in nowdays hot topics, but old days topics are also welcome.
Let me try to mention what I heard about.
Hot topic in error-correction is finding LDPC codes with very low "error-floor" for code lengths dozens thoursands bits, this might be useful for optic transmission. However it is not clear for me what kind of math playing role here ? ("Error-floor" is related with codewords with small Hamming weight. So the code might be quite good - means majority of codewords have big Hamming weight, so in most case code performs well, but very small number having small Hamming weight will cause small number of errors - it can be seen on the BER/SNR plot as a "floor".)
There is certain number of papers applying number theory (lattices in algebraic number fields) to consruct good codes. One may see papers by F. Oggier, G. Rekaya-Ben Othman, J.-C. Belfiore, E. Viterbo: e.g. this one : http://arxiv.org/abs/cs/0604093. I am not aware how "hot" is this topic and how far it is from practical applications...
Polar codes is a hot topic. What kind of math is playing role here ?
Probably most classical example is the Golay code (1948) and sporadic simple Mathieu groups. Let me quote Wikipedia: http://en.wikipedia.org/wiki/Binary_Golay_code : "The automorphism group of the binary Golay code is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of W." By the way - is it occasional coincidence of there is something behind it ?