Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability in the classical setting. This interest also translates to the insdel metric setting, as the Guruswami-Sudan decoding algorithm can be utilized to provide a deletion correcting algorithm in the insdel metric. Nevertheless, there have been few studies on the insdel error-correcting capability of Reed-Solomon codes. Our main contributions in this article are explicit constructions of two families of 2-dimensional Reed-Solomon codes with insdel error-correcting capabilities asymptotically reaching those provided by the Singleton bound. The first construction gives a family of Reed-Solomon codes with insdel error-correcting capability asymptotic to its length. The second construction provides a family of Reed-Solomon codes with an exact insdel error-correcting capability up to its length. Both our constructions improve the previously known construction of 2-dimensional Reed-Solomon codes whose insdel error-correcting capability is only logarithmic on the code length.

中文翻译：

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高插入和删除噪声状态下二维 Reed-Solomon 码的显式构造


插入和删除（简称insdel）错误是通信系统中由于消息中位置信息丢失而引起的同步错误。里德-所罗门码因其编码简单、结构良好以及经典环境中的列表解码能力而引起了广泛关注。这种兴趣也转化为insdel度量设置，因为可以利用Guruswami-Sudan解码算法来提供insdel度量中的删除校正算法。然而，关于Reed-Solomon码的insdel纠错能力的研究还很少。我们在本文中的主要贡献是显式构造了两个二维 Reed-Solomon 码族，其 insdel 纠错能力渐近达到了单例边界所提供的能力。第一个构造给出了一系列具有渐近其长度的 insdel 纠错能力的 Reed-Solomon 码。第二种结构提供了一系列里德-所罗门码，在其长度范围内具有精确的插入错误纠正能力。我们的两种结构都改进了先前已知的二维里德所罗门码的结构，其插入错误校正能力仅是码长度的对数。