On cosets weight distributions of the doubly-extended Reed-Solomon codes of codimension 4

We consider the $[q+1,q-3,5{]}_{q}3$ generalized doubly-extended Reed-Solomon code of codimension $4$ as the code associated with the twisted cubic in the projective space $\mathrm{P}\mathrm{G}(3,q)$. Basing on the point-plane incidence matrix of $\mathrm{P}\mathrm{G}(3,q)$, 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.