The extended coset leader weight enumerator of a twisted cubic code

TL;DR

This paper computes the extended coset leader weight enumerator of a specific Reed-Solomon code using finite geometry, classifying geometric objects in projective space and analyzing rational functions and their double point schemes.

Contribution

It introduces a geometric approach to determine the weight enumerator of a twisted cubic code, linking finite geometry, rational functions, and algebraic curves.

Findings

Classification of points, lines, and planes under projectivities leaving the twisted cubic invariant.

Connection between lines in space and rational functions of degree at most three.

Existence of a 3-plane containing a specific passant, based on the Hasse-Weil bound.

Abstract

The extended coset leader weight enumerator of the generalized Reed-Solomon $[q + 1, q - 3, 5]_q$ code is computed. The computation is considered as a question in finite geometry. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse-Weil bound on Fq-rational points we show that there is a 3-plane containing the passant.