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

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