Before revision:  The deletion channel is a noise model where each bit of the string is independently deleted with some fixed probability, and the retained bits are concatenated. We note that if \ell = 1, i.e., we are trying to learn a single string , learning the distribution is equivalent to the famous trace reconstruction problem.
After revision:  This problem generalizes the famous trace reconstruction problem, where one wishes to learn a single string , learning the distribution is equivalent to the famous trace reconstruction problem.