Dual of BCH is an independent source
A certain family of BCH codes have a particularly useful property, which is that treated as linear operators, their dual operators turns their input into an -wise independent source. That is, the set of vectors from the input vector space are mapped to an -wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a -approximation to MAXEkSAT.
Let be a linear code such that has distance greater than . Then is an -wise independent source.
Proof of Lemma
It is sufficient to show that given any matrix M, where k is greater than or equal to l, such that the rank of M is l, for all , takes every value in the same number of times.
Since M has rank l, we can write M as two matrices of the same size, and , where has rank equal to l. This means that can be rewritten as for some and .
If we consider M written with respect to a basis where the first l rows are the identity matrix, then has zeros wherever has nonzero rows, and has zeros wherever has nonzero rows.
Now any value y, where , can be written as for some vectors .
We can rewrite this as:
Fixing the value of the last coordinates of (note that there are exactly such choices), we can rewrite this equation again as:
for some b.
Since has rank equal to l, there is exactly one solution , so the total number of solutions is exactly , proving the lemma.
Recall that BCH2,m,d is an linear code.
Let be BCH2,log n,ℓ+1. Then is an -wise independent source of size .
Proof of Corollary
The dimension d of C is just . So .
So the cardinality of considered as a set is just , proving the Corollary.