  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 earlier work the author
developed methods to construct linear error correcting codes from toric
varieties and derive the code parameters using the cohomology and the
intersection theory on toric varieties. This method is generalized in section
to construct linear codes suitable for constructing quantum codes by the
Calderbank-Shor-Steane method. Essential for the theory is the existence and
the application of a dualizing differential form on the toric surface. A.R.
Calderbank, P.W. Shor and A.M. Steane produced stabilizer codes from linear
codes containing their dual codes. These two constructions are merged to obtain
results for toric surfaces. Similar merging has been done for algebraic curves
with different methods by A. Ashikhmin, S. Litsyn and M.A. Tsfasman.
