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TitleAlgebraic geometry codes from polyhedral divisors
Author(s) Nathan Owen Ilten, Hendrik Süß
TypeArticle in Journal
AbstractA 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.
KeywordsAG codes, Evaluation codes, Toric varieties
URL http://www.sciencedirect.com/science/article/pii/S0747717110000453
JournalJournal of Symbolic Computation
Pages734 - 756
NoteAlgebraic Coding Theory and Applications
Translation No
Refereed No