发表机构
Northeastern University; Massachusetts Institute of Technology（东北大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究列表解码中似然排名与解码正确性后验概率的关系，证明后续列表条目携带额外可靠性信息，并刻画了首个条目的渐近可靠性仅取决于第一、第二及最后排名。
AI 中文摘要
在纠错中，码字与接收词之间的差异，即候选噪声序列，携带了关于该码字作为提议解码的置信度信息。对于加性信道，已知若噪声序列按似然递减排序，则较低排名与较高的解码置信度相关。在此，我们通过探索列表解码中出现的似然排名与解码正确性的后验概率之间的关系，大幅扩展了考虑范围，确立了与后续列表条目相关的排名携带了仅凭第一个排名无法获得的可靠性信息。基于软输出猜测随机加性噪声解码的最新进展，我们首先证明，在随机码本模型下，列表解码的软输出表达式与解码正确性的精确有限块长条件概率一致。对于硬判决最大似然解码，我们刻画了第一个列表条目的渐近可靠性。我们确定了一个决策函数，其符号决定列表中第一个条目的后验正确概率是收敛到一还是零，其绝对值给出收敛的指数速率。或许令人惊讶的是，第一个列表条目的渐近可靠性仅依赖于第一、第二和最后一个排名。对于列表大小为1的情况，该结果恢复了先前已知的可靠解码的似然排名阈值。对于列表大小为2的情况，足够迟到的第二个条目可以意味着第一个条目渐近正确，即使其排名单独看来会指示相反的结果。
英文摘要
In error correction, the difference between a codeword and a received word, a candidate noise sequence, carries information on the confidence of the codeword as a proposed decoding. For additive channels, it is known that if noise sequences are ranked in decreasing order of likelihood, a lower rank correlates with higher confidence in a decoding. Here we substantially expand considerations by exploring the relationship between the likelihood ranks arising in list decoding and the posterior probability of decoding correctness, establishing that the ranks associated with subsequent list entries carry reliability information that is not obtainable from the first rank alone. Building on the recent development of Soft-Output Guessing Random Additive Noise Decoding, we first show that, under a random codebook model, soft-output expressions for list decoding coincide with the exact finite-blocklength conditional probabilities of decoding correctness. For hard-decision maximum likelihood decoding, we characterize the asymptotic reliability of the first list entry. We identify a decision function whose sign determines whether the posterior correctness probability of the first entry in the list converges to one or zero and whose absolute value gives the exponential rate of convergence. Perhaps surprisingly, the asymptotic reliability of the first list entry depends only on the first, second and last ranks. For list size one, the result recovers the previously known likelihood-rank threshold for reliable decoding. For list size two, a sufficiently late second entry can imply that the first entry is asymptotically correct even when its rank, viewed in isolation, would indicate otherwise.
