Posted by: atri | February 2, 2009

## Lecture 9: Gilbert-Varshamov Bound

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 $\mathbf{m}\in\mathbb{F}_q^k$ and a random $k\times n$ matrix $\mathbf{G}$ over $\mathbb{F}_q$, the vector $\mathbf{m}\cdot\mathbf{G}$ is uniformly distributed over $\mathbb{F}_q^n$. 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 $(j,i)$th entry in $\mathbf{G}$ ($1\le j\le k, 1\le i\le n$) be denoted by $g_{ji}$. Note that as $\mathbf{G}$ is a random $k\times n$ matrix over $\mathbb{F}_q$, each of the $g_{ji}$ is a uniformly random element from $\mathbb{F}_q$. Now, note that we would be done if we can show that for every $1\le i\le n$, the $i$th entry in $\mathbf{m}\cdot\mathbf{G}$  (call it $b_i$) is a uniformly random element from $\mathbb{F}_q$. To finish the proof, we prove this latter fact. If we denote $\mathbf{m}=(m_1,\dots,m_k)$, then $b_i=\sum_{j=1}^k m_jg_{ji}$. (Recall that the $g_{ji}$‘s are uniformly random elements from $\mathbb{F}_q$.) The rest of the proof is a generalization the argument we used in class to show that every non-zero codeword in ${\sc Had}_r$ has Hamming weight $2^{r-1}$ to the general $\mathbb{F}_q$.

Note that to show that $b_i$ is uniformly distributed over $\mathbb{F}_q$, it is sufficient to prove that $b_i$ takes every value in $\mathbb{F}_q$ equally often over all the choices of values that can be assigned to $g_{1i},g_{2i},\dots,g_{ki}$. Now as $\mathbf{m}$ is non-zero, at least one of the its element is non-zero: with loss of generality assume that $m_1\neq 0$. Thus, we can write $b_i= m_1g_{1i}+\sum_{j=2}^k m_jg_{ji}$. Now for every fixed assignment of values to $g_{2i},g_{3i},\dots,g_{ki}$ (note that there are $q^{k-1}$ such assignments), $b_i$ takes a different value for each of the $q$ distinct possible assignments to $g_{1i}$ (this is where we use the assumption that $m_{1i}\neq 0$). Thus, over all the possible assignments of $g_{1i},\dots,g_{ki}$, $b_i$ takes each of the values in $\mathbb{F}_q$ exactly $q^{k-1}$ times, which proves our claim.

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