We discussed first order Reed-Muller codes, and contrasted the two extremes of Hadamard and Hamming codes. We defined asymptotically good codes families, and posed the question about their existence. We proved the Gilbert-Varshamov bound in two ways, by picking a code greedily and by picking a random linear code. This in particular showed the existence of asymptotically good codes of any rate. En route proving the asymptotic version of the Gilbert-Varshamov bound, we proved entropy estimates for the volume of Hamming balls. We concluded with some remarks on attaining the GV bound explicitly, mentioning the “better than random” algebraic-geometric code constructions, and the conjectured optimality of the GV bound on rate for binary codes.

## January 21, 2010

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