Posted by: atri | January 21, 2009

## Lecture 4: Hamming bound and probability basics

In today’s lecture, we proved the Hamming bound. That is,  any $q$-ary code of block length $n$ and distance $d$ has dimension $k\le n-\log_q\left(\sum_{i=0}^{\lfloor (d-1)/2\rfloor} \binom{n}{i}(q-1)^i\right)$.  For more details see these lecture notes.

Codes that satisfy this bound are called perfect  codes. Perfect binary codes have been exactly characterized. See  from Fall 07 for more details.

At the end of the lecture, we did a quick review of ideas/tools from (discrete) probability theory that we will use in this course.