Posted by: atri | March 25, 2009

## Lecture 26: Construction of l-wise independent sources

In the first part of today’s lecture, we proved the following general result. Any linear code $S\subseteq \mathbb{F}_2^n$ with dual distance at least $\ell+1$, is an $\ell$-wise independent source. As a corollary, we saw that the dual of the $BCH_{2,\log{n},\ell+1}$ code is an $\ell$-wise independent source of size $O\left(n^{\lfloor\frac{\ell}{2}\rfloor}\right)$.

Next, we returned to the question of an explicit asymptotically good code. The best construction we have seen so far is the trivial conversion of a RS code over $\mathbb{F}_{2^m}$ into a binary code using any map $f:\mathbb{F}_{2^m}\rightarrow \mathbb{F}_2^m$. We finally observed that we need to replace $f$ by a code with large distance. Next lecture, we will see a general code composition technique called code concatenation that does exactly this.

In the lecture today, I totally messed up the statement of the Weil-Carlitz-Uchiyama bound. Here is the correct statement: Any non-zero codewords in the dual of the $BCH_{2,m,2t+1}$ has Hamming weight $w$ such that

$2^{m-1}-(t-1)2^{m/2}\le w\le 2^{m-1}+(t-1)2^{m/2},$

provided $2t-2 < 2^{m/2}$.