  This paper studies second-order coding rates for memoryless channels with a
state sequence known non-causally at the encoder. In the case of finite
alphabets, an achievability result is obtained using constant-composition
random coding, and by using a small fraction of the block to transmit the type
of the state sequence. For error probabilities less than 1/2, it is shown that
the second-order rate improves on an existing one based on i.i.d. random
coding. In the Gaussian case (dirty paper coding) with an almost-sure power
constraint, an achievability result is obtained used using random coding over
the surface of a sphere, and using a small fraction of the block to transmit a
quantized description of the state power. It is shown that the second-order
asymptotics are identical to the single-user Gaussian channel of the same input
power without a state.
