Function-Correcting b-symbol Codes for Locally $(λ, ρ,b)$-Functions

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 b≥1.