Counting plane cubic curves over finite fields with a prescribed number of rational intersection points

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