On Function-Correcting Codes in the Lee Metric
Gyanendra K. Verma, Abhay Kumar Singh
TL;DR
The paper develops function-correcting codes under the Lee metric over the ring $\mathbb{Z}_m$, introducing irregular Lee-distance codes to bound the optimal redundancy necessary to recover function evaluations rather than entire messages. It establishes a tight equivalence between $(f,t)$-FCLMCs and irregular-distance codes via distance matrices $D_f^L$, and provides both lower and upper bounds on redundancy through generalized Plotkin/Gilbert–Varshamov-type results. The authors analyze special function classes, including locally defined functions, the Lee weight function, and the Lee weight distribution function, deriving concrete redundancy bounds and showing attainability in the locally binary Lee case. They also derive a Plotkin-like bound for linear, onto function maps, and discuss reductions to classical error-correcting codes in the bijective or binary-Hamming limits, highlighting the framework’s unifying power across Lee and Hamming settings with practical implications for nonbinary alphabets.
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 Lee-local 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 $\mathbb{Z}_m$. 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 $\mathbb{Z}_2$. Furthermore, when the underlying function is bijective, function-correcting codes reduce to classical error-correcting codes. In parallel, our bound correspondingly reduces to the classical Plotkin bound for error-correcting codes, both for the Lee metric over $\mathbb{Z}_m$ and for the Hamming metric over $\mathbb{Z}_2$.