Cyclic
code
Cyclic
code are subset of the class of linear codes that satisfy the following cyclic
shift property: if C=[cn-1cn-2…….c1co] is a code word of acyclic code then C=[cn-2 cn-3…….c0cn-1] obtained by a cyclic shift of the element of C, is also
a code word . that is , all cyclic shift of C are code word. As a consequence
of the cyclic property , the code process a considerable a mount of structure
which can be exploited in the encoding and decoding operation . a number of
efficient encoding and hard- decision decoding algorithms have been devised for
cyclic code that makes it possible to implement long block coders with a large
number of a code word in practical communication systems.
In dealing
with cyclic code it is convenient to associate with a code word C=[cn-1cn-2…….c1co] a
polynomial C(p) of degree ? n-1 define as
C(p)= cn-1 pn + cn-2 pn-1+…….c1p2 + co ] ------- (1)
For
a binary code , each of the coefficient of the polyniomial is either zero or
one .
Now suppose we form the
polynomial
Note
that the polynomial C1(p)
represent the code word C1 =[cn-2 cn-3…….c0cn-1]whch is just the code word C shifted cyclic code by one postion . since
the C1 (p)is the reminder obtained by dividing pC(p) mod (pn +1)