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A certain family of [[BCH code]]s have a particularly useful property, which is that
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treated as [[linear operator]]s, their [[dual basis|dual operators]] turns their input into an <math>\ell</math>-wise [[pairwise independence|independent source]]. That is, the set of vectors from the input [[vector space]] are mapped to an <math>\ell</math>-wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a <math>1-2^{-\ell}</math>-approximation to [[MAXEkSAT]].


==Lemma==
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Let <math>C\subseteq F_2^n</math> be a linear code such that <math>C^\perp </math> has distance greater than <math> \ell +1</math>. Then <math>C</math> is an <math>\ell</math>-wise independent source.
 
==Proof of Lemma==
 
It is sufficient to show that given any <math>k \times l</math> matrix ''M'', where ''k'' is greater than or equal to ''l'', such that the rank of ''M'' is ''l'', for all <math>x\in F_2^k</math>, <math>xM</math> takes every value in <math>F_2^l</math> the same number of times.
 
Since ''M'' has rank ''l'', we can write ''M'' as two matrices of the same size, <math>M_1</math> and <math>M_2</math>, where <math>M_1</math> has rank equal to ''l''.  This means that <math>xM</math> can be rewritten as <math>x_1M_1 + x_2M_2</math> for some <math>x_1</math> and <math>x_2</math>.
 
If we consider ''M'' written with respect to a basis where the first ''l'' rows are the identity matrix, then <math>x_1</math> has zeros wherever <math>M_2</math> has nonzero rows, and <math>x_2</math> has zeros wherever <math>M_1</math> has nonzero rows.
 
Now any value ''y'', where <math>y=xM</math>, can be written as <math>x_1M_1+x_2M_2</math> for some vectors <math>x_1, x_2</math>.
 
We can rewrite this as:
 
<math>x_1M_1 = y - x_2M_2</math>
 
Fixing the value of the last <math>k-l</math> coordinates of
<math>x_2\in F_2^k</math> (note that there are exactly <math>2^{k-l}</math>
such choices), we can rewrite this equation again as:
 
<math>x_1M_1 = b</math> for some ''b''.
 
Since <math>M_1</math> has rank equal to ''l'', there
is exactly one solution <math>x_1</math>, so the total number of solutions is exactly <math>2^{k-l}</math>, proving the lemma.
 
==Corollary==
 
Recall that BCH<sub>2,''m'',''d''</sub> is an <math> [n=2^m, n-1 -\lceil {d-2}/2\rceil m, d]_2</math> linear code.
 
Let <math>C^\perp</math> be BCH<sub>2,log&nbsp;''n'',''ℓ''+1</sub>.  Then <math>C</math> is an <math>\ell</math>-wise independent source of size <math>O(n^{\lfloor \ell/2 \rfloor})</math>.
 
==Proof of Corollary==
 
The dimension ''d'' of ''C'' is just <math>\lceil{(\ell +1 -2)/{2}}\rceil \log n +1 </math>. So <math>d = \lceil {(\ell -1)}/2\rceil \log n +1 = \lfloor \ell/2 \rfloor \log n +1</math>.
 
So the cardinality of <math>C</math> considered as a set is just
<math> 2^{d}=O(n^{\lfloor \ell/2 \rfloor})</math>, proving the Corollary.
 
== References ==
[http://www.cse.buffalo.edu/~atri/courses/coding-theory/ Coding Theory notes at University at Buffalo]
 
[http://people.csail.mit.edu/madhu/FT01/course.html/ Coding Theory notes at MIT]
 
[[Category:Article proofs]]

Latest revision as of 01:49, 29 November 2014

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