We study one-shot detection under isotropic multivariate Cauchy noise using finite constellations, with emphasis on the geometric mechanisms governing symbol-level reliability. Under isotropic Cauchy noise, the maximum-likelihood rule induces the same Euclidean Voronoi decision regions as in the Gaussian case, so the distinction lies not in the decision geometry itself but in how probability mass is distributed over these fixed regions. In the small-noise regime, we derive a reciprocal distance-spectrum upper bound for the symbol error probability (SEP), showing that this bound, and the associated reliability descriptor, retain a longer-range dependence on the global constellation geometry than under additive white Gaussian noise. In the large-noise regime, we prove that the correct-decision probability converges to a limit determined solely by the angular measure of the associated Voronoi recession cone. These results formalize a regime-dependent transition from bound-based distance descriptors to angle-based reliability descriptors under heavy-tailed noise. Beyond asymptotic characterization, we show that these descriptors also admit a lightweight design interpretation for planar constellations under a common average power budget. The theory is further illustrated through an asymmetric four-point example exhibiting geometric collapse, a standard four-point Quadrature Amplitude Modulation (4QAM) sanity check, and finite-$γ$ numerical validation for both asymptotic regimes, together with descriptor-guided design comparisons that reveal collapse avoidance and reciprocal-distance burden as practically meaningful screening criteria.

翻译：摘要：本文研究了在各向同性多元柯西噪声下，利用有限星座进行的单次检测，重点关注控制符号级可靠性的几何机制。在各向同性柯西噪声下，最大似然准则与高斯情形诱导出相同的欧几里得Voronoi决策区域，因此差异不在于决策几何本身，而在于概率质量在这些固定区域上的分布方式。在小噪声条件下，我们推导了符号错误概率（SEP）的倒数距离谱上界，表明该上界及其关联可靠性描述符对全局星座几何的依赖范围比加性高斯白噪声情形更长。在大噪声条件下，我们证明正确决策概率收敛于一个仅由关联Voronoi退缩锥的角测度决定的极限。这些结果形式化了重尾噪声下从基于距离描述符到基于角度可靠性描述符的、依赖于信噪比区间的转变。除渐近特性刻画外，我们进一步表明这些描述符在常见平均功率预算下对平面星座具有轻量级设计解释。该理论通过一个展现几何坍缩的非对称四点示例、标准四点正交幅度调制（4QAM）的合理性检验，以及两种渐近区间的有限γ数值验证加以阐明，同时结合描述符引导的设计比较，揭示了坍缩避免与倒数距离负担作为具有实际意义的筛选准则。