Need Help: Cyclic Codes. Performance of coded systems.

[peterh01]

Good Morning,

I need help understanding and solving the following problems:
Problem 1. Topic: Cyclic Codes

*Given n=6

-For the value k an (n,k) cyclic code exist? List all possible K's with corresponding generator polynomials.

Hint: The generator polynomial of degree n-k has to divide x^n + 1. The polynomial x^n + 1 can be factorize with primitive polynomials as x^n + 1 = (x+1)(x+1)(x^2+x+1)(x^2+x+1).

Problema 2. Topic: Cyclic Codes. Performance of coded systems.

Desing a (6,2) cyclic code by choosing the shortest possible generator polynomial. (Use the result from Problem 1)

1. Determine the generator matrix G (in the systematic form) for this code and find all possible code words.
2. How many errors can be corrected by this code?
3. Find the syndrome polynomial if received word is 101110.
4. Compare the performance (BER) of coded systems with that of the same system using uncoded transmission, where Eb/N0 = 7 dB given that Differential Phase Shift Keying (DPSK) modulation is used.

Any help would be greatly appreciated.

Thanks in advanced,
-Peter

 

[oldsirhippy]

A book that I used extensively when doing CRC codes is "Error Correcting codes by Peterson and Weldon". This explains the theory and I found it very helpful and practical.