The Hamming distance between two strings of equal length is the number of positions at which their symbols differ (Black 2006). For binary strings, this is the number of coordinates at which the corresponding vertices of a hypercube graph differ. For example, abc and bca have Hamming distance 3.
For binary strings 
and 
of length 
,
 |
where 
is the Hamming weight and 
denotes bitwise XOR. Indeed, a
coordinate of 
is 1 iff the corresponding coordinates of 
and 
differ. For any binary string 
of length 
, applying the same XOR mask to both
arguments gives
 |
Thus common XOR masking is an isometry of the binary string space.
