Emerging applications in manufacturing, wireless communication, and molecular data storage require robust coding schemes that remain effective under physical distortions where codewords may be arbitrarily fragmented and partially missing. To address such challenges, we propose a new family of error-correcting codes, termed $(t,s)$-break-resilient codes ($(t,s)$-BRCs). A $(t,s)$-BRC guarantees correct decoding of the original message even after up to~$t$ arbitrary breaks of the codeword and the complete loss of some fragments whose total length is at most~$s$. This model unifies and generalizes previous approaches, extending break-resilient codes (which handle arbitrary fragmentation without fragment loss) and deletion codes (which correct bit losses in unknown positions without fragmentation) into a single information-theoretic framework. We develop a theoretical foundation for $(t,s)$-BRCs, including a formal adversarial channel model, lower bounds on the necessary redundancy, and explicit code constructions that approach these bounds.

翻译：在制造、无线通信和分子数据存储等新兴应用中，需要能够在物理畸变下保持有效的鲁棒编码方案，此类畸变可能导致码字被任意分段且部分缺失。为应对此类挑战，我们提出了一类新的纠错码，称为$(t,s)$-抗断裂编码（$(t,s)$-BRC）。$(t,s)$-BRC能够保证即使在码字发生最多$t$次任意断裂，且部分片段完全丢失（其总长度不超过$s$）的情况下，仍能正确解码原始消息。该模型统一并推广了先前的方法，将抗断裂编码（处理无片段丢失的任意分段）与删除码（纠正未知位置上的比特丢失而无分段）纳入统一的信息论框架。我们为$(t,s)$-BRC建立了理论基础，包括形式化的对抗信道模型、冗余度的必要下界，以及逼近这些下界的显式编码构造。