Perfect LRCs and k-optimal LRCs
Perfect LRCs and k-optimal LRCs
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DOI：
10.1007/s10623-022-01148-7
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发表时间：
2022-11
期刊：
影响因子：
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通讯作者：
Weijun Fang;Bin Chen;Shu-Tao Xia;Fang-Wei Fu;Xiangyu Chen
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文献类型：
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作者：
Weijun Fang;Bin Chen;Shu-Tao Xia;Fang-Wei Fu;Xiangyu Chen
如果一个线性码可以通过访问其它码元来恢复被擦除的码元，则称之为具有局部性的局部可修复码。近年来，LRC的构建得到了广泛的研究。在本文中，我们在这个方向上迈出了一步。首先，我们提出了一个新的概念，完美的LRC的大小正好达到汉明类型的界，类似于完美的代码，达到汉明界的经典编码理论。利用奇偶校验矩阵的方法，我们分别建立了LRC的存在性与有限几何和有限域的某些子集的存在性之间的一些重要联系.利用射影几何中的q-Steiner系统和向日葵以及有限域上的差集，我们得到了两个新的具有可变参数的完美LRC的构造，并给出了几个新的在整数约束下达到另一Hamming型界的k-最优LRC的构造.此外，对于固定的qandr，本文所构造的所有q-aryr-LRC的码长可以是任意大的，并且码率可以渐近地达到上界。
A linear code is called a locally repairable code (LRC) with localityrif one can recover an erased code symbol by accessing at mostrother code symbols. Constructions of LRCs have been widely investigated in recent years. In this paper, we give a step forward in this direction. Firstly, we propose a novel concept of perfect LRCs whose size exactly achieves the Hamming-type bound, similar to the perfect codes that achieving the Hamming bound in classical coding theory. By the parity-check matrix approach, we establish some important connections between the existence of LRCs and the existence of some subsets of finite geometry and finite fields with certain properties, respectively. By employingq-Steiner systems and sunflowers in projective geometry and difference sets in finite fields, we obtain two new constructions of perfect LRCs with flexible parameters and present several new constructions ofk-optimal LRCs achieving another Hamming-type bound under the integers restriction. Moreover, for fixedqandr, the code lengths of all theq-aryr-LRCs constructed in this paper can be arbitrarily large and the code rates can asymptotically achieve the upper bound.