Codes from varieties over finite fields

Abstract: There are $q^{20}$ homogeneous cubic polynomials in four variables with coefficients in the finite field $F_q$. How many of them define a cubic surface with $q^2+7q+1$ $F_q$-rational points? What about other numbers of rational points? How many of the $q^{20}$ pairs of homogeneous cubic polynomials in three variables define cubic curves that intersect in 9 $F_q$-rational points? The goal of this talk is to explain how ideas from the theory of error-correcting codes can be used to study families of varieties over a fixed finite field. We will not assume any previous familiarity with coding theory. We will start from the basics and emphasize examples.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar