Posted by: atri | March 23, 2009

## Lecture 25: l-wise independent sources

In today’s lecture, we defined the notion of an $\ell$-wise independent sources (as I mentioned in the lecture, these are generally called $k$-wise independent sources but for us $k$ is already taken). We also saw how given an $\ell$-wise independent sources, we can come up with a $1-\frac{1}{2^{\ell}}$-approximation algorithm for the the MAXE$\ell$SAT problem. (In the problem, we are given a bunch of clauses where each clause has exactly $\ell$ literals. The goal is to come up with an assignment to the variables that satisfies as many clauses as possible.)

Next lecture, we will see that the dual of the $BCH_{2,\log{n},\ell+1}$ code is an $\ell$-wise independent source. By the bounds on the dimension of these codes, this means that we can get an $\ell$-wise independent source of size $O\left(n^{\lfloor \frac{\ell}{2}\rfloor}\right)$. Further, as these codes are linear codes, each codeword can be generated in time $O(n^2)$. This implies, that we have an $1-2^{-\ell}$-approximation algorithm for MAXE$\ell$SAT that runs in time $O\left(n^{2+\lfloor \frac{\ell}{2}\rfloor}\right)$.