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Summary

In an influential paper Calderbank, Rains, Shor and N.J.A.Sloane showed that error-correction is possible in the context of quantum computations. The dominating construction (quantum stabilizer codes) makes use of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic bilinear form. We describe a geometric approach to quantum stabilizer codes. This is a generalization of the classical geometric description of linear codes in terms of (multi)sets of points in projective spaces. Quantum codes are described as sets of lines. This description is particularly efficient when the minimum distance "d" is small. In particular in case "d=3" this leads to spreads of lines, a well-studied concept in finite geometries. As an application we obtain simple descriptions of large classes of quantum codes as well as constructions of new quantum codes. All existence problems of "d=3" quantum codes left open in the range of the data base are resolved. In case "d=4" the problem leads to the concept of a quantum cap.