810, Math Spring 2009 HW 2
Due Friday, 13 February 2009
Make sure you justify your statements completely and carefully. 1. Problem 2.2.4 from the Notes. 2. Use Bernoullis Weak Law of Large Numbers (as given and discussed in class) to prove that the set of all binary repetition codes is a Shannon family for the binary symmetric channel BSC(p) with 0 p < 1/2. (Note that this is not a consequence of Shannons Theorem.) 3. We have now proven as Shannons Theorem (on the BSC(p) with p [0, 1/2) and on the BEC(p) with p [0, 1) ) that () for every > 0 and < Cap(p) there is a binary code C of rate at least a and decoding algorithm AC such that avgxC Px (C, AC ) < for the appropriate capacity functions Cap(p). Use () to prove the stronger () for every > 0 and < Cap(p) there is a binary code C of rate at least and a decoding algorithm AC such that maxxC Px (C, AC ) < . Hint: Choose 0 with < 0 < Cap(p). Then () guarantees codes D of rate at least 0 ...