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Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen’s conjecture. (English) Zbl 1155.94022

The author considers codes defined by polynomials of degree 2 on the set of points of an Hermitian surface. The key remark is that the weight of a codeword can be computed by studying the intersection between a Hermitian surface and a quadric. The core of the paper is the analysis of the cardinalities of the configurations which might thus arise. A nice result is the description of all the codewords of minimum weight and second minimum weight for the codes under consideration. This also shows that in the case under consideration Sørensen’s conjecture [A. B. Sørensen, Rational points on hypersurfaces, Reed-Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark (1991)] on the minimum weight is satisfied.

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50 Applications to coding theory and cryptography of arithmetic geometry
11E39 Bilinear and Hermitian forms
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