The Hamming scheme 
,
where 
and 
,
is the association scheme on 
for an alphabet 
of 
symbols. For 
, 1, ..., 
, its relations are
 |
where 
is the Hamming distance.
The relation 
is the adjacency relation
of the Hamming graph 
, and 
consists of the pairs at graph
distance 
.
Each relation 
has valency
 |
The adjacency matrices of its relations belong to its Bose-Mesner algebra and have
eigenvalues given by values of 
-ary Krawtchouk polynomials
(Bannai and Ito 1984, Brouwer et al. 1989).
The binary Hamming scheme 
is the ambient association
scheme for binary codes of length 
(Schrijver 1979). At a base point, its Terwilliger
algebra can be block diagonalized to obtain semidefinite
programming upper bounds for binary
codes and constant-weight codes (Schrijver 2005).
