Page 338 - Satellite Communications, Fourth Edition

318 Chapter Eleven

TABLE 11.1 Even Parity Codewords
Modulo-2 addition
Dataword of dataword Codeword
000 0 0000
001 1 0011
010 1 0101
011 0 0110
100 1 1001
101 0 1010
110 0 1100
111 1 1111

The Hamming distance between two codewords is defined as the
number of positions by which the two codewords differ. Thus the code-
words 0000 and 1111 differ in four positions, so their Hamming dis-
tance is four. The minimum Hamming distance, usually just referred to
as the minimum distance is the smallest Hamming distance between
any two codewords. It can be shown that the minimum distance is given
by the minimum number of binary 1s in any codeword, excluding the
all-zero codeword. By inspection it will be seen that the minimum dis-
tance of the code in Table 11.1 is two. The greater the minimum distance
the better the code, as this reduces the chances of one codeword being
converted to another by noise.
The properties of linear block codes are best formulated in terms of
matrices. Only a summary of some of these results are presented
here, as background to aid in the understanding of coding methods
used in satellite communications. A dataword (or message block) of
size k is denoted by a row vector d, for example the sixth dataword
in Table 11.1 is d [101]. Denoting the codeword by row vector c, the
6
corresponding codeword is c [1010]. In general, the codeword is
6
generated from the dataword by use of a generator matrix denoted by
G, where

c dG (11.2)

Design of the generator matrix forms part of coding practice and will
not be gone into here. However an example will illustrate the proper-
ties. One example of a generator matrix for a (7, 4) code is
1 0 0 0 1 1 1
0 1 0 0 1 1 0
G ≥ ¥ (11.3)
0 0 1 0 1 0 1
0 0 0 1 0 1 1