In class today I mentioned that the Shannon’s capacity theorem is proved using random codes. This is going to be recurring theme: almost always the best parameters are achieved by random codes. One notable exception is a specific class of codes called algebraic geometric codes which have better distance than random codes (for ).

For a gentle introduction consult the notes from Sudan’s course on coding theory. A more detailed exposition can be found in the chapter on Algebraic geometric codes by Høholdt, van Lint and Pellikaan (in Handbook of Coding theory, Eds: Pless, Huffman and Brualdi).

This project can be done jointly by two students. Warning: Algebraic geometric codes is based on some deep (and difficult) math so do not take on this project lightly.

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