In this paper, we construct variable-length prefix-free codes that are optimal (or near-optimal) in the sense that no (or few) other codes of the same cardinality can achieve a smaller expected energy per code symbol for the same resolution rate. Under stringent constraints of 4096 codewords or below per codebook, the constructed codes yield an energy per code symbol within a few tenths of a dB of the unconstrained theoretic lower bound, across a wide range of resolution rates with fine granularity. We also propose a framing method that allows variable-length codes to be transmitted using a fixed-length frame. The penalty caused by framing is studied using simulations and analysis, showing that the energy per code symbol is kept within 0.3 dB of the unconstrained theoretic limit for some tested codes with a large frame length. When the proposed method is used to implement probabilistic constellation shaping for communications in the additive white Gaussian noise channel, simulations show that between 0.21 dB and 0.98 dB of shaping gains are achieved relative to uniform 4-, 8-, 16-, and 32-ary quadrature amplitude modulation.

中文翻译：

---

用于概率星座整形的无前缀代码分布匹配

在本文中，我们构造了最优（或接近最优）的可变长度无前缀码，因为没有（或很少）具有相同基数的其他代码可以在相同分辨率下实现每个代码符号的较小预期能量速度。在每个码本 4096 个码字或以下的严格约束下，构建的码在无约束理论下界的十分之几 dB 范围内产生每个码元的能量，跨越宽范围的细粒度分辨率。我们还提出了一种成帧方法，允许使用固定长度的帧传输可变长度的代码。使用模拟和分析研究了由成帧引起的损失，表明对于一些具有大帧长度的测试代码，每个代码符号的能量保持在无约束理论限制的 0.3 dB 以内。