Since the birth of digital communications, it has been known that the optimal signal distribution for an additive white Gaussian noise channel is not uniform. There are two primary approaches to generate the non-uniform distribution for digital communications systems—geometric shaping, whereby equiprobable constellation points are arranged in a non-uniform manner in order to maximize performance; and probabilistic shaping, whereby the probabilities of constellation points are optimized in order to maximize performance. While it is generally accepted that the performance of probabilistic shaping is superior to the performance of geometric shaping for equal cardinality, methods for mapping sequences of uniformly distributed bits of information (such as those which we would like to transmit) onto sequences of non-equiprobable symbols has proven extremely challenging. The most commonly used method is that of constant composition distribution matching (CCDM), which maps equiprobable bits onto a sequence which is a permutation of the “typical sequence,” which has the desired symbol probability mass function (PMF). While this method can achieve good performance (achieving arbitrarily low rate loss for asymptotically long symbol sequences), it has two critical flaws: the ability to achieve low rate-loss requires very long sequences, which causes high complexity and latency; and the only known efficient mapping and de-mapping algorithms are sequential in symbols (that is, need to decode each symbol in-turn in a symbol sequence), which also leads to prohibitively high complexity and latency. Some of those problems are addressed in U.S. Pat. No. 10,069,519.