Minimum distance of a code

The minimum distance of a Binary Code is the smallest Hamming distance between distinct codewords.

An Binary Code with minimum distance is sometimes called

Lemma

Let be a code with minimum distance

1. Then is -Error detecting, but cannot detect errors
2. is -Error correcting, but cannot correct all sets of errors

Proof

If and with then so errors detected
Suppose with . Then can be corrupted to with just errors, so this set of errors will not be detected.

Let , so
Let . If there is some with we want to show for all and
By triangle inequality this just works.
Suppose with . Let differ from in precisely places where and differ.
Then while so couldn’t correct these errors