In the second part of today’s lecture, we saw the first lower bound on rate (given a fixed relative distance) called the Gilbert-Varshamov bound: for . The Gilbert bound follows by a greedy construction of a (general) code and the Varshamov bound is satisfied with high probability by a random linear code.

In class I used the following statement without proof: for a random generator matrix , and any fixed non-zero vector , the vector is an uniformly random vector in . I left it as an exercise.

You can prove this in more than one way. I’ll sketch out one way here. As a start, first try and prove the following. Given any () such that for some and some , show that , where is chosen uniformly at random (and the choice is independent for every ). Then using this (and the fact that all entries of are independent random elements from ) prove the claimed statement.

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