Lower Bound:

otherwise

For and , set

Returning to our original equality,

Putting everything together,

**Remarks:**

- If , the Plotkin bound can be improved (p. 59). If , then
- Asymptotic Plotkin Bound says:
Let
. Then,
ififThe proof, for is to apply the usual Plotkin Bound. For , shorten enough so that usual Plotkin applies.
- if is even. if is odd. if is odd. These can be shown by stretching the Plotkin Bound with the simple facts learned in the previous section.