Page 340 - Satellite Communications, Fourth Edition

320 Chapter Eleven

The number of rows in H is equal to the number of parity bits, n k, and
the number of columns is n, that is the parity check matrix is a (n k, n)
matrix. A fundamental property of these code matrices is that

T
GH 0 (11.6)
When a codeword is received it can be verified as being correct on mul-
T
T
tiplying it by H . The product cH should be equal to zero. This follows
T
T
since c dG and therefore cH dGH 0. If a result other than zero
is obtained, then an error has been detected. In general terms, the prod-
T
uct cH gives what is termed the syndrome and denoting this by s:
s cH T (11.7)
A received codeword can be represented by the transmitted codeword
plus a possible error vector. For example if the transmitted codeword is
[1010010] and the received codeword is [1010110] the error is in the fifth
bit position from the left and this can be written as
[1010110] [1010010] [0000100]

More generally, if c is the received codeword, c the transmitted
R
T
codeword and e the error vector then, with modulo-2 addition
c c e (11.8)
T
R
Substituting c for c in Eq. (11.7) gives
R
s (c e)H T
T
T
c H eH T
T
T
T
But as shown earlier, the product cH , which is c H in this notation,
T
is zero, hence,
T
s eH (11.9)
This shows that the syndrome depends only on the error vector and is
independent of the codeword transmitted. Since the error vector has the
n
same number of bits n as the codeword there will be 2 possible error vec-
tors. Not all of these can be detected since the syndrome has only n k bits
n–k
(determined by the number of rows in the H matrix), giving as 2 the
number of different syndromes. One of these will be the all zero syndrome,
n–k
and hence the number of errors that can be detected is just 2 – 1. In prac-
tice the decoder is designed to correct the most likely errors, for example
those with only 1-bit error. The received syndrome may be compared with