  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.
