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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 define the distance requirement matrix $\mathcal{D}$ for GFCPCs and use it to characterize the optimal redundancy in terms of the shortest length of an associated $\mathcal{D}$-code. We derive general upper and lower bounds on the optimal redundancy, including an upper bound that considers the join of different combinations of the partitions. For two partitions of the message space over the binary field, we establish improved lower bounds on the optimal redundancy under specific neighborhood conditions on the partitions. We develop a linear programming bound on the optimal redundancy of GFCPCs. The bound specializes to function-correcting codes and function-correcting codes with data protection, for which, to the best of our knowledge, no such bound was previously known. We present a multi-step construction procedure for these codes and illustrate it with some examples. 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).

Submission history

From: Charul Rajput [view email]
[v1] Tue, 5 May 2026 05:22:56 UTC (22 KB)
[v2] Mon, 14 Sep 2026 17:28:45 UTC (36 KB)