## 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://repository.lsu.edu/gradschool_disstheses/6947

## ISBN

9780599372566

## Pages

50

## DOI

10.31390/gradschool_disstheses.6947