On varieties defined by large sets of quadrics and their application to error-correcting codes

Abstract

Let U be a (k−12−1)-dimensional subspace of quadratic forms defined on Fk 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 Glynn (1986) 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.