In the torn paper channel, a transmitted codeword is broken at random locations into fragments that arrive at the decoder in an unordered manner. A central theoretical challenge within this model is global alignment -- the task of determining each fragment's original position -- in order to faithfully reconstruct the entire codeword. Prior work by Shomorony and Vahid introduced an interleaved-pilot scheme that successfully achieved a vanishing error probability. However, their alignment strategy relies heavily on global statistics, requiring fragments to exceed a minimum length and effectively discarding many shorter ones as erasures, which results in rates significantly below capacity. To address this gap, we propose an improved coding scheme that achieves a provable rate increase through a novel approach we call \textit{local alignment}. This approach identifies global alignment bits within each fragment using only local information, allowing the decoder to determine the positions of fragments that are shorter than those used in previous work. Consequently, the decoder can extract information from a much larger fraction of the channel output than in previous work, yielding significantly higher rates. Furthermore, we extend our analysis to torn paper coding with lost pieces (TPC-LP), a generalized model that accounts for length-dependent fragment deletion. For a class of TPC-LP channels that delete all fragments below a logarithmic length threshold while allowing arbitrary length-dependent deletion probabilities for longer fragments, we show that the proposed local alignment strategy achieves an arbitrarily small additive gap to capacity as the threshold increases.

翻译：在碎纸信道中，传输的码字在随机位置被撕成碎片，并以无序方式到达译码器。该模型中的一个核心理论挑战是全局对齐——即确定每个碎片的原始位置以忠实重构整个码字的任务。Shomorony 和 Vahid 的先前工作引入了一种交错导频方案，成功实现了可忽略的误差概率。然而，其对齐策略严重依赖全局统计信息，要求碎片超过最小长度，并将许多较短碎片视为擦除而有效丢弃，这导致速率显著低于信道容量。为弥补这一不足，我们提出了一种改进的编码方案，通过一种称为局部对齐的新方法实现可证明的速率提升。该方法仅利用局部信息识别每个碎片内的全局对齐比特，使译码器能够确定比先前工作中所用碎片更短的碎片位置。因此，与先前工作相比，译码器可从信道输出的更大比例中提取信息，从而获得显著更高的速率。此外，我们将分析扩展到带丢失碎片的碎纸编码（TPC-LP），这是一种考虑长度依赖碎片删除的广义模型。对于一类 TPC-LP 信道——该类信道删除所有低于对数长度阈值的碎片，同时允许较长碎片具有任意长度依赖的删除概率——我们证明，随着阈值增加，所提出的局部对齐策略可实现与信道容量之间任意小的加性差距。