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Hyperelliptic Curves and Quadratic Residue Codes

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

For an odd prime p and a nonempty subset SGF(p), consider the hyperelliptic curve X S defined by
$$y^2=f_S(x),$$
where f S (x)=∏aS(xa). Since the days of E. Artin in the early 1900s, mathematicians have searched for good estimates for the number of points on such curves. In the late 1940s and early 1950s, A. Weil developed good estimates when the genus is small relative to the size of the prime p. When the genus is large compared to p, good estimates are still unknown.

A long-standing problem has been to develop “good” binary linear codes to be used for error-correction. For example, is the Gilbert–Varshamov bound asymptotically exact in the case of binary codes?

This chapter is devoted to explaining a basic link between these two unsolved problems. Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this chapter investigates how coding theory bounds give rise to bounds such as the following example: for all sufficiently large primes p, there exists a subset SGF(p) for which the bound |X S (GF(p))|>1.39p holds.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentUS Naval AcademyAnnapolisUSA
  2. 2.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

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