We consider the 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, ext {q}). Basing on the point-plane incidence matrix of mathrm {PG}(3, ext {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 < ext {w}le ext {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.
On Cosets Weight Distribution of Doubly-Extended Reed-Solomon Codes of Codimension 4
Marcugini S.;Pambianco F.
2021
Abstract
We consider the 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, ext {q}). Basing on the point-plane incidence matrix of mathrm {PG}(3, ext {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 < ext {w}le ext {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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
