Toric residue codes. I. (Q620931)

Formerly, algebraic codes have been built over toric varieties as evaluation codes of sections of line bundles at rational points. Intersection theory has been very useful in the construction of evaluation codes. In this paper, the codes are obtained as residues of differential forms over rational points producing codes close to the dual codes of the former evaluation codes. Firstly, the authors establish several properties of toric residues over finite fields and they extend some results from intersection theory. The Riemann-Roch theorem is used to compute the parameters of the dual of evaluation codes, and using the residue theorem, the authors compare the toric residues codes with those duals. The developed techniques indeed allow to build quantum stabilizer codes from higher dimensional projective smooth toric varieties. The authors provide illustrative examples of their construction. The paper is hard to read since it requires a high level of specialization in algebraic geometry. However, a careful self-contained exposition and sufficient references to the literature provide the reader with a beautiful construction of toric codes.