  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.
