  In this paper, we develop a new channel model, which we name the $q$-ary
partial erasure channel (QPEC). QPEC has a $q$-ary input, and its output is
either one symbol or a set of $M$ possible values. This channel mimics
situations when current/voltage levels in measurement channels are only
partially known, due to high read rates or imperfect current/voltage sensing.
Our investigation is concentrated on the performance of low-density
parity-pheck (LDPC) codes when used over this channel, due to their low
decoding complexity with iterative-decoding algorithms. We give the density
evolution equations of this channel, and develop its decoding-threshold
analysis. Part of the analysis shows that finding the exact decoding threshold
efficiently lies upon a solution to an open problem in additive combinatorics.
For this part we give bounds and approximations.
