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

814 ERROR CORRECTING CODES
One of the interesting properties of code polynomials is that when x ; c(x) is divided by x n + 1,
the remainder is c < il (x). We can verify this property as follows:
xc(x) = c1x n + c2x n-l + · · · + C n X
CJ
:'.'. .... ±J.Jc 1x n + c2x n-l + · · · + C nX
C)X n +
CJ
c2x n-l + c3x 11 - 2 + · · · + C n X + C)
remainder
The remainder is clearly c <ll (x). In deriving this result, we have used the fact that subtraction
amounts to summation when modulo-2 operations are involved. Continuing in this fashion,
we can show that the remainder of x ; c(x) divided by x n + I is c < il (x).
We now introduce the concept of code generator polynomial g(x). Since each (n, k)
codeword can be represented by a code polynomial
c(x) = C)X n-l + c2x n- 2
+ · · · + C n
g(x) is a code generator polynomial (of degree n - k), if for a data polynomial d(x) of
degree k - l
we can generate code polynomial via
c(x) = d (x) g(x) (14. 14)
Notice that there are 2 k distinct code polynomials ( or codewords). For cyclic code, a codeword
after cyclic shift is still a codeword.
We shall now prove an important theorem in cyclic codes:
Cyclic Linear Block Code Theorem: If g(x) is a polynomial of degree n - k and is
a factor of x n + I (modulo-2), then g (x) is a generator polynomial that generates an (n, k)
linear cyclic block code.
Proof- For a data vector (d1 , d2, ... , d k ), the data polynomial is
Consider k polynomials
(14. 15)
g(x),
xg (x),
which have degrees n - k, n - k + I, ... , n - I, respectively. Hence, a linear combination of
these polynomials equals
(14.16)