Posted by: atri | March 21, 2009

## Lecture 24: BCH codes

In Friday’s lecture, we started by wondering if the GV bound is tight for binary codes? We looked at the special case of odd constant distance (i.e. $d=2t+1$). We revisited some of the bounds for the constant distance regime. In particular, we saw that the GV bound in this case states

$k\ge n-(2t)\log{n}.$

On the other hand, the Hamming bound states:

$k \le n-t\log{n}+O(1).$

For $t=1$, we had already seen the Hamming code, which meets the Hamming bound (and hence, beats the GV bound). In the lecture we studied the $BCH_{2,m,d}=[n=2^m,k_{BCH},d]_{\mathbb{F}_2}$ code, which is the subfield sub-code of the $[n,n-d+1,d]_{\mathbb{F}_{2^m}}$ RS code. We argued in the lecture that

$k_{BCH}\ge n-(2t)\log{n}.$

In fact, it can be show that

$k_{BCH}\ge n-1-t\log{n}.$

For the proof of the above see these lecture notes from Madhu Sudan‘s coding theory course (the argument we used to show the weaker lower bound on $k_{BCH}$ is also from the same notes).

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