基于Dickson多项式的Reed-Solomon码中的深空
Deep Holes in Reed-Solomon Codes Based on Dickson Polynomials
摘要：
对于$[n,k]$里德-所罗门码$\mathcal{C}$，可以证明任何接收的字$r$与$\mathcal{C}$之间的距离至多为$n-k$，表示为$d(r,\mathcal{C})\leq n-k$。任何满足等式的单词$r$都被称为深孔。Guruswami和Vardy（2005）表明，对于特定类别的代码，确定一个单词是否是一个深孔是NP难的。他们顺便建议，当$\mathcal{C}$的评估集较大或结构化时，可能会更容易。根据这一思想，我们研究的情况下，评估集是一个迪克森多项式的图像，其值出现一个特殊的均匀性。要找到家庭收到的话，不是深洞，我们减少到一个子集和问题（或等效的，一个迪克森多项式变化的华林的问题），并找到解决方案的条件，通过应用参数使用估计的字符和索引的评估集。
英文摘要：
For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
