  We consider the $[q+1,q-3,5]_q3$ generalized doubly-extended Reed-Solomon
code of codimension $4$ as the code associated with the twisted cubic in the
projective space $\mathrm{PG}(3,q)$. Basing on the point-plane incidence matrix
of $\mathrm{PG}(3,q)$, we obtain the number of weight 3 vectors in all the
cosets of the considered code. This allows us to classify the cosets by their
weight distributions and to obtain these distributions. The weight of a coset
is the smallest Hamming weight of any vector in the coset. For the cosets of
equal weight having distinct weight distributions, we prove that the difference
between the $w$-th components, $3<w\le q+1$, of the distributions is uniquely
determined by the difference between the $3$-rd components. This implies an
interesting (and in some sense unexpected) symmetry of the obtained
distributions.
