On cosets weight distributions of the doubly-extended Reed-Solomon codes of codimension 4
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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 PG(3,q). Basing on the point-plane incidence matrix of 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. For the cosets of equal weight having distinct weight distributions, we prove that the difference between the w-th components, 3<w≤ q+1, of the distributions is unambiguously determined by the difference between the 3-rd components. This implies an interesting (and in some sense unexpected) symmetry of the obtained distributions. To obtain the property of differences we prove a useful relation for the Krawtchouck polynomials. Also, we describe an alternative way of obtaining the cosets weight distributions on the base of the integral weight spectra over all the cosets of the fixed weight.
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