AG码在$C_{ab}$曲线上的快速编码
Fast Encoding of AG Codes over $C_{ab}$ Curves
摘要：
我们调查的算法编码的一点代数几何（AG）代码在某些平面曲线称为$C_{ab}$曲线，以及算法反转的编码映射，我们称之为“unencoding”。有些$C_{ab}$曲线有许多点，甚至是最大的，例如厄米曲线。我们的编码响应。解编码算法分别具有复杂度$\tilde{O}(n^{3/2})$。$\tilde{O}(qn)$的AG代码在任何$C_{ab}$曲线满足非常温和的假设，其中$n$是代码长度和$q$的基本字段大小，和$\tilde{O}$忽略常数和对数因子的估计。对于曲线上的代码，其评估点位于一个网格状的结构，特别是埃尔米特曲线和范数跟踪曲线，我们证明了我们的算法具有准线性的时间复杂度$\tilde{O}(n)$这两个操作。对于无限族的曲线，其点的数量是一个常数因子远离Hasse-Weil界，我们的编码算法的复杂性$\tilde{O}(n^{5/4})$，而unencoding有$\tilde{O}(n^{3/2})$。
英文摘要：
We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $\tilde{O}(n^{3/2})$ resp. $\tilde{O}(qn)$ for AG codes over any $C_{ab}$ curve satisfying very mild assumptions, where $n$ is the code length and $q$ the base field size, and $\tilde{O}$ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, notably the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity $\tilde{O}(n)$ for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse--Weil bound, our encoding algorithm has complexity $\tilde{O}(n^{5/4})$ while unencoding has $\tilde{O}(n^{3/2})$.
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