Binary cyclic codes
Most of the linear block codes that have proved to be useful in practical applications belong to a class of codes called cyclic codes, and most of the research on block codes has been concentrated on them. Cyclic codes are attractive for two reasons. First, cyclic codes are easy to encode. Secondly, they have a well-defined algebraic structure, which has led to the development of very efficient decoding schemes. Cyclic codes include a broad and important subclass called BCH codes. The BCH codes are efficient multiple-error-correcting codes .
Definition : An (n , k) LBC C is called a cyclic code if it has the following property : if an n-tuple v = (v0,v1,v2, . . . , vn-1) is a code vector of C , the n-tuple v(1) = (vn-1,v0,v1, . . . , vn-2) obtained by shifting v cyclically one place to the right is also a code vector of C.
From the definition, it is clear that v(i) = vn-i ,vn-i+1, . . . , vn-1, v0, v1, . . . , vn-i-1) obtained by shifting v to the right cyclically I places is also a code vector . Therefore, cyclic codes can be decomposed into classes, each class consists of code words that are cyclic shifts of each other and then, all code words in each class are of the same weight. Such classes are called cycles. The size of a class is termed its period .
We shall treat the components of a code vector as coefficients of a polynomial as follows :
v=(v0,v1, . . . ,vn-1) çè v(x)= v0 + v1x + . . . + vn-1xn-1
Thus, each code vector has one-to-one correspondence to a polynomial of degree of v(x) is less than n-1. We call v(x) the code polynomial .
Theorem 2-4 : In an (n,k) cyclic code, there exists one and only one code polynomial g(x) of degree n-k,
g(x)= 1+g1x+g2x2+. . . +gn-k-1xn-k-1+xn-k , where g=0 or 1 .
Every code polynomial v(x) is a multiple of g(x) and every polynomial of degree n-1 or less which is a multiple of g(x) must be a code polynomial.
Since the polynomial g(x) can be used to form all the code words in the code, it is called the generator polynomial of the cyclic code. The degree n-k of g(x) is equal to the number of parity check digits of the code. The k information digits can be encoded by multiplying the message m(x) by g(x).
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