The Performance of Coded Modulation with Gallager Mapping in the Finite Length Regime

This paper investigates the capacity of coded modulation (CM) schemes with Gallager mapping based constellations in the finite length regime. We first present the channel capacity for M-ary modulations with a finite block-length. Then, we discuss the performance of Gallager mapping based constellation with probabilistic shaping. We show that at an asymptotic high signal-to-noise-ratio (SNR), the channel capacity of equiprobable constellations with a finite block-length can reach the capacity with infinite block-length while the channel capacity of nonequiprobable constellations can not approach the capacity in the infinite length regime. Finally, we discuss the channel capacity loss of non-equiprobable constellations in the finite length regime. At an asymptotic high SNR, the channel dispersion of nonequiprobable constellations equals to a positive constant, which causes the channel capacity loss for non-equiprobable constellations. We also derive an upper bound and a tight approximation of the asymptotic capacity loss (ACL) of the constellations in the finite length regime.