  Write-Once-Memory (WOM) is a model for many modern non-volatile memories,
such as flash memories. Recently, several capacity-achieving WOM coding schemes
have been proposed based on polar coding. Due to the fact that practical
computer memory systems always contain noises, a nature question to ask next is
how may we generalize these coding schemes, such that they may also have the
error-control capabilities. In this paper, we discuss a joint WOM and
error-control coding scheme, which is a generalization of the
capacity-achieving WOM codes based on source polarization. In this paper, we
prove a sufficient and necessary condition for the noisy reading channel being
less noisy than the test channel in data encoding in the polar WOM coding. Such
a sufficient and necessary condition is usually satisfied in reality. As a
consequence of the sufficient and necessary condition, the high entropy set
related to the noisy channel is usually strictly contained in the high entropy
set related to the test channel in data encoding. Therefore the low-complexity
polar joint WOM and error-control codes are sufficient for most practical
coding scenarios.
