A covering code of radius 
is a code 
of length 
over an alphabet of size 
such that every word is at Hamming
distance at most 
from some codeword in 
. The least such 
is its covering radius, and 
denotes the smallest possible number of codewords
in a covering code of radius at most 
.
The covering density is
 |
It is at least one, with equality for a perfect code.
Maharaj (2026) obtained 
for every odd integer 
and 
, using quaternary Preparata
codes. In particular, the octacode gives 
, where the upper bound improves 352.
A separate construction gives 
, improving 52. Maharaj (2026) states that Claude
assisted with the searches and proof and that independent programs verified the computational
claims.
