Function-correcting codes are a coding framework designed to minimize redundancy while ensuring that specific functions or computations of encoded data can be reliably recovered, even in the presence of errors. The choice of metric is crucial in designing such codes, as it determines which computations must be protected and how errors are measured and corrected. Previous work by Liu and Liu [6] studied function-correcting codes over $\mathbb{Z}_{2^l},\ l\geq 2$ using the homogeneous metric, which coincides with the Lee metric over $\mathbb{Z}_4$. In this paper, we extend the study to codes over $\mathbb{Z}_m,$ for any positive integer $m\geq 2$ under the Lee metric and aim to determine their optimal redundancy. To achieve this, we introduce irregular Lee distance codes and derive upper and lower bounds on the optimal redundancy by characterizing the shortest possible length of such codes. These general bounds are then simplified and applied to specific classes of functions, including locally bounded functions, Lee weight functions, and Lee weight distribution functions. We extend the bounds established by Liu and Liu [6] for codes over $\mathbb{Z}_4$ in the Lee metric to the more general setting of $\mathbb{Z}_m$. Moreover, we give explicit constructions of function-correcting codes in Lee metric. Additionally, we explicitly derive a Plotkin-like bound for linear function-correcting codes in the Lee metric. As the Lee metric coincides with the Hamming metric over the binary field, we demonstrate that our bound naturally reduces to a Plotkin-type bound for function-correcting codes under the Hamming metric over $\mathbb{Z}_2$.

翻译：函数校正码是一种编码框架，旨在最小化冗余的同时，确保即使存在错误，也能可靠地恢复编码数据的特定函数或计算结果。度量的选择对此类码的设计至关重要，因为它决定了哪些计算必须受到保护，以及如何度量和纠正错误。Liu和Liu[6]先前的工作研究了在$\mathbb{Z}_{2^l},\ l\geq 2$上使用齐次度量的函数校正码，该度量与$\mathbb{Z}_4$上的Lee度量一致。本文将此研究扩展到任意正整数$m\geq 2$下$\mathbb{Z}_m$上的Lee度量码，旨在确定其最优冗余。为此，我们引入了不规则Lee距离码，并通过刻画此类码的最短可能长度，推导了最优冗余的上界和下界。这些一般性界随后被简化并应用于特定的函数类，包括局部有界函数、Lee重量函数以及Lee重量分布函数。我们将Liu和Liu[6]为$\mathbb{Z}_4$上Lee度量码建立的界推广到更一般的$\mathbb{Z}_m$情形。此外，我们给出了Lee度量下函数校正码的显式构造。另外，我们显式推导了Lee度量下线性函数校正码的一个类Plotkin界。由于Lee度量在二进制域上与汉明度量一致，我们证明了我们的界自然地退化为$\mathbb{Z}_2$上汉明度量下函数校正码的一个类Plotkin型界。