Nonlinear codes from points of bounded height

Authors:
Chris Hurlburt;Jeffrey Lin Thunder
Affiliations:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA;Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA
Venue:
Finite Fields and Their Applications
Year:
2007

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Excellent codes from modular curves

STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Algebraic-Geometric Codes

Algebraic-Geometric Codes

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Abstract

This paper generalizes Elkies' construction of error-correcting nonlinear codes found in [N. Elkies, Excellent nonlinear codes from modular curves, in: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, STOC'01, Hersonissos, Crete, Greece, 2001, pp. 200-208]. The generalization produces a precise average code size over codes in the new construction. The result is a larger family of codes with similar transmission rates and error detection rates to the nonlinear codes found in [N. Elkies, Excellent nonlinear codes from modular curves, in: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, STOC'01, Hersonissos, Crete, Greece, 2001, pp. 200-208]. Moreover, we exhibit a connection between these nonlinear codes and solutions to simple homogeneous linear equations defined over the function field of a curve.