A Hamming ball of radius 
about a word 
is the set of words
at Hamming distance at most 
from 
. Here 
is an alphabet with 
symbols, and 
is a nonnegative integer.
Its size is independent of its center and, for 
,
is
 |
since a word at distance 
differs in a choice of 
coordinates, with 
alternatives at each.
For a code of minimum Hamming distance 
, the Hamming balls of radius 
about distinct codewords
are disjoint, where 
is the floor function (Hamming 1950). The Gilbert-Varshamov
bound instead uses a greedy covering by radius-
Hamming balls (Gilbert 1952).
