Comparison and Optimization of Enumerative Coding Techniques for Amplitude Shaping

The energy efficiency of enumerative coding algorithms is investigated for amplitude shaping in the context of binary transmission. First, a simple method for the calculation of the amplitude distribution is derived for enumerative sphere shaping (ESS). For ultra-short blocklengths, ESS—which orders sequences lexicographically—is shown to be less energy-efficient than algorithms that use energy-based ordering such as shell mapping. Second, ESS is optimized heuristically such that its energy efficiency is improved. Simulations show that optimized ESS achieves the same error probabilities as energy-based ordering methods for the additive white Gaussian noise (AWGN) channel, even at ultra-short blocklengths