Work

Subquadratic Time Encodable Codes Beating the Gilbert–Varshamov Bound

Year: 2019

Type: article

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.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2019.2930538

Authors: Anand Kumar Narayanan, Matthew Weidner

Institutions: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, LIP6, California Institute of Technology

Language: English

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Simons Foundation, European Union Agency for Network and Information Security

Awards: AF: Small: Algorithms for Matrix Multiplication, Polynomial Factorization and Generalized Fourier Transform, G6996087912