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}.



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