RS-like locally recoverable (LRC) codes have construction based on the classical construction of Reed–Solomon (RS) codes, where codewords are obtained as evaluations of suitably chosen polynomials. These codes were introduced by Tamo and Barg (IEEE Trans Inf Theory 60(8):4661–4676, 2014) where they assumed that the length n of the code is divisible by $$r+1$$ r + 1 , where r is the locality of the code. They also proposed a construction with this condition lifted to $$n \ne 1 \bmod (r+1)$$ n ≠ 1 mod ( r + 1 ) . In a recent paper, Kolosov et al. (Optimal LRC codes for all lenghts $$n \le q$$ n ≤ q , arXiv:1802.00157, 2018) have given an explicit construction of optimal LRC codes with this lifted condition on n . In this paper we remove any such restriction on n completely, i.e., we propose constructions for q -ary RS-like LRC codes of any length $$n \le q$$ n ≤ q . Further, we show that the codes constructed by the proposed construction are optimal LRC codes for their parameters.

中文翻译：

---

任意长度的最佳类 RS LRC 代码

RS-like locally recoverable (LRC) codes have construction based on the classical construction of Reed–Solomon (RS) codes, where codewords are obtained as evaluations of suitably chosen polynomials. 这些代码是由 Tamo 和 Barg 引入的（IEEE Trans Inf Theory 60(8):4661–4676, 2014），他们假设代码的长度 n 可以被 $$r+1$$ r + 1 整除，其中 r是代码的位置。他们还提出了一个将这个条件提升到 $$n \ne 1 \bmod (r+1)$$ n ≠ 1 mod ( r + 1 ) 的构造。在最近的一篇论文中，Kolosov 等人。（所有长度的最佳 LRC 代码 $$n \le q$$ n ≤ q ，arXiv:1802.00157, 2018）已经给出了在 n 上具有这种提升条件的最佳 LRC 代码的明确构造。在本文中，我们完全消除了对 n 的任何此类限制，即，我们建议构造任何长度 $$n \le q$$ n ≤ q 的 q -ary RS 类 LRC 代码。此外，我们表明由所提出的构造构造的代码是其参数的最佳 LRC 代码。