Function-correcting codes were introduced in the work "Function-Correcting Codes" (FCC) by Lenz et al. 2023, which provides a graphical representation for the problem of constructing function-correcting codes. We use this function dependent graph to get a lower bound on the redundancy required for function correction. By considering the function to be a bijection, such an approach leads to a lower bound on the redundancy required for classical systematic error correcting codes (ECCs) of small distances. We propose a range of parameters for which the bound is tight. For single error correcting codes, we show that this bound is at least as good as a bound proposed by Zinoviev, Litsyn, and Laihonen in 1998. Thus, this framework helps to study systematic classical error correcting codes. Further, we study the structure of this function dependent graph for linear functions, which leads to bounds on the redundancy of linear-function correcting codes. We show that the Plotkin-like bound for Function-Correcting Codes that was proposed by Lenz et.al 2023 is simplified for linear functions. Also, 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 and also identify a class of functions for which coset-wise coding is equivalent to a lower dimensional classical error correction problem.

翻译：函数校正码由Lenz等人于2023年在《函数校正码》一文中提出，该工作为构建函数校正码的问题提供了图形化表示。我们利用这种函数依赖图来获得函数校正所需冗余度的下界。通过考虑函数为双射，该方法可导出小距离经典系统纠错码所需冗余度的下界。我们提出了一系列使该界紧致的参数范围。对于单纠错码，我们证明该界至少与Zinoviev、Litsyn和Laihonen于1998年提出的界同样优越。因此，该框架有助于研究经典系统纠错码。进一步，我们研究了线性函数下该函数依赖图的结构，从而得到线性函数校正码冗余度的界。我们证明Lenz等人2023年提出的类Plotkin界在线性函数情形下可得到简化。同时，我们提出了线性函数校正码的球包界版本。我们识别出一类线性函数使得Lenz等人提出的上界是紧的，并识别出一类函数使得陪集编码等价于低维经典纠错问题。