发表机构
Washington University in St Louis（华盛顿大学圣路易斯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对撕纸信道现有译码方案的局限，提出多级导频回收的撕纸码构造，以随机线性码实现该信道容量，降低译码复杂度并支持更短片段对齐
AI 中文摘要
在撕纸信道中，码字会在随机位置被切割，产生的无差错片段以无序多重集的形式传递给译码器。尽管该信道的容量可通过随机码实现，但这类码的译码通常需要指数时间。Shomorony和Vahid提出的导频交织构造方法，将De Bruijn序列嵌入到移位删除码的符号中，仅通过全局统计唯一性对齐足够长的片段。Liu和Raviv后续提出的局部对齐方案，采用游程受限约束和专用全零标记从局部结构识别导频位置，大幅降低了可对齐的最小片段长度。我们通过将其固定导频序列替换为多级连续局部对齐和导频回收流程，进一步改进了Liu和Raviv的方法。在Liu和Raviv的方案中，导频序列仅选择一次，必须同时平衡导频序列长度与短片段对齐能力；我们的构造通过在连续译码级间复用导频序列，消除了这一限制。首先使用带独立随机线性码的导频序列对齐并译码最长片段，该阶段恢复的信息随后作为更大的导频序列用于下一级，该过程在多个级上重复，从而在导频序列长度保持较小时，恢复越来越短的片段。我们的构造依赖于选择一系列随机线性码，证明对于任意ε>0，存在这样的码选择，当块长趋于无穷时，以高概率达到比容量低至少ε的速率，因此我们的构造实现了撕纸信道的容量。
英文摘要
In the torn paper channel, a codeword is cut at random locations, and the resulting error-free fragments are delivered to the decoder as an unordered multiset. Although the capacity of this channel can be achieved using random code, decoding such codes generally requires exponential time. The interleaved-pilot construction of Shomorony and Vahid embeds a De Bruijn sequence among the symbols of a shifted erasure code and aligns only fragments that are sufficiently long through global statistical uniqueness. A subsequent local-alignment scheme by Liu and Raviv employs run-length-limited constraints and exclusive all-zero markers to identify pilot positions from local structure, substantially reducing the minimum fragment length that can be aligned. We further improve the method of Liu and Raviv by replacing its fixed pilot sequence with a multilevel successive local alignment and pilot-recycling procedure. In Liu and Raviv, the pilot sequence is chosen once and must simultaneously balance the length of the pilot sequence against the ability to align short fragments. Our construction removes this limitation by reusing pilot sequences across successive decoding levels. A pilot sequence with an independent random linear code is first used to align and decode the longest fragments. The information recovered at this stage is then recycled as a larger pilot sequence for the next stage.This process is repeated over multiple levels, so that progressively shorter fragments are recovered while the length of the pilot sequence remains small. Our construction depends on choosing a series of random linear codes, and we show that for any $\varepsilon>0$, there exists a choice of such codes which attains rate of at least $\varepsilon$ below the capacity, with high probability as the block length goes to infinity. Therefore, our construction achieves the capacity of the torn-paper channel.
