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

Abstract

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 ) , 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 ⩽ t (where q = t 2 ) the following result: # X Z ( f ) ( F q ) ⩽ h ( t 3 + t 2 − t ) + t + 1 which should give the exact value of the minimum distance of the functional code C h ( X ) . In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. h = 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.