Function-Correcting Codes with Optimal Data Protection for Hamming Code Membership

TL;DR

This paper develops an optimal systematic construction of single-error-correcting function-correcting codes for Hamming code membership, improving data protection and BER performance over AWGN channels.

Contribution

It introduces a bipartite graph-based systematic construction that maximizes sum-distance, achieving optimal error correction properties for Hamming code membership functions.

Findings

Max-sum SEFCCs achieve the largest minimum distance of 2.

The construction uniquely maximizes the sum of pairwise distances.

Simulation shows significantly improved BER performance.

Abstract

This paper investigates single-error-correcting function-correcting codes (SEFCCs) for the Hamming code membership function (HCMF), which indicates whether a vector in $\mathbb{F}_2^7$ belongs to the [7,4,3]-Hamming code. Necessary and sufficient conditions for valid parity assignments are established in terms of distance constraints between codewords and their nearest non-codewords. It is shown that the Hamming-distance-3 relations among Hamming codewords induce a bipartite graph, a fundamental geometric property that is exploited to develop a systematic SEFCC construction. By deriving a tight upper bound on the sum of pairwise distances, we prove that the proposed bipartite construction uniquely achieves the maximum sum-distance, the largest possible minimum distance of 2, and the minimum number of distance-2 codeword pairs. Consequently, for the HCMF SEFCC problem, sum-distance maximisation is not merely heuristic-it exactly enforces the optimal distance-spectrum properties relevant to error probability. Simulation results over AWGN channels with soft-decision decoding confirm that the resulting max-sum SEFCCs provide significantly improved data protection and Bit Error Rate (BER) performance compared to arbitrary valid assignments.