AI 中文总结
本文受函数校正码和RT度量启发，引入函数校正RT码，通过建立与不规则RT距离码的联系推导冗余界，对任意函数及特定函数类分析最优冗余界，给出RT权重分布函数的显式构造，扩展了函数校正码理论，为并行信道通信提供新工具。
AI 中文摘要
函数校正码最近成为一种有效方法，用于在仅需可靠恢复传输信息的规定函数值而非完整消息本身时减少编码冗余。作为汉明度量的自然推广，罗森布鲁姆 - 茨法斯曼度量（RT - 度量）因其与并行信道通信的相关性而备受关注。受这两个研究方向的启发，本文引入函数校正RT码（FCRTCs），将函数校正码框架从汉明度量扩展到更一般的RT度量。为研究最优冗余的基本问题，引入一类新的矩阵码——不规则RT距离码，并建立FCRTCs冗余优化问题与此类码构造之间的直接对应关系。这一联系为推导RT度量下的冗余界提供了统一框架。对于任意函数，建立了FCRTCs最优冗余的一般上下界。然后针对几种重要函数类进行分析，通过利用其特定结构性质获得了更精确的界。此外，给出了RT权重分布函数的FCRTCs显式构造，并证明这些构造在几种情况下达到了最优冗余界。所提出的框架将函数校正码理论显著扩展到RT度量，为研究并行信道上的可靠通信提供了新的理论工具，并为超越经典汉明设置的冗余高效编码开辟了新视角。
英文摘要
Function-correcting codes have recently emerged as an effective approach to reducing coding redundancy when the objective is to reliably recover only the prescribed function values of the transmitted information, rather than the complete message itself. As a natural generalization of the Hamming metric, the Rosenbloom--Tsfasman metric (RT-metric) has attracted considerable attention due to its relevance to communication over parallel channels. Motivated by these two research directions, we introduce in this paper function-correcting RT codes (FCRTCs), which extend the framework of function-correcting codes from the Hamming metric to the more general RT metric. To investigate the fundamental problem of optimal redundancy, we introduce a new class of matrix codes, called irregular RT-distance codes, and establish a direct correspondence between the redundancy optimization problem for FCRTCs and the construction of such codes. This connection provides a unified framework for deriving redundancy bounds under the RT metric. For arbitrary functions, we establish general upper and lower bounds on the optimal redundancy of FCRTCs. We then specialize our analysis to several important classes of functions, namely RT-weight functions, RT-weight distribution functions, and RT-locally-two-valued binary functions, for which substantially sharper bounds are obtained by exploiting their specific structural properties. Furthermore, we present explicit constructions of FCRTCs for RT-weight distribution functions and prove that these constructions attain the optimal redundancy bounds in several cases. The proposed framework considerably extends the theory of function-correcting codes to the RT metric, provides new theoretical tools for studying reliable communication over parallel channels, and opens new perspectives for redundancy-efficient coding beyond the classical Hamming setting.
