Function-Correcting Codes (FCCs) enable reliable computation of a function of a $k$-bit message over noisy channels without requiring full message recovery. In this work, we study optimal single-error correcting FCCs (SEFCCs) for maximally-unbalanced Boolean functions, where $k$ denotes the message length and $t$ denotes the error-correction capability. We analyze the structure of optimal SEFCC constructions through their associated codeword distance matrices and identify distinct FCC classes based on this structure. We then examine the impact of these structural differences on error performance by evaluating representative FCCs over the additive white Gaussian noise (AWGN) channel using both soft-decision and hard-decision decoding. The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy.

翻译：函数校正码（FCCs）使得能够在噪声信道上可靠地计算一个$k$比特消息的函数，而无需完全恢复消息。在本工作中，我们研究了针对最大不平衡布尔函数的最优单错误校正FCC（SEFCC），其中$k$表示消息长度，$t$表示纠错能力。我们通过其关联的码字距离矩阵分析了最优SEFCC构造的结构，并基于此结构识别了不同的FCC类别。随后，我们通过在加性高斯白噪声（AWGN）信道上使用软判决和硬判决译码评估代表性FCC，研究了这些结构差异对错误性能的影响。结果表明，具有不同距离矩阵结构的FCC可能表现出显著不同的数据误比特率和函数错误行为，并且码结构的影响在很大程度上取决于译码策略。