A theory for constructing quantum error correcting codes from Toric surfaces by the Calderbank-Shor-Steane method is presented. In particular, we study the method on toric Hirzebruch surfaces. The results are obtained by constructing a dualizing differential form for the toric surface and by using the cohomology and the intersection theory of toric varieties.

In the decoding of one-point algebraic-geometry codes with one defining equation in two variables we generalize the Forney formula for the Bose-Chaudhuri-Hocqungham (BCH) codes.

A sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field GF(2^{3}). In the sequence there is, for instance, a code of length 21 and dimension10with minimal distance9, and a code of length21and dimension16with minimal distance3. The codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by Goppa. The curve used in the present work is the Klein quartic. This curve has the maximal number of rational points over GF(2^{3})allowed by Serre's improvement of the Hasse-Weil bound, which, together with the low genus, accounts for the good code parameters. The Klein quartic has the Frobenius groupGof order21acting as a group of automorphisms which accounts for the particular symmetries of the codes. In fact, the codes are given alternative descriptions as left ideals in the group-algebra GF(2^{3})[G]. This description allows for easy decoding. For instance, in the case of the single error correcting code of length21and dimension16with minimal distance3. decoding is obtained by multiplication with an idempotent in the group algebra.