#### Date of Award

1999

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Robert F. Lax

#### Abstract

We prove that elements of the Weierstrass gap set of a pair of points may be used to define a geometric Goppa code that has minimum distance greater than the usual lower bound. We determine the Weierstrass gap set of a pair of any two Weierstrass points on a Hermitian curve and use this to increase the lower bound on the minimum distance of certain codes defined using a linear combination of the two points. In particular, we obtain some two-point codes on a Hermitian curve that have better parameters than the one-point code on this curve with the same dimension. These results generalize to certain codes defined using an m-tuple of points on a smooth projective absolutely irreducible curve.

#### Recommended Citation

Matthews, Gretchen L., "Weierstrass Pairs and Minimum Distance of Goppa Codes." (1999). *LSU Historical Dissertations and Theses*. 6947.

https://digitalcommons.lsu.edu/gradschool_disstheses/6947

#### ISBN

9780599372566

#### Pages

50