On Function-Correcting Codes in the Lee Metric

Abstract

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 $\mathbb{Z}_2$.