Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published October 2019 | public
Journal Article

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 identified with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degreeone places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic ω/2 < 1.19 runtime exponent encoding and 1 + ω/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here ω < 2.373 is the matrix multiplication exponent. If ω = 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.

Additional Information

© 2019 IEEE. Manuscript received August 11, 2018; accepted June 23, 2019. Date of publication July 23, 2019; date of current version September 13, 2019. This work was supported in part by the NSF under Grant CCF-1423544, in part by Chris Umans' Simons Foundation Investigator Grant, in part by the European Union's H2020 Programme under Grant ERC-669891, and in part by Caltech's Rita A. and Øistein Skjellum SURF Fellowship. We would like to thank the reviewers for their suggestions.

Additional details

Created:
August 19, 2023
Modified:
October 20, 2023