On Optimization and Analysis of Enumerative Sphere Shaping for Short Blocklengths

Enumerative sphere shaping (ESS) has drawn considerable attention in recent years, yielding a practical solution for the application of probabilistic shaping in the short-blocklength regime. However, ESS suffers from some energy-efficiency degradation for binary transmission, because it orders sequences lexicographically. In this work, we propose the optimum ESS (OESS), as an optimized version of ESS, which can always achieve the lowest average energy at any blocklength. We conduct a comprehensive study on ESS and OESS in terms of energy efficiency, distribution, rate loss and practical performance. Moreover, we propose to use the reverse trellis to obtain the sequence energy distribution of ESS and OESS. Numerical analysis shows that OESS leads to better energy efficiency, as well as more Maxwell-Boltzmann-like distributions compared with ESS, yielding considerably lower rate loss. The effectiveness of OESS is verified by the Monte-Carlo simulation in the linear additive white Gaussian noise channel, and an experimental transmission in a single sideband discrete-Fourier-transform spread discrete multi-tone system. Compared with conventional constant composition distribution matcher, both ESS and OESS performs remarkably better for the short blocklengths. Moreover, OESS achieves observable gains over ESS for ultrashort blocklengths of 20 and 40.