Within the Guessing Random Additive Noise Decoding (GRAND) family, ordered reliability bits GRAND (ORBGRAND) has received considerable attention for its hardware-friendly exploitation of soft information. Existing information-theoretic results for ORBGRAND are asymptotic in blocklength and do not quantify its performance at short-to-moderate blocklengths. This paper develops a finite-blocklength analysis for ORBGRAND over general bit channel, addressing the key challenge that the rank-induced decoding metric is non-additive and coupled across symbols. We first derive an ORBGRAND-specific random-coding union (RCU)-type achievability (ORB-RCU) bound on the ensemble-average error probability. We then characterize two governing decoding metrics: the transmitted-codeword metric is treated as a U-statistic and analyzed via Hoeffding decomposition, while the competing-codeword metric is reduced to a weighted sum of independent and identically distributed Bernoulli random variables and analyzed through strong large-deviation analysis. Combining these ingredients with a Berry-Esseen argument yields a second-order achievable-rate expansion and the associated normal approximation, whose first-order term is shown to equal the ORBGRAND generalized mutual information and whose second-order term defines an ORBGRAND dispersion with a single-letter variance representation. Numerical results for BPSK-modulated additive white Gaussian noise channel validate the tightness of ORB-RCU relative to the maximum-likelihood based RCU benchmark and the accuracy of the normal approximation in the operating regime of practical interest.

翻译：在猜测随机加性噪声解码（GRAND）算法族中，有序可靠性比特GRAND（ORBGRAND）因其对软信息的硬件友好型利用而受到广泛关注。现有关于ORBGRAND的信息论结果均基于分组长度渐近假设，未能量化其在短至中等分组长度下的实际性能。本文针对通用比特信道建立了ORBGRAND的有限分组长度分析框架，其核心挑战在于由排序机制导出的解码度量具有非加性且符号间耦合的特性。我们首先推导出适用于ORBGRAND的随机编码联合（RCU）型可达性界（ORB-RCU），该界限定了系统平均错误概率的上界。随后，我们刻画了两个关键的解码度量：将发送码字度量建模为U统计量并通过霍夫丁分解进行分析，而将竞争码字度量简化为独立同分布伯努利随机变量的加权和，并采用强偏差分析技术进行研究。结合贝里-埃森逼近论证，我们推导出二阶可达速率展开式及其对应的正态近似，其中一阶项被证明等于ORBGRAND广义互信息，二阶项则定义了具有单字母方差表示的ORBGRAND色散系数。针对BPSK调制加性高斯白噪声信道的数值仿真结果表明：ORB-RCU界相对于基于最大似然的RCU基准具有紧致性，且正态近似在实际工作区间内具有较高的精确度。