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.

翻译：本文研究了用于汉明码成员资格函数（HCMF）的单错误校正功能校正码（SEFCC），该函数指示 $\mathbb{F}_2^7$ 中的向量是否属于 [7,4,3]-汉明码。我们依据码字与其最近非码字之间的距离约束，建立了有效奇偶校验分配的必要和充分条件。研究表明，汉明码字之间的汉明距离-3关系诱导出一个二分图，这一基本几何性质被用于开发系统性的SEFCC构造方法。通过推导出成对距离之和的紧上界，我们证明了所提出的二分图构造方法唯一地实现了最大和距离、可能的最大最小距离2以及最少的距离-2码字对。因此，对于HCMF SEFCC问题，和距离最大化并非仅仅是启发式的——它精确地强制实现了与错误概率相关的最优距离谱特性。在采用软判决解码的AWGN信道上的仿真结果证实，与任意有效分配方案相比，所得的最大和SEFCC提供了显著改善的数据保护能力和误码率（BER）性能。