On Cosets Weight Distribution of Doubly-Extended Reed-Solomon Codes of Codimension 4

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