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

Abstract

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)