  The problem of communication over binary dirty paper (DP) using nested polar
codes is considered. An improved scheme, focusing on low delay, short to
moderate blocklength communication is proposed. Successive cancellation list
(SCL) decoding with properly defined CRC is used for channel coding, and SCL
encoding without CRC is used for source coding. The performance is compared to
the best achievable rate of any coding scheme for binary DP using nested codes.
A well known problem with nested polar codes for binary DP is the existence of
frozen channel code bits that are not frozen in the source code. These bits
need to be retransmitted in a second phase of the scheme, thus reducing
transmission rate. We observe that the number of these bits is typically either
zero or a small number, and provide an improved analysis, compared to that
presented in the literature, on the size of this set and on its scaling with
respect to the blocklength when the power constraint parameter is sufficiently
large or the channel crossover probability sufficiently small.
