Coding Theory

Third Assignment: due Fri. Mar 9

1. CT 3.1.11 codewords with k zeros (over an arbitrary field).
   
2. a. Find a code with minimum distance d = 5 , redundancy r = n - k = 8 and length n as large as you can.
   b. Compare your result with the Gilbert-Varshimov bound.
   c. Do the same for r = 9 if you want to impress me.
   
3. CT 3.3.10, decoding for a Hamming code.
   
4. CT 3.4.4, SDA for decoding an extended Hamming code.
   
5. CT 3.4.5, extended Hamming code is self dual.
   
6. Write down a parity check matrix for the ternary Hamming code with redundancy r = 3.
   
7. CT 3.5.2, another generator for the Golay code.
   
8. CT 3.5.3, Golay code is self dual.
   
9. CT 3.6.5 f, g, h, i, decoding for the extended Golay code.
   
10. Find the number of codewords of each possible weight, 0,1,..,24 for the binary extended Golay code (CT 3.7.7-9).
    
11. a. What is the difference between puncturing a code and shortening a code?
    b. Use the ternary Golay code to illustrate the difference by puncturing and shortening with respect to the last coordinate. The ternary Golay has matrix [I_6 A] where A is
    011111
    101221
    110122
    121012
    122101
    112210
    c. Which of the two, the shortened Golay or the punctured Golay, is perfect?