In today’s class we proved the Gilbert-Varshamov bound. The relevant lecture notes from fall 07 is lecture 15.

In class today I made the following claim without proof: given a non-zero vector and a random matrix over , the vector is uniformly distributed over . I realized later that this might not be so obvious if you have not fiddled with finite fields much. So below is a proof of this statement.

Let the th entry in () be denoted by . Note that as is a random matrix over , each of the is a uniformly random element from . Now, note that we would be done if we can show that for every , the th entry in  (call it ) is a uniformly random element from . To finish the proof, we prove this latter fact. If we denote , then . (Recall that the ’s are uniformly random elements from .) The rest of the proof is a generalization the argument we used in class to show that every non-zero codeword in has Hamming weight to the general .  

Note that to show that is uniformly distributed over , it is sufficient to prove that takes every value in equally often over all the choices of values that can be assigned to . Now as is non-zero, at least one of the its element is non-zero: with loss of generality assume that . Thus, we can write . Now for every fixed assignment of values to (note that there are such assignments), takes a different value for each of the distinct possible assignments to (this is where we use the assumption that ). Thus, over all the possible assignments of , takes each of the values in exactly times, which proves our claim.

Please use the comments section for any questions you might have about today’s lecture or the proof above.