A Bound on Equitable Partitions of the Hamming Space

We denote Q
_n
the set of binary words of length n. A partition {C
₁
,C
₂
,
_¿
,C
_r
} of Q
_n
with quotient matrix B = (b
_ij
)
_r×r
is equitable if for all i and j, any word in d has exactly bij neighbors in C
_j
. The equitable partitions of Q
_n
can be obtained from completely regular codes. We derive a bound on equitable partitions of Q
_n
that does not depend on the size of the partition.