Any integral convex polytope P in ℝN provides an N-dimensional toric variety XP and an ample divisor DP on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on XP, obtained by evaluating global section of the line bundle corresponding to DP on every rational point of XP. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope P and an algorithmic technique to get a lower bound on the minimum distance is described.

中文翻译：

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投影复曲面码

任何积分凸多面体磷在ℝñ提供了一个ñ维复曲面变体X磷和一个充足的除数D磷在这个品种上。本文给出了代数几何纠错码的显式构造X磷，通过评估对应于的线束的全局截面获得D磷在每一个理性点上X磷. 这项工作通过评估整个变体而不是仅考虑具有非零坐标的点，将类似于 Reed-Muller 码的复曲面码扩展到投影码。代码的维度是根据多面体中的积分点数给出的磷并描述了一种获得最小距离下限的算法技术。