3 research outputs found
Singularities of Symmetric Hypersurfaces and Reed-Solomon Codes
We determine conditions on q for the nonexistence of deep holes of the standard Reed-Solomon code of dimension k over Fq generated by polynomials of degree k + d. Our conditions rely on the existence of q-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over Fq. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of q-rational points is established.Fil: Cafure, Antonio Artemio. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Privitelli, Melina Lorena. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentin
On the number of nonequivalent propelinear extended perfect codes
The paper proves that there exist an exponential number of nonequivalent
propelinear extended perfect binary codes of length growing to infinity.
Specifically, it is proved that all transitive extended perfect binary codes
found by Potapov are propelinear. All such codes have small rank, which is one
more than the rank of the extended Hamming code of the same length. We
investigate the properties of these codes and show that any of them has a
normalized propelinear representation