Coding Theory

Third Assignment: due Mon. Oct. 4

1. CT 3.1.11 codewords with k zeros (over an arbitrary field).
   
2. Use the Gilbert-Varshamov bound for binary codes.
   a. Find the largest k for which the GV bound says there exists a code of length n=20 , minimum distance d=5 and dimension at least k.
   b. Find the smallest n such that the GV bound says there exists a code of rate 1/2 that can correct 3 errors.
   c. Find the largest d such that the GV bound guarantees a code of length n=50, dimension k=20 and minimum distance at least d.
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. The extended ternary Golay has parameters [12,6,6 ] and generator matrix [I_6 A] where A is
   011111
   101221
   110122
   121012
   122101
   112210
   a. Find a generator matrix for the puncturing of this code at the last coordinate.
   b. Find a generator matrix for the shortening of this code at the last coordinate.
   c. Either the shortened code or the punctured code is perfect. Which?