On the floor and the ceiling of a divisor

Given a divisor A of a function field, there is a unique divisor of minimum degree that defines the same vector space of rational functions as A and there is a unique divisor of maximum degree that defines the same vector space of rational differentials as A. These divisors are called the floor and the ceiling of A. A method is given for finding both the floor and the ceiling of a divisor. The floor and the ceiling of a divisor give new bounds for the minimum distance of algebraic geometry codes. The floor and the ceiling of a divisor supported by collinear places of the Hermitian function field are determined. Finally, we find the exact code parameters for a large class of algebraic geometry codes constructed from the Hermitian function field.