Nonlinear codes from points of bounded height

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. © 2005 Published by Elsevier Inc.