3.13 Hamming Code

For a Hamming code of length 2 m - 1 , its parity-check matrix is
a matrix whose columns consist of the entire set of the non-zero
binary m-tuples.
3.14 Golay Code
The (23,12) Gloay code is the only known multiple-errorcorrecting
binary perfect code, which is capable of correcting 3
or fewer random errors in a block of 23 digits, d min
= 7 .
(Discovered by Golay in 1949).
The (23, 12) Gloay code is either generated by
g +
2 4 5 6 10 11
1(
x)
= 1 + x + x + x + x + x x
or by
g +
5 6 7 9 11
2
( x)
= 1 + x + x + x + x + x x
Both g 1
(x)
and g 2
(x)
are factors of x 23 + 1
23
and + 1 = ( 1 + x)
g ( x)
g ( x)
x
1 2