Erasures Repair for Decreasing Monomial-Cartesian and Augmented Reed-Muller Codes of High Rate
Erasures Repair for Decreasing Monomial-Cartesian and Augmented Reed-Muller Codes of High Rate
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DOI：
10.1109/tit.2021.3130096
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发表时间：
2021-07
影响因子：
2.5
通讯作者：
H. L'opez;Gretchen L. Matthews;Daniel Valvo
中科院分区：
文献类型：
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作者：
H. L'opez;Gretchen L. Matthews;Daniel Valvo
在这项工作中，我们提出了一两个擦除的线性精确修复方案，以减少单一 - 牙周代码（DM-CC），这是一个为极地代码提供框架的代码家族。在两个擦除的情况下，擦除的位置应满足一定的限制。我们介绍了增强的Reed-Muller（ARM）和增强的笛卡尔法规（ACAR）的家族，这些家族分别是通过策略性地将向量添加到Reed-Muller和笛卡尔代码中获得的评估代码家族。我们为这些增强法规家庭开发了一两个擦除的维修方案。与DM-CC的两种擦除的维修计划不同，两个用于增强代码的擦除的维修方案对擦除位置没有任何限制。当尺寸和基地固定时，我们提供了示例，其中ARM和ACAR代码与Reed-Solomon（hermitian）代码相比，ARM和ACAR代码提供了较低的带宽（分别，位于位宽）。当固定长度和基地时，我们提供了与ARM相比提供较低带宽的示例。最后，当增强代码达到最大速率时，我们分析了渐近行为。
In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes (DM-CC), a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.