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

2026-02-25

AI Summary

This work proposes a systematic method for constructing single-error-correcting function error-correcting codes (SEFCC) to protect the outputs of Hamming code member functions (HCMF) against transmission noise. By analyzing the distance constraints between valid Hamming codewords and their nearest non-codewords, the authors construct a bipartite graph induced by Hamming distance 3, revealing its underlying geometric structure. Leveraging this insight, they non-heuristically optimize codeword assignments to maximize the total pairwise distance. Integrating algebraic coding theory, graph-theoretic modeling, and soft-decision decoding, the approach achieves an optimal distance spectrum under the constraint of minimum distance 2, significantly enhancing bit error rate (BER) performance and data protection capability over additive white Gaussian noise (AWGN) channels.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• function-correcting codes
• Hamming code membership
• single-error-correcting
• data protection
• distance spectrum

Innovation

• function-correcting codes
• Hamming code membership
• bipartite graph construction
• sum-distance maximization
• distance spectrum optimization