We investigate the fundamental limits of the unsourced random access over the binary-input Gaussian channel. By fundamental limits, we mean the minimal energy per bit required to achieve the target per-user probability of error. The original method proposed by Y. Polyanskiy (2017) and based on Gallager's trick does not work well for binary signaling. We utilize Fano's method, which is based on the choice of the so-called ``good'' region. We apply this method for the cases of Gaussian and binary codebooks and obtain two achievability bounds. The first bound is very close to Polyanskiy's bound but does not lead to any improvement. At the same time, the numerical results show that the bound for the binary case practically coincides with the bound for the Gaussian codebook. Thus, we conclude that binary modulation does not lead to performance degradation, and energy-efficient schemes with binary modulation do exist.

翻译：我们研究了二进制输入高斯信道上无源随机接入的基本极限。所谓基本极限，是指在满足每用户目标错误概率条件下所需的最小单位比特能量。Y. Polyanskiy（2017）提出的基于Gallager技巧的原始方法不适用于二进制信号传输。我们采用了基于所谓"好"区域选择的Fano方法，将其分别应用于高斯码本和二进制码本的情形，得到了两个可达性界。第一个界与Polyanskiy界非常接近但未带来任何改进。同时，数值结果表明二进制码本的界与高斯码本的界实际上完全一致。因此我们得出结论：二进制调制不会导致性能退化，且存在采用二进制调制的节能方案。