The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Toric Surface Codes and Minkowski Sums
SIAM Journal on Discrete Mathematics
Toric Surface Codes and Minkowski Length of Polygons
SIAM Journal on Discrete Mathematics
On the parameters of r-dimensional toric codes
Finite Fields and Their Applications
Error-Correcting Codes from Higher-Dimensional Varieties
Finite Fields and Their Applications
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A description of complete normal varieties with lower-dimensional torus action has been given by Altmann et al. (2008), generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T-varieties to define new evaluation codes called T-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticated T-varieties. These results suggest looking further for good codes on T-varieties.