Posted by: atri | March 27, 2009

Lecture 27: Code Concatenation

In today’s lecture, we defined the code composition method of code concatenation. By picking the outer code to be on the Singleton bound and the inner code to be on the GV bound, we obtained the Zyablov bound. That is, for relative distance \delta>0, we can obtain a rate of at least

\displaystyle\max_{0<r<1-H_q(\delta+\epsilon)}r\left(1-\frac{\delta}{H_q^{-1}(1-r)-\epsilon}\right),

where \epsilon>0. In the class today I presented the formula with \epsilon=0. However, the form above is more accurate since the Varshamov bound achieves a rate of 1-H_q(\delta)-\gamma for any \gamma>0.

Further, we presented a polynomial time construction of codes that achieves the Zyablov bound: the outer code is the strongly explicit RS code while the inner code on the GV bound can be constructed in (singly) exponential time. (The latter result was problem 4 in the homework.)

The material covered in today’s lecture can be found in the scribed notes of Lecture 24 from Fall 07.

Next lecture, we will look at a strongly explicit construction of an asymptotically good binary code due to Justesen.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Categories

%d bloggers like this: