In this paper, we study linear-function correcting codes, a class of codes designed to protect linear function evaluations of a message against errors. The work "Function-Correcting Codes" by Lenz et al. 2023 provides a graphical representation for the problem of constructing function-correcting codes. We use this graph to get a lower bound the on redundancy required for function correction. By considering the function to be a bijection, such an approach also provides a lower bound on the redundancy required for classical systematic error correcting codes. For linear-function correction, we characterise the spectrum of the adjacency matrix of this graph, which gives rise to lower bounds on redundancy. The work "Function-Correcting Codes" gives an equivalence between function-correcting codes and irregular-distance codes. We identify a structure imposed by linearity on the distance requirement of the equivalent irregular-distance code which provides a simplified Plotkin-like bound. We propose a version of the sphere packing bound for linear-function correcting codes. We identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight. We also identify a class of functions for which coset-wise coding is equivalent to a lower dimensional classical error correction problem.

翻译：本文研究了线性函数纠正码，这是一类旨在保护消息的线性函数评估免受错误影响的码。Lenz等人2023年的工作《函数纠正码》为构建函数纠正码的问题提供了一种图形化表示方法。我们利用该图得到了函数纠正所需冗余度的下界。通过将函数视为双射，这种方法也为经典系统纠错码所需的冗余度提供了下界。对于线性函数纠正，我们刻画了该图邻接矩阵的谱，从而推导出冗余度的下界。《函数纠正码》的工作揭示了函数纠正码与不规则距离码之间的等价关系。我们识别了线性性在等价不规则距离码的距离要求上施加的结构，该结构提供了简化的Plotkin型界。我们提出了线性函数纠正码的球堆积界版本。我们识别了一类线性函数，对于这些函数，Lenz等人提出的上界是紧的。我们还识别了一类函数，对于它们，陪集编码等价于一个低维的经典纠错问题。