论文标题
来自Deligne-Lusztig曲线的环状扩展的局部性代码
Codes with locality from cyclic extensions of Deligne-Lusztig curves
论文作者
论文摘要
最近，Skabelund定义了新的最大曲线，这些曲线是铃木和REE曲线的循环扩展。以前，现在众所周知的GK曲线是遗传曲线的循环扩展。在本文中，我们考虑了从这些新曲线构建的本地可回收代码，这补充了为GK曲线所做的。可局部可恢复的代码允许通过仅访问其他几个形成所谓恢复集的其他符号来恢复单个符号。如果每个符号至少都有两个不交织的恢复集，则据说该代码具有可用性。将三个结构描述为最适合特定情况。第一次使用Tamo和Barg从曲线中使用了本地可回收的代码的原始结构。第二个通过呼吁使用Haymaker，Malmskog和Matthews的纤维产品来产生可用性的代码，而第三个则可以通过使用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比文献中通常发现的代码更小。
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.
