Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture

We study the functional codes Ch(X)Ch(X) defined by Lachaud in [G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in: Arithmetic, Geometry, and Coding Theory, Luminy, France, 1993, de Gruyter, Berlin, 1996, pp. 77–104] where X⊂PNX⊂PN is an algebraic projective variety of degree d and dimension m. When X   is a Hermitian surface in PG(3,q)PG(3,q), Sørensen in [A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for h⩽th⩽t (where q=t2q=t2) the following result:#XZ(f)(Fq)⩽h(t3+t2−t)+t+1#XZ(f)(Fq)⩽h(t3+t2−t)+t+1 which should give the exact value of the minimum distance of the functional code Ch(X)Ch(X). In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. h=2h=2), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight.