Digital Communications 2

8Study of a Coding and Decoding System by Cyclic Codes

8.1. Objective

The objective of this lab work is the study, then the complete simulation under MATLAB and Simulink, of a coding and decoding application using cyclic codes.

8.2. Recall of the principles of cyclic coding and decoding

Cyclic codes are block codes where the n symbols which constitute a word are considered as coefficient of a polynomial of degree n – 1 :

Any circular permutation on the symbols of a codeword gives a codeword:

The addition of two codewords is a codeword:

The set of all the words of the code constitutes an algebra, while the set of words having a meaning constitutes an ideal.

The benefits of these codes are multiple:

• – they are well suited to the detection of independent and packet errors;
• – their implementation is easy, since the coding and decoding procedures can be made automatic by means of shift registers;
• – their basic principle is based on the theory of polynomials and algebra.

8.3. Coding by division: systematic code

Let the information word represented by its polynomial:

The same word ...