Codes from families of varieties over a fixed finite field

Nathan Kaplan
University of California, Irvine

Date(s) : 18/03/2021   iCal
17 h 00 min - 18 h 00 min

How many of the $q^{10}$ homogeneous cubic polynomials in $x,y,z$ with coefficients in $\mathbb{F}_q$ define a cubic curve with a given number of points?  How many of the $q^{20}$ pairs of these polynomials have exactly $k$ common zeros?  We will show how to phrase these kinds of questions in terms of the weight enumerators of projective Reed-Muller codes.  We will then see how ideas from the theory of error-correcting codes, in particular the MacWilliams identity and its variations, can help answer them.
Results from number theory and algebraic geometry play an important role in modern coding theory, but it is less common to see examples where ideas from coding theory are used to solve problems in number theory.  This talk will not assume any previous background in coding theory.  We will start with the basics of coding theory and emphasize concrete examples.
Meeting ID : 921 9584 8065
Passcode : voir mail
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