Applications of algebraic geometry to error-correcting codes
Posted on:2006-01-06
Degree:Ph.D
Type:Dissertation
University:University of Colorado at Boulder
Candidate:Massman, John Douglas, III
Full Text:PDF
GTID:1458390005495621
Subject:Mathematics
Abstract/Summary:
We apply algebraic geometry to two problems in error-correcting codes. We construct error-correcting codes in two ways from nonsingular projective varieties and discuss the possible duality between them. The problem of classifying space-time codes leads one to the problem of classifying non-associative division algebras over a finite field. This leads one to investigate homogeneous polynomials of degree n in n variables over Fq with no nontrivial Fq -rational points. The case n = 3 was first investigated by Dickson, and for n = 4 we give a complete geometrical classification of all possible forms.