We construct families of locally recoverable codes with availability \(t\ge 2\) using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.

中文翻译：

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局部可恢复代码的最小距离和参数范围，可从曲线的纤维产品中获得

我们使用曲线的纤维积构建具有可用性\(t\ge 2\)的局部可恢复代码族，确定许多族的精确最小距离，并证明此类代码的最小距离的一般定理。本文最后探讨了这些系列的代码参数和更普遍的纤维产品结构。我们表明，纤维产品代码可以实现任意大的速率和任意小的相对缺陷，并与文献中的已知界限和重要结构进行比较。