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In this work, we construct energy-efficient variable-to-fixed length (V2F),fixed-to-variable length (F2V), and variable-to-variable length (V2V)prefix-free codes, which are optimal (or near-optimal) in the sense that no (orfew) other codes with the size can achieve a smaller energy per code letter forthe same entropy rate. Under stringent constraints of 4096 entries or below percodebook, the constructed codes yield an energy per code letter within a fewtenths of a dB of the unconstrained theoretic lower bound, across a wide rangeof entropy rates with a very fine granularity. We also propose a framing methodthat allows variable-length codes to be transmitted using a fixed-length frame.The penalty caused by framing is studied using simulations and analysis,showing that the energy per code letter is kept within 0.2 dB of theunconstrained theoretic limit for some tested codes with a large frame length.When framed prefix-free codes are used to implement probabilistic constellationshaping (PCS) for communications in the additive white Gaussian noise channel,simulations show that 1.1 dB and 0.65 dB of shaping gains are achieved relativeto uniform 8- and 16-quadrature amplitude modulation (QAM), respectively.

Prefix-Free Code Distribution Matching for Probabilistic Constellation Shaping