Folded Reed-Solomon (FRS) and univariate multiplicity codes are prominent polynomial codes over finite fields, renowned for achieving list decoding capacity. These codes have found a wide range of applications beyond the traditional scope of coding theory. In this paper, we introduce improved bounds on the list size for list decoding of these codes, achieved through a more streamlined proof method. Additionally, we refine an existing randomized algorithm to output the codewords on the list, enhancing its success probability and reducing its running time. Lastly, we establish list-size bounds for a fixed decoding parameter. Notably, our results demonstrate that FRS codes asymptotically attain the generalized Singleton bound for a list of size $2$ over a relatively small alphabet, marking the first explicit instance of a code with this property.

翻译：折叠Reed-Solomon（FRS）码与单变量多重码是有限域上著名的多项式码，因其达到列表译码容量而闻名。这些码在编码理论传统范围之外已获得广泛应用。本文通过更简洁的证明方法，引入了这些码在列表译码中列表尺寸的改进界。此外，我们优化了现有随机化算法以输出列表中的码字，提高了其成功概率并降低了运行时间。最后，我们针对固定译码参数建立了列表尺寸界。值得注意的是，我们的结果表明，FRS码在相对较小的字母表上，对于尺寸为$2$的列表渐近地达到广义Singleton界，这是首个具有该性质的显式码实例。