Subquadratic time encodable codes beating the Gilbert-Varshamov bound
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.
Keywords
Cite
@article{arxiv.1712.10052,
title = {Subquadratic time encodable codes beating the Gilbert-Varshamov bound},
author = {Anand Kumar Narayanan and Matthew Weidner},
journal= {arXiv preprint arXiv:1712.10052},
year = {2019}
}