  Fountain codes are rateless erasure-correcting codes, i.e., an essentially
infinite stream of encoded packets can be generated from a finite set of data
packets. Several fountain codes have been proposed recently to minimize
overhead, many of which involve modifications of the Luby transform (LT) code.
These fountain codes, like the LT code, have the implicit assumption that the
probability distribution is fixed throughout the encoding process. In this
paper, we will use the theory of posets to show that this assumption is
unnecessary, and by dropping it, we can achieve overhead reduction by as much
as 64% lower than LT codes. We also present the fundamental theory of
probability distribution designs for fountain codes with non-constant
probability distributions that minimize overhead.
