函数校正符号对码中的数据保护：冗余界与保护轮廓
Data Protection in Function-Correcting Symbol-Pair Codes: Redundancy Bounds and Protection Profiles
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- Indian Institute of Technology (ISM), Dhanbad（印度理工学院（ISM），丹巴德）
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中文总结 AI 辅助
针对符号对错误场景，提出带数据保护的函数校正符号对码，推导冗余界、给出构造，并引入不变量刻画保护性质，推广经典界。
中文摘要 AI 辅助
在包括DNA存储和闪存在内的若干存储系统中，错误会联合影响相邻符号，而汉明度量无法充分刻画此类错误模式。由Cassuto和Blaum提出的符号对读取信道通过读取连续符号对而非单个符号来解决这一问题。受此启发，我们引入了带数据保护的函数校正符号对码（FCSPC-DP），该码在保护消息本身免受符号对错误影响的同时，保证所需消息函数的可靠恢复。我们推导了此类码最优冗余的界，并建立了其与联合对距离矩阵的关系。我们还针对局部对有界函数和符号对权重函数给出了FCSPC-DP的显式构造。我们引入了函数的对分离常数，即共享同一函数值的消息之间的最小符号对距离，并证明当该常数足够大时，数据保护不需要额外冗余：最优冗余与不带数据保护的对应码的最优冗余一致。考虑α-距离图的符号对模拟，我们引入了两个码不变量，即生成轮廓和断连阈值，并用它们来刻画码的保护性质。通过这些不变量关联两种度量，得到了符号对阈值相对于其汉明对应物的上下界，且这两个界均可达到。我们进一步将经典的Plotkin界和球填充界推广到该设置中。
英文摘要
In several storage systems, including DNA storage and flash memory, errors affect neighbouring symbols jointly, and the Hamming metric does not adequately capture such error patterns. The symbol-pair read channel, introduced by Cassuto and Blaum~\cite{cassuto2011codes}, addresses this by reading consecutive pairs of symbols rather than individual symbols. Motivated by this, we introduce function-correcting symbol-pair codes with data protection (FCSPC-DP), which guarantee reliable recovery of a desired function of the message while simultaneously protecting the message itself against symbol-pair errors. We derive bounds on the optimal redundancy of such codes and establish a relationship with joint-pair distance matrices. We also give explicit constructions of FCSPC-DP for locally pair-bounded functions and symbol-pair weight functions. We introduce the pair-separation constant of a function, the minimum symbol-pair distance between messages sharing a function value, and show that when it is sufficiently large, data protection requires no additional redundancy: the optimal redundancy coincides with that of the corresponding code without data protection. Considering the symbol-pair analogue of the $α$-distance graph, we introduce two code invariants, the generation profile and the disconnection threshold, and use them to characterise a code's protection properties. Relating the two metrics through these invariants yields upper and lower bounds on the symbol-pair threshold in terms of its Hamming counterpart, both of which are attained. We further extend the classical Plotkin and sphere-packing bounds to this setting.
