Function-Correcting $b$-symbol Codes for Locally $(\lambda, \rho,b)$-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 $(\lambda,\rho)$-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 $(\lambda,\rho,b)$-functions. First, we discuss the values of $\lambda$ and $\rho$ for which a function can be considered as a locally $(\lambda,\rho)$-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 $(\lambda,\rho,b)$-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 ($\lambda,\rho$, $b$)-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 $(\lambda,2t,b)$-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 $b\geq1$.