In this paper, we further extend the study of function-correcting codes in the homogeneous metric over a chain ring $\mathbb{Z}_{2^s}$ for broader classes of functions, namely, locally bounded functions and linear functions, and for weight functions, modular sum functions. e define locally bounded functions in the homogeneous metric over $\mathbb{Z}_{2^s}^k$ and investigate the locality of weight functions. We derive a Plotkin-like bound for irregular homogeneous distance code over $\mathbb{Z}_4$, which improves the existing bound. Using locality properties of functions, we establish upper and lower bounds on the optimal redundancy. We provide several explicit constructions of function-correcting codes for locally bounded functions, weight functions, and weight distribution functions. Using these constructions, we further discuss the tightness of the derived bound. We explicitly derive a Plotkin-like bound for linear function-correcting codes that reduces to the classical Plotkin bound when the linear function is bijective, we further discuss a construction of function-correcting linear codes over $\mathbb{Z}_{2^s}$.

翻译：本文进一步扩展了链环 $\mathbb{Z}_{2^s}$ 上齐次度量下函数纠错码的研究，针对更广泛的函数类别——局部有界函数与线性函数，以及权函数、模和函数。我们定义了 $\mathbb{Z}_{2^s}^k$ 上齐次度量下的局部有界函数，并研究了权函数的局部性质。我们推导了 $\mathbb{Z}_4$ 上非规则齐次距离码的一个类普洛特金界，改进了现有界。利用函数的局部性质，我们建立了最优冗余度的上界与下界。我们为局部有界函数、权函数以及权分布函数提供了若干函数纠错码的显式构造。基于这些构造，我们进一步讨论了所得界的紧性。我们显式推导了线性函数纠错码的一个类普洛特金界，当线性函数为双射时该界退化为经典普洛特金界；我们进一步讨论了 $\mathbb{Z}_{2^s}$ 上线性函数纠错码的一种构造。