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

818 ERROR CORRECTING CODES
TABLE 14.5
d
1111
1110
1101
1100
1011
1010
1001
1000
0111
0110
0101
0100
0011
0010
0001
0000
C
1111111
1110010
1101000
1100101
1011100
1010001
1001011
1000110
0111001
0110100
0101110
0100011
0011010
0010111
0001101
0000000
so on. Hence,
G =
1 0 0 0 1 1 O
0 1 0 0 0 1 1
[
0 0 1 0 1 1 1
J
0 0 0 1 1 0 1
Now, we can use c = dG to construct the rest of the code table. This is an efficient
method because it allows us to construct the entire code table from the knowledge of only
k codewords.
Table 14.5 shows the complete code. Note that drnin, the minimum distance between
two codewords, is 3. Hence, this is a single-error correcting code, and 14 of these codewords
can be obtained by successive cyclic shifts of the two codewords 1110010 and
1101000. The remaining two codewords, 1111111 and 0000000, remain unchanged under
cyclic shift.
Generator Polynomial and Generator Matrix of Cyclic Codes
Cyclic codes can also be described by a generator matrix G (Probs. 14.3-6 and 14.3-7). It can
be shown that Hamming codes are cyclic codes. Once the generator polynomial g(x) has been
given, it is simple to find the systematic code generator matrix G = [I P] by determining the
parity submatrix P:
1st row of P:
2nd row of P:
x n-1
Rem--
g(x)
x n-2
Rem--
g(x)
kth row of P:
x n-k
Rem --
g(x)
(14.19)