Constructions of optimal locally recoverable codes via Dickson polynomials
Constructions of optimal locally recoverable codes via Dickson polynomials
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通过 Dickson 多项式构造最优局部可恢复代码
DOI：
10.1007/s10623-020-00731-0
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Deng Tang
中科院分区：
文献类型：
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作者：
Jian Liu;Sihem Mesnager;Deng Tang
2014 年，Tamo 和 Barg 在一篇非常引人注目的论文中提出了一系列最优线性局部可恢复码（LRC 码），这些码可以达到最大可能距离（给定码长、基数和局部性）。构造这种最佳线性 LRC 码的关键要素是所谓的良好多项式，其中 等于 LRC 码的局部性。 2018 年，刘等人。提出了使用函数组合设计良好多项式的两种通用方法，从而产生了良好多项式的三种新构造。接下来，米歇尔提供了一个伽罗瓦理论框架，允许构造良好的多项式。著名的迪克森多项式是一类重要的多项式，近年来在不同的背景下得到了广泛的研究。在本文中，我们提供了基于迪克森多项式的设计良好多项式的新方法。这种良好的多项式提供了最优 LRC 码的新构造。
In 2014, Tamo and Barg have presented in a very remarkable paper a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality). The key ingredients for constructing such optimal linear LRC codes are the so-calledr-good polynomials, whereris equal to the locality of the LRC code. In 2018, Liu et al. presented two general methods of designingr-good polynomials by using function composition, which led to three new constructions ofr-good polynomials. Next, Micheli provided a Galois theoretical framework which allows to constructr-good polynomials. The well-known Dickson polynomials form an important class of polynomials which have been extensively investigated in recent years in different contexts. In this paper, we provide new methods of designingr-good polynomials based on Dickson polynomials. Suchr-good polynomials provide new constructions of optimal LRC codes.