Algebraic Geometric Codes on Anticanonical Surfaces

Authors

Jennifer A. Davis, University of Nebraska, LincolnFollow

Date of this Version

June 2007

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy. Major: Mathematics. Under the Supervision of Professors Brian Harbourne and Judy Walker. Lincoln, Nebraska; June 2007.
Copyright 2007 Jennifer A. Everson.

Abstract

Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance.

We study AG codes on anticanonical surfaces coming from blow-ups of P² at points on a line and points on a conic. We can compute the dimension of such codes exactly due to known results. For certain families of these codes, we prove an exact result on the minimum distance. For other families, we obtain lower bounds on the minimum distance. We also investigate and obtain some results for codes on blow-ups of P^r, where r is at least 3. We include tables of code parameters as well as Magma functions which can be used to generate the codes.