A deep hole of an error-correcting code is a word whose Hamming
distance to its nearest codeword equals the code
covering radius. For the full-length Reed-Solomon
code of dimension 
over 
,
this radius is 
.
Each word has a unique interpolation polynomial 
of polynomial
degree at most 
.
Chen and Huang (2026) report that, for odd 
and 
, the deep holes are exactly the words
with 
. For even 
and 
, the same description holds except that 
has the additional family 
, where 
and 
. Thus an arbitrary degree-
polynomial is not included
in this extra family.
For even 
with 
, Chen and Huang (2026) instead parametrize
the deep holes as 
,
with 
, 
, and 
a normalized o-polynomial
of polynomial degree at most 
. This parametrization does not classify all hyperovals
explicitly.
Fang (2026) reports that for every prime power 
and 
, a word is a deep
hole iff 
, where 
is the floor function.
Together with the previously known high-dimensional cases, this removes the restriction
for the nonexceptional cases
. The exceptional even-characteristic
case 
and the separate 
description above remain.
Fan (2026) also reports an exact classification when 
, 
, 4, ..., 7, and 
. There are no deep holes in these ranges for
odd 
. The only exceptions for even 
have 
and 
: for 
they are 
with odd 
, and for 
they are 
with 
. Here the coefficients belong to 
and 
.
Fang (2026) reports ChatGPT assistance with checking some main proofs.
Fan (2026) reports ChatGPT assistance with exploratory proof
development and computational checks. These reported classifications await further
independent verification.
