Subquadratic Time Encodable Codes Beating the Gilbert-Varshamov Bound

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function-field tower beating 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 omega/2 < 1.19 runtime exponent encoding and 1 + omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here omega < 2.373 is the matrix multiplication exponent. If omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding time of code families beating the Gilbert-Varshamov bound were quadratic or worse.