Function-Correcting RT Codes: A Coding Framework for Parallel Channels

Abstract

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.