Nearly linear time encodable codes beating the Gilbert-Varshamov bound

We construct explicit nearly linear time encodable error-correcting codes beating the Gilbert-Varshamov bound. Our codes are algebraic geometry codes built from the Garcia-Stichtenoth function field tower and beat the Gilbert-Varshamov bound for alphabet sizes at least $19^2$. Messages are identified with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic nearly linear time encoding algorithm and nearly quadratic expected time randomized (unique and list) decoding algorithms.