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 $$\ell$$-wise independent source. That is, the set of vectors from the input vector space are mapped to an $$\ell$$-wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a $$1-2^{-\ell}$$-approximation to MAXEkSAT.

Lemma
Let $$C\subseteq F_2^n$$ be a linear code such that $$C^\perp $$ has distance greater than $$ \ell +1$$. Then $$C$$ is an $$\ell$$-wise independent source.

Proof of lemma
It is sufficient to show that given any $$k \times l$$ matrix M, where k is greater than or equal to l, such that the rank of M is l, for all $$x\in F_2^k$$, $$xM$$ takes every value in $$F_2^l$$ the same number of times.

Since M has rank l, we can write M as two matrices of the same size, $$M_1$$ and $$M_2$$, where $$M_1$$ has rank equal to l. This means that $$xM$$ can be rewritten as $$x_1M_1 + x_2M_2$$ for some $$x_1$$ and $$x_2$$.

If we consider M written with respect to a basis where the first l rows are the identity matrix, then $$x_1$$ has zeros wherever $$M_2$$ has nonzero rows, and $$x_2$$ has zeros wherever $$M_1$$ has nonzero rows.

Now any value y, where $$y=xM$$, can be written as $$x_1M_1+x_2M_2$$ for some vectors $$x_1, x_2$$.

We can rewrite this as:

$$x_1M_1 = y - x_2M_2$$

Fixing the value of the last $$k-l$$ coordinates of $$x_2\in F_2^k$$ (note that there are exactly $$2^{k-l}$$ such choices), we can rewrite this equation again as:

$$x_1M_1 = b$$ for some b.

Since $$M_1$$ has rank equal to l, there is exactly one solution $$x_1$$, so the total number of solutions is exactly $$2^{k-l}$$, proving the lemma.

Corollary
Recall that BCH2,m,d is an $$ [n=2^m, n-1 -\lceil {d-2}/2\rceil m, d]_2$$ linear code.

Let $$C^\perp$$ be BCH2,log n,ℓ+1. Then $$C$$ is an $$\ell$$-wise independent source of size $$O(n^{\lfloor \ell/2 \rfloor})$$.

Proof of corollary
The dimension d of C is just $$\lceil{(\ell +1 -2)/{2}}\rceil \log n +1 $$. So $$d = \lceil {(\ell -1)}/2\rceil \log n +1 = \lfloor \ell/2 \rfloor \log n +1$$.

So the cardinality of $$C$$ considered as a set is just $$ 2^{d}=O(n^{\lfloor \ell/2 \rfloor})$$, proving the Corollary.