30-07-2012, 03:07 PM
CYCLIC CODES
6 Cyclic Codes.PPT (Size: 1.05 MB / Downloads: 200)
Motivation & Properties of cyclic code
Cyclic code are a class of linear block codes. Thus, we can find generator matrix (G) and parity check matrix (H).
The reason is that they can be easily implemented with externally cost effective electronic circuit.
More on Code Polynomials
The nonzero code polynomial of minimum degree in a cyclic code C is unique.
(If not, the sum of the two polynomials will be a code polynomial of degree less than the minimum. Contradiction)
Let g(X) = g0 + g1X +….+ gr-1Xr-1 +Xr be the nonzero code polynomial of minimum degree in an (n,k) cyclic code. Then the constant term g0 must be equal to 1.
(If not, then one cyclic shift to the left will produce a code polynomial of degree less than the minimum. Contradiction)
For the (7,4) code given in the Table, the nonzero code polynomial of minimum degree is g(X) = 1 + X + X3
Association of Syndrome to Error Pattern
Look-up table implemented via a combinational logic circuit (CLC). The complexity of the CLC tends to grow exponentially with the code length and the number of errors to correct.
Cyclic property helps in simplifying the decoding circuit.
The circuit is designed to correct the error in a certain location only, say the last location. The received word is shifted cyclically to trap the error, if it exists, in the last location and then correct it. The CLC is simplified since it is only required to yield a single output e telling whether the syndrome, calculated after every cyclic shift of r(X), corresponds to an error at the highest-order position.
The received digits are thus decoded one at a time.