On varieties defined by large sets of quadrics and their application to error-correcting codes
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 li...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/343831 |
| Acesso em linha: | https://hdl.handle.net/2117/343831 https://dx.doi.org/10.1016/j.disc.2020.112007 |
| Access Level: | acceso abierto |
| Palavra-chave: | Discrete mathematics Computer science -- Mathematics Arcs Tracks MDS codes AMDS codes Informàtica -- Matemàtica Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta |
| Resumo: | 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. |
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