Information Theory and Coding - Codes: The Guide to Secrecy from Ancient to Modern Times - page 455

which told him that there might exist a perfect [23 , 12] binarycode that had the
potential to correct up to three errors. In 1949, he made the discoveryof G 23 ,

which indeed satisfies these properties, and is to this day, the only one known
to be capable of correcting anycombination of up to three random errors in a
vector of length 23 (see Footnote 11.8 on page 447). Later,

G 23 was extended

G

to what is now known as the

24 -code, which is a (nonperfect) [24 , 12 , 8]-code.

G

24 proved to be extremelyuseful in satellite transmission. In fact during the
years 1979-1981, Voyager I and Voyager II spacecrafts sent back signals from
Jupiter and Saturn that were error-corrected using G 24 .

It turns out that it is veryproductive to first define the G 24 code and derive

the G 23 code from it. Since a linear code maybe characterized via its generat-

ing matrix, we provide the

24
matrix, we first provide the initial twelve columns that are given bythe iden-
titymatrix I 12 , so it has the form G =[ I 12 |

G 24 generator matrix first. To construct this 12 ×

M 12 × 12 ]. The matrix M 12 × 12 is
constructed as follows. The first row of the matrix is given by

R 1 =(1 , 1 , 1 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 1 , 0)=( r 1 ,r 2 ,...,r 12 ) ,

and rows 2 through 11 are given byfixing the first element of this row and cycli-
callypermuting the remaining elements to the right, namely, for j =2 , 3 ,..., 11,

R j =(1 ,r 12 − j +2 ,r 12 − j +3 ,...,r 12 ,r 2 ,r 3 ,...,r 12 − j +1 ) ,

and row 12 is given by

R 12 =(0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1) .

Hence, we achieve, G =

100000000000111011100010
010000000000101101110001
001000000000110110111000
000100000000101011011100
000010000000100101101110
000001000000100010110111
000000100000110001011011
000000010000111000101101
000000001000111100010110
000000000100101110001011
000000000010110111000101
000000000001011111111111

.

Properties of G 24

The matrix G is a generator matrix for the G 24 linear [24 , 12 , 7]-code, and

this code satisfies the following properties.

G 24 = 1. G 24 is self dual. In other words, G 24 . (See page 448.)

2. If c is a codeword of G 24 , then the weight w ( c ) satisfies that

w ( c ) ≡ 0(mod 4) ,

and w ( c ) > 4. (See page 446.)