论文标题
投影式复曲面代码
Projective toric codes
论文作者
论文摘要
任何积分凸polytope $ p $ in $ \ mathbb {r}^n $都提供$ n $ dimensional toric品种$ x_p $和一个充分的除数$ d_p $。本文在$ x_p $上对代数几何误差校正代码进行了明确的构造，该代码通过评估$ \ \ natercal {l}（d_p）$的全局部分获得$ x_p $的每个理性点。这项工作通过评估整个品种，而不是仅考虑具有非零坐标的点，将类似于Reed-Muller代码之一的曲曲面代码扩展为投影。代码的尺寸是根据polytope $ p $中的积分点的数量和算法技术来给出的，以在最小距离上进行较低范围。
Any integral convex polytope $P$ in $\mathbb{R}^N$ provides a $N$-dimensional toric variety $X_P$ and an ample divisor $D_P$ on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on $X_P$ , obtained by evaluating global section of $\mathcal{L}(D_P)$ on every rational point of $X_P$. 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 non-zero 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 lowerbound on the minimum distance is described.
