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Lett."],"published-print":{"date-parts":[[2026,7,1]]},"abstract":"<jats:p>This Letter proposes a hierarchical mapping distribution matching (HMDM) scheme for probabilistic shaping (PS) systems. Using a lowest-energy first scheduling (LEFS) approach, the proposed scheme achieves superior energy efficiency over enumerative sphere shaping (ESS) without increasing complexity. A trellis construction approach is also proposed to support LEFS. Experimental and simulation results demonstrate that, under short blocklengths, HMDM approximates the Maxwell\u2013Boltzmann distribution more closely than ESS, achieving superior performance over both ESS and uniform QAM. 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