Size of Largest Code

is the largest possible size of a Binary Code.

Lemma

Proof

Let and let be a code.
Let be two distinct codewords such that .
Let differ from in exactly one of the places where and differ.
Then , and for any we have

thus , so by replacing with we get a code

Proposition

Proof

Hamming’s Bound
Gilbert-Shannon-Varshamov Bound

Example

discovered in 1999.