The family of functions plays a central role in the design and effectiveness of function-correcting codes. By focusing on a well-defined family of functions, function-correcting codes can be constructed with minimal length while still ensuring full error detection and correction within that family. In this work, we explore the concept of locally (λ, ρ) -functions for b-symbol read channels and investigate the optimal redundancy of the corresponding function-correcting b-symbol codes (FCBSC) by introducing the notions of locally (λ, ρ, b) -functions. First, we discuss the values of λ and ρ for which a function can be considered as a locally (λ, ρ) -function in b-symbol metric. The findings improve some known results in the Hamming metric and present several new results in the b-symbol metric. Then we investigate the optimal redundancy of (f, t) -FCBSCs for locally (λ, ρ, b) -functions. We establish a recurrence relation between the optimal redundancy of (f, t) -function-correcting codes for the (b+1) -symbol read and b-symbol read channels. We present an upper bound on the optimal redundancy of (f, t) -function-correcting b-symbol codes for general locally (λ, ρ, b) -functions by associating it to the minimum achievable length of b-symbol error-correcting codes and traditional Hamming-metric codes, given a fixed number of codewords and a specified minimum distance. We derive some explicit upper bounds on the redundancy of (f, t) -function-correcting b-symbol codes for locally (λ, 2t, b) -functions. Moreover, for the case where b=1, we show that a locally (3, 2t, 1) -function achieves the optimal redundancy of 3t. Additionally, we explicitly investigate the locality and optimal redundancy of FCBSCs for the b-symbol weight function and weight distribution function for b1.
Verma et al. (Wed,) studied this question.