Posted by: atri | October 2, 2007

Lecture 15: Gilbert-Varshamov Bound

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: R\ge 1-H_q(\delta)-\epsilon for 0\le \delta<1-\frac{1}{q}. 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 k\times n generator matrix \mathbf{G}\in\mathbb{F}_q^{k\times n}, and any fixed non-zero vector \mathbf{y}\in\mathbb{F}_q^k, the vector \mathbf{y}\cdot \mathbf{G} is an uniformly random vector in \mathbb{F}_q^n. 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 y_i\in\mathbb{F}_q (1\le y\le k) such that y_i\neq 0 for some i and some a\in\mathbb{F}_q, show that \mathrm{Pr}\left[\sum_{i=1}^k y_i\cdot r_i=a\right]=\frac{1}{q}, where r_i\in \mathbb{F}_q is chosen uniformly at random (and the choice is independent for every 1\le i\le k). Then using this (and the fact that all entries of \mathbf{G} are independent random elements from \mathbb{F}_q) prove the claimed statement.


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