AbstractThe extended coset leader weight enumerator of the generalized Reed–Solomon $$[q+1,q-3,5]_q$$ [ q + 1 , q - 3 , 5 ] q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. 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 $${\mathbb F}_q$$ F q -rational points we show that there is a 3-plane containing the passant.