Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points

2020/03/31 by Nathan Kaplan, Nathan O. Kaplan, Vlad Matei · 1 citation
Computer Science · #Coding theory and cryptography #Cellular Automata and Applications #Error Correcting Code Techniques

paper · pdf · doi:10.1007/s40879-021-00472-x

Abstract

For each integer k ∈ [0,9], we count the number of plane cubic curves defined over a finite field \mathbbF_q that do not share a common component and intersect in exactly k \mathbbF_q-rational points. We set this up as a problem about a weight enumerator of a certain projective Reed-Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.

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