Golay Codes
Knowing only about the (7,4) Hamming code, Marcel J.E. Golay generalised Hamming's idea to perfect singleerror correcting
codes based on any prime number. Having done that, Golay began searching for perfect multierror correcting codes. One of
the Golay codes is of special interest since it was later used to generate a packing in the 24dimensional space.
Recall that an
x
errorcorrecting code must have a minimum distance of at least 2
x
+1. For the code to be perfect, the number of
vertices of the unit
n
cube inside a packing sphere of radius
x
must be a power of
r
, where
r
is the radix of the code. In the
binary case
for some integer
k
. This is the sum of the first
x
+1 entries of the
n
th row of the Pascal triangle. Golay found two such numbers,
In the case
n
=90 Golay showed that no perfect doubleerror correcting (90,78) code could exist. For
n
=23 Golay found a 3
error correcting (23,12) code and gave a matrix for it, shown in Figure
6
.
Figure 6:
A matrix for the (23,12) Golay code
By attaching
I
11
to this matrix we obtain a matrix of the same form as that of the matrices for the singleerror correcting codes.
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To see how the code is constructed, let [
a
i
,
j
] be the matrix in Figure
6
. The check digits
are determined from the
message digits
by
or, since the code is binary
The check digits are placed directly after the message digits, so a codeword has the form
Golay supplied only a partial account of the way he constructed this matrix. As it is not at all trivial, and given that we are only
interested in the result itself, it will be skipped.
In
mathematics
and
electronics engineering
, a
binary Golay code
is a type of
errorcorrecting code
used in
digital
communications
. The binary Golay code, along with the
ternary Golay code
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 Fall '10
 Dr. Yaan
 Coding theory, Error detection and correction, Binary Golay Code, Marcel J. E. Golay

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