  In this paper, we study the finite blocklength limits of state-dependent
discrete memoryless channels where the discrete memoryless state is known
noncausally at the encoder. For the point-to-point case, this is known as the
Gel'fand-Pinsker channel model. We define the (n,\epsilon)-capacity of the
Gel'fand-Pinsker channel as the maximal rate of transmission of a message
subject to the condition that the length of the block-code is n and the average
error probability is no larger than \epsilon. This paper provides a lower bound
for the (n,\epsilon)-capacity of the Gel'fand-Pinsker channel model, and hence
an upper bound on the dispersion, a fundamental second-order quantity in the
study of the performance limits of discrete memoryless channels. In addition,
we extend the work of Y. Steinberg (2005), in which the (degraded) broadcast
channel extension of the Gel'fand-Pinsker model was studied. We provide and
inner bound to the (n,\epsilon)-capacity region for this broadcast channel
model using a combination of ideas of Gel'fand-Pinsker coding, superposition
coding and dispersion (finite blocklength) analysis.
