On planes through points off the twisted cubic in $\mathrm{PG}(3,q)$ and multiple covering codes

TL;DR

This paper explores the geometric structure of points and planes related to a twisted cubic in projective space, using this to analyze and identify asymptotically optimal multiple covering codes derived from Reed-Solomon codes.

Contribution

It characterizes the point-plane incidence matrix in PG(3,q) relative to a twisted cubic and applies this to demonstrate the optimality of certain multiple covering codes.

Findings

Incidence matrix structure described for points and planes under group action.

Generalized Reed-Solomon codes shown to be asymptotically optimal multiple covering codes.

Provides a geometric framework linking algebraic codes and projective geometry.

Abstract

Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes.