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208 13 — Binary Codes
13.3 Perfect codes
1 2
A t-sphere (or a sphere of radius t) in Hamming space, centred on a point x,
is the set of points whose Hamming distance from x is less than or equal to t.
The (7, 4) Hamming code has the beautiful property that if we place 1spheres
about each of its 16 codewords, those spheres perfectly fill Hamming
space without overlapping. As we saw in Chapter 1, every binary vector of
length 7 is within a distance of t = 1 of exactly one codeword of the Hamming
code.
A code is a perfect t-error-correcting code if the set of t-spheres centred
on the codewords of the code fill the Hamming space without overlapping.
(See figure 13.4.)
Let’s recap our cast of characters. The number of codewords is S = 2 K .
The number of points in the entire Hamming space is 2 N . The number of
points in a Hamming sphere of radius t is
t
w=0
t
. . .
t
t
N
. (13.1)
w
For a code to be perfect with these parameters, we require S times the number
of points in the t-sphere to equal 2 N :
for a perfect code, 2 K
or, equivalently,
t
N
w
t
N
w
w=0
w=0
= 2 N
(13.2)
= 2 N−K . (13.3)
For a perfect code, the number of noise vectors in one sphere must equal
the number of possible syndromes. The (7, 4) Hamming code satisfies this
numerological condition because
1 +
7
= 2
1
3 . (13.4)
Figure 13.4. Schematic picture of
part of Hamming space perfectly
filled by t-spheres centred on the
codewords of a perfect code.