When we put "a" in front of "notice", we tend to mean the notice is written on a piece of paper, or something in the same vein. You will normally use "notice" without an article: to give notice, to receive late notice, to be put on notice, at such late notice... in all cases "notice" is a noun, and is the general act of notifying a person.

Noun. But are you sure these two sentences are right ?