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.