B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

14.3 Linear Block Codes 809
where the error word (or error vector) e, is also a row vector of n elements. For example, if
the data word 100 in Example 14.1 is transmitted as a codeword 100101 (see Table 14.2), and
the channel noise causes a detection error in the third digit, then
and
r = 101101
C = 100101
e = 001000
Thus, an element 1 in e indicates an error in the corresponding position, and O indicates no
error. The Hamming distance between r and c is simply the number of ls in e.
Suppose the transmitted codeword is c; and the channel noise causes an error e;, making
the received word r = c; +e;. If there were no errors, that is, if e; were 000000, then we would
have rH T = 0. But because of possible channel errors, rH T is in general a nonzero row vector
s, called the syndrome:
s = rH T
(14.lla)
= (c; EB e;)H T
= c;H T EB e;H T
= e;H T (14.l lb)
Receiving r, we can compute the syndrome s [Eq. (14.lla)] and presumably we can compute
e; from Eq. (14.1 lb). Unfortunately, knowledge of s does not allow us to solve e; uniquely.
This is because r can also be expressed in terms of codewords other than c;. Thus,
r = CJ EB e1
j i= i
Hence,
Because there are 2 k possible codewords,
s =eH T
is satisfied by 2 k error vectors. In other words, the syndrome s by itself cannot define a unique
error vector. For example, if a data word d = 100 is transmitted by a codeword 100101 in
Example 14.1, and if a detection error is caused in the third digit, then the received word is
101101. In this case we have c = 100101 and e = 001000. But the same word could have
been received if c = 101011 and e = 000110, or if c = 010011 and e = 111110, and so
on. Thus, there are eight possible error vectors (2 k error vectors) that all satisfy Eq. (14. llb).
Which vector shall we choose? For this, we must define our decision criterion. One reasonable
criterion is the maximum likelihood rule according to which, if we receive r, then we decide
in favor of that c for which r is most likely to be received. In other words, we decide "c ;
transmitted" if
all ki=-i
For a binary symmetric channel (BSC), this rule gives a very simple answer. Suppose the
Hamming distance between r and c; is d ; that is, the channel noise causes errors in d digits.