It is a legal document guaranteeing a natural right. The Bill of Rights doesn't codify much of anything, it lays out certain things further laws can't tread on without extraordinary permission granted.
It guarantees much more than a natural right, and does so with the backing power of a legal institution. For example, the UDHR doesn't even mention the word "warrant", and it doesn't do anything to compel enforcement.
The BoR is, by plain definition, a codification of rights. To codify means to make into law, which is what it is: a law, implementing the formal recognition rights... and it does so by making illegal specific abuses.
