A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.

翻译：在有限域 $\mathbb{F}_{q}$ 上，一个具有局部性 $r$ 的局部可恢复码是指该码的每个码字坐标均可通过该码字至多 $r$ 个其他坐标的值来恢复。局部可恢复码在恢复受损消息和数据方面具有高效性，这使其在分布式存储系统中具有高度适用性。长度为 $n=m\ell$、指数为 $\ell$ 的准循环码是一类在循环移位 $\ell$ 个位置下保持不变的线性码。本文中，我们将准循环局部可恢复码分解为一系列分量码之和，其中每个分量码是定义在 $\mathbb{F}_q$ 的域扩张上的线性码。利用这些具有设定参数的分量码，我们提出了确保具有更大维度和码长的几乎最优及最优准循环局部可恢复码存在的条件。