Function-Correcting Codes with Optimal Data Protection for Hamming Code Membership
Swaraj Sharma Durgi, Anjana A. Mahesh, Anupriya Kumari, Rajlaxmi Pandey, and B. Sundar Rajan

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 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…
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Taxonomy
TopicsError Correcting Code Techniques · Advanced Wireless Communication Techniques · Advanced Data Storage Technologies
