Divergence-Optimal Fixed-to-Fixed Length Distribution Matching With Shell Mapping

Distribution matching (DM) transforms independent and Bernoulli(1/2) distributed bits into a sequence of output symbols with a desired distribution. A fixed-to-fixed length, invertible DM architecture based on shell mapping (SM) is presented. It is shown that SM for DM (SMDM) is the optimum DM for the informational divergence metric and that finding energy optimal sequences is a special case of divergence minimization. Additionally, it is shown how to find the required SM weight function to approximate arbitrary output distributions. SMDM is combined with probabilistic amplitude shaping to operate close to the Shannon limit. SMDM exhibits excellent performance for short blocklengths as required by ultra-reliable low-latency applications. SMDM outperforms constant composition DM by 0.7 dB when used with 64-QAM at a spectral efficiency of 3 bits/channel use and a 5G low-density parity-check code with a short blocklength of 192 bits.