Before the formal definition is given, let us go straight to an example.

*``What?''*

*``Hi Bob'' (louder).*

*``What?''*

*``Hi Bob'' (even louder).*

*``Oh, hi.''*

*This illustrates a ``repetition code''. More precisely, the -ary repetition code of length is the set of all -tuples of the form , for . (We leave it as an exercise to verify that this is a vector space over .) We think of as representing information you want to send. It could be the ``greyness'' of a pixel in a picture or a letter (represented in ASCII code) in a word, for example. Since the channel might contain noise, we send instead, with the understanding that the receiver should perform a ``majority vote'' to decode the vector. (For example, if was received then 0 ``wins the vote''). *

This wasn't a very efficient example. Let's try again.

*First, write the number in binary, 0 as , as , ..., as . If for , then define*

*The problem with this example boils down to the fact that the first and second column of are the same. We shall explain this type of deficency more later.*

David Joyner 2007-09-03