Generalized Function-Correcting Partition Codes

Abstract: We introduce generalized function-correcting partition codes (GFCPCs) that simultaneously protect multiple partitions of the message space against different numbers of errors. Given partitions with respective distance requirements, a GFCPC is a systematic encoding that guarantees, for each partition, a specified minimum Hamming distance between codewords whose messages lie in different blocks. This framework unifies and generalizes both function-correcting partition codes, which protect multiple functions with a common error-correction level, and function-correcting codes with data protection, which assign different levels of protection to data and a single function. We present a multi-step construction procedure for these codes and demonstrate it with some examples. We derive general upper and lower bounds on the optimal redundancy, including the upper bound which considers the join of different combinations of the partitions. We define the distance requirement matrix $\mathcal{D}$ for the GFCPCs and use it to characterize the optimal redundancy in terms of the shortest length of an associated $\mathcal{D}$-code. For two partitions of message space over the binary field, we establish improved lower bounds on the optimal redundancy under specific neighborhood conditions on the partitions. Through several examples, we demonstrate that the proposed framework can yield strictly smaller redundancy than both the sum of the individual FCPC redundancies and the redundancy of a single FCPC designed for the join partition with the highest distance (strongest protection required).