2026-04-13T18:23:17Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/3438312026-01-27T02:28:24Zcom_2072_1033col_2072_452950

urn:hdl:2117/343831 On varieties defined by large sets of quadrics and their application to error-correcting codes Ball, Simeon Michael Pepe, Valentina Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta Discrete mathematics Computer science -- Mathematics Arcs Tracks MDS codes AMDS codes Informàtica -- Matemàtica Let U be a ( k-1 2 - 1)-dimensional subspace of quadratic forms defined on F k with the property that U does not contain any reducible quadratic form. Let V (U) be the points of PG(k - 1, F) which are zeros of all quadratic forms in U. We will prove that if there is a group G which fixes U and no line of PG(k - 1, F) and V (U) spans PG(k - 1, F) then any hyperplane of PG(k - 1, F) is incident with at most k points of V (U). If F is a finite field then the linear code generated by the matrix whose columns are the points of V (U) is a k-dimensional linear code of length |V (U)| and minimum distance at least |V (U)| - k. A linear code with these parameters is an MDS code or an almost MDS code. We will construct examples of such subspaces U and groups G, which include the normal rational curve, the elliptic curve, Glynn’s arc from [8] and other examples found by computer search. We conjecture that the projection of V (U) from any k - 4 points is contained in the intersection of two quadrics, the common zeros of two linearly independent quadratic forms. This would be a strengthening of a classical theorem of Fano, which itself is an extension of a theorem of Castelnuovo, for which we include a proof using only linear algebra. Postprint (author's final draft) 2020-10-01 Article https://www.sciencedirect.com/science/article/abs/pii/S0012365X2030193X info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-82166-P/ES/COMBINATORIA GEOMETRICA, ALGEBRAICA Y PROBABILISTICA/ http://creativecommons.org/licenses/by-nc-nd/3.0/es/ Open Access Attribution-NonCommercial-NoDerivs 3.0 Spain