Subquadratic time encodable codes beating the Gilbert-Varshamov bound

TL;DR

This paper presents explicit algebraic geometry codes from Garcia-Stichtenoth towers that surpass the Gilbert-Varshamov bound, with subquadratic encoding and decoding algorithms exploiting algebraic structures.

Contribution

It introduces novel algebraic geometry codes with subquadratic encoding and decoding times that outperform previous codes in terms of efficiency.

Findings

Codes beat the Gilbert-Varshamov bound for large alphabets.

Encoding runtime is nearly linear under the matrix multiplication conjecture.

Decoding algorithms have sub-quadratic expected runtime.

Abstract

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied 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 (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse.