Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are described, as each best fits a particular situation. The first employs the original construction of locally recoverable codes from curves by Tamo and Barg. The second yields codes with availability by appealing to the use of fiber products as described by Haymaker, Malmskog, and Matthews, while the third accomplishes availability by taking products of codes themselves. We see that cyclic extensions of the Deligne–Lusztig curves provide codes with smaller locality than those typically found in the literature.

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来自 Deligne-Lusztig 曲线循环扩展的局部性代码

最近，Skabelund 定义了新的最大曲线，它们是 Suzuki 和 Ree 曲线的循环扩展。以前，现在众所周知的 GK 曲线被发现是 Hermitian 曲线的循环扩展。在本文中，我们考虑从这些新曲线构建的局部可恢复代码，补充为 GK 曲线所做的。本地可恢复代码允许通过仅访问形成所谓的恢复集的少数其他符号来恢复单个符号。如果每个符号至少有两个不相交的恢复集，则称该代码具有可用性。描述了三种结构，因为每种结构最适合特定情况。第一个使用由 Tamo 和 Barg 对曲线进行局部可恢复代码的原始构造。第二个通过吸引使用 Haymaker 描述的纤维产品来产生具有可用性的代码，Malmskog 和 Matthews，而第三个则通过获取代码产品本身来实现可用性。我们看到 Deligne-Lusztig 曲线的循环扩展提供的代码具有比文献中通常发现的更小的局部性。