Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum linear locality of linear codes. In addition, the minimum linear locality of many known families of linear codes is not studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum linear locality of linear codes, and investigates the minimum linear locality of a number of families of linear codes, such as $q$-ary Hamming codes, $q$-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.
翻译：大约九年前,为在分布式和云储存系统中恢复数据,提出了可在当地回收的代码(LRCs),目前,LRCs的研究取得了很大进展,然而,在线性代码的最低线性位置方面缺乏一般理论;此外,文献中未研究许多已知线性代码家族的最低线性位置,根据这两个事实,本文件发展了一些关于线性代码最低线性位置的一般理论,并调查了一些线性代码家庭的最低线性线性位置,如$q$-ary Hamming代码、$q$-mery simplex代码、通用Reed-Muller代码、One codes、最高弧码、扩展超声波性代码和接近MDS代码。本文介绍了许多远程最佳和尺寸最佳LRC的类别。许多线性代码家庭的最低线性位置与本文中制定的一般理论一致。
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