A binary Hamming code 
of length 
(with 
)
is a linear code with parity-check matrix 
whose columns consist of all nonzero binary vectors of length
, each used once (MacWilliams and Sloane
1977). 
is an 
code. Hamming codes are perfect single error-correcting
codes.
For 
, the family gives the Hamming
(7,4) code, which has parameters 
.
