Toric Varieties and Codes, Error-correcting Codes, Quantum Codes, Secret Sharing and Decoding

Abstract: Toric varieties and their associated toric codes, as well as determination of their parameters with intersection theory, are presented in the two dimensional case. Linear Secret Sharing Schemes with strong multiplication are constructed from toric varieties and codes by the J. L. Massey construction. Asymmetric Quantum Codes are obtained from toric codes by the A.R. Calderbank, P.W. Shor and A.M. Steane construction of stabilizer codes from linear codes containing their dual codes. Decoding of a class of toric codes is presented.