Asymptotically Good Generalized Algebraic Geometry Codes

Preface In any form of electronic communication, errors can occur and hence there is a need to deal with them. Before error-correcting codes were used, only once a message had been received and could not be understood, one could conclude that errors had occurred. There was no way to resolve these errors other than asking for retransmission. This changed when Claude Shannon in 1948 proved a theorem 1 which says that even with a noisy channel, there exist ways to encode messages in such a way that they have an arbitrarily good chance of being transmitted safely, provided that one does not exceed the capacity of the channel by trying to transmit too much information too quickly. This theorem, which marks the start of coding theory, means that with long enough codes we can achieve communication that is as safe as we like. These codes do not have to be linear, and the proof does not construct them. All we know is that they exist. The main problem of coding theory is the construction of these error-correcting codes. One of the first error-correcting codes was constructed at Bell Labs, by a mathematician named Richard Hamming. He became fed up with a computer which could detect errors in his input during weekend runs, but would then just dump the program, wasting the entire run. He devised ways to encode the input so that the computer could correct isolated errors and continue running, and his work led him to discover what are now called Hamming codes. Soon after, Marcel Golay generalized Hamming's construction from binary codes to codes using an alphabet of p symbols for p prime. He also constructed two very remarkable codes that correct multiple errors and that now bear his name. However, it is a curious fact of history that one of the very same Golay codes appeared a few years earlier, in a Finnish magazine dedicated to betting on soccer games [2]. This ternary Golay code was first discovered by a Finn who was determining good strategies for betting on blocks of 11 soccer games. Here, one places a bet by predicting a win, lose, or tie for all 11 games, and as long as you do not miss more than two of them, you get a payoff. If a group gets together in a pool and makes multiple bets to cover all the options, so that no …