This paper investigates guesswork over ordered statistics and formulates the complexity of ordered statistics decoding (OSD) in binary additive white Gaussian noise (AWGN) channels. It first develops a new upper bound of guesswork for independent sequences, by applying the Holder's inequity to Hamming shell-based subspaces. This upper bound is then extended to the ordered statistics, by constructing the conditionally independent sequences within the ordered statistics sequences. We leverage the established bounds to formulate the best achievable decoding complexity of OSD that ensures no loss in error performance, where OSD stops immediately when the correct codeword estimate is found. We show that the average complexity of OSD at maximum decoding order can be accurately approximated by the modified Bessel function, which increases near-exponentially with code dimension. We also identify a complexity saturation threshold, where increasing the OSD decoding order beyond this threshold improves error performance without further raising decoding complexity. Finally, the paper presents insights on applying these findings to enhance the efficiency of practical decoder implementations.

翻译：本文研究了有序统计上的猜测行为，并系统阐述了二进制加性高斯白噪声（AWGN）信道中有序统计解码（OSD）的复杂度。首先，通过将霍尔德不等式应用于汉明壳子空间，推导出独立序列猜测的一个新上界。随后，通过在有序统计序列内构建条件独立序列，将该上界扩展到有序统计场景。利用所建立的边界，本文公式化了OSD在不损失译码性能前提下可实现的最优解码复杂度，即当找到正确码字估计时立即停止解码。研究表明，最大解码阶数下OSD的平均复杂度可由修正贝塞尔函数精确近似，该函数随码长维度呈近指数增长。同时，本文识别出一个复杂度饱和阈值：当OSD解码阶数超过该阈值时，进一步提高阶数可在不增加解码复杂度的前提下改善译码性能。最后，本文为提升实际解码器实现效率提供了基于这些发现的应用洞见。