  BATS codes were proposed for communication through networks with packet loss.
A BATS code consists of an outer code and an inner code. The outer code is a
matrix generation of a fountain code, which works with the inner code that
comprises random linear coding at the intermediate network nodes. In this
paper, the performance of finite-length BATS codes is analyzed with respect to
both belief propagation (BP) decoding and inactivation decoding. Our results
enable us to evaluate efficiently the finite-length performance in terms of the
number of batches used for decoding ranging from 1 to a given maximum number,
and provide new insights on the decoding performance. Specifically, for a fixed
number of input symbols and a range of the number of batches used for decoding,
we obtain recursive formulae to calculate respectively the stopping time
distribution of BP decoding and the inactivation probability in inactivation
decoding. We also find that both the failure probability of BP decoding and the
expected number of inactivations in inactivation decoding can be expressed in a
power-sum form where the number of batches appears only as the exponent. This
power-sum expression reveals clearly how the decoding failure probability and
the expected number of inactivation decrease with the number of batches. When
the number of batches used for decoding follows a Poisson distribution, we
further derive recursive formulae with potentially lower computational
complexity for both decoding algorithms. For the BP decoder that consumes
batches one by one, three formulae are provided to characterize the expected
number of consumed batches until all the input symbols are decoded.
