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

We study the functional codes C"h(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@?P^N is an algebraic projective variety of degree d and dimension m. When X is a Hermitian surface in PG(3,q), Sorensen in [A.B. Sorensen, Rational points on hypersurfaces, Reed-Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for h=

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