Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Plotkin-like Bound and Explicit Function-Correcting Code Constructions for Lee Metric Channels

Hareesh K., Rashid Ummer N. T., B. Sundar Rajan

TL;DR

This work addresses efficient protection of function evaluations over Lee metric channels by developing function-correcting Lee codes (FCLCs). It introduces a Plotkin-like bound for irregular-Lee-distance codes and an exact connection between FCLCs and irregular-Lee-distance codes via distance requirement matrices, enabling explicit redundancy analysis. The authors present explicit FCLC constructions for three function classes—Lee weight, Lee weight distribution, and modular sum—with tight redundancy bounds in several parameter regimes, and they demonstrate substantial redundancy reductions compared with classical Lee ECCs and ECCs on function values. Collectively, the results advance practical designs for reliable function evaluation with reduced redundancy in non-binary, Lee-metric environments, and they lay groundwork for extending these techniques to additional function classes.

Abstract

Function-Correcting Codes (FCCs) are a novel class of codes designed to protect function evaluations of messages against errors while minimizing redundancy. A theoretical framework for systematic FCCs to channels matched to the Lee metric has been studied recently, which introduced function-correcting Lee codes (FCLCs) and also derived upper and lower bounds on their optimal redundancy. In this paper, we first propose a Plotkin-like bound for irregular Lee-distance codes. We then construct explicit FCLCs for specific classes of functions, including the Lee weight, Lee weight distribution, modular sum, and locally bounded function. For these functions, lower bounds on redundancy are obtained, and our constructions are shown to be optimal in certain cases. Finally, a comparative analysis with classical Lee error-correcting codes and codes correcting errors in function values, demonstrates that FCLCs can significantly reduce redundancy while preserving function correctness.

Plotkin-like Bound and Explicit Function-Correcting Code Constructions for Lee Metric Channels

TL;DR

This work addresses efficient protection of function evaluations over Lee metric channels by developing function-correcting Lee codes (FCLCs). It introduces a Plotkin-like bound for irregular-Lee-distance codes and an exact connection between FCLCs and irregular-Lee-distance codes via distance requirement matrices, enabling explicit redundancy analysis. The authors present explicit FCLC constructions for three function classes—Lee weight, Lee weight distribution, and modular sum—with tight redundancy bounds in several parameter regimes, and they demonstrate substantial redundancy reductions compared with classical Lee ECCs and ECCs on function values. Collectively, the results advance practical designs for reliable function evaluation with reduced redundancy in non-binary, Lee-metric environments, and they lay groundwork for extending these techniques to additional function classes.

Abstract

Function-Correcting Codes (FCCs) are a novel class of codes designed to protect function evaluations of messages against errors while minimizing redundancy. A theoretical framework for systematic FCCs to channels matched to the Lee metric has been studied recently, which introduced function-correcting Lee codes (FCLCs) and also derived upper and lower bounds on their optimal redundancy. In this paper, we first propose a Plotkin-like bound for irregular Lee-distance codes. We then construct explicit FCLCs for specific classes of functions, including the Lee weight, Lee weight distribution, modular sum, and locally bounded function. For these functions, lower bounds on redundancy are obtained, and our constructions are shown to be optimal in certain cases. Finally, a comparative analysis with classical Lee error-correcting codes and codes correcting errors in function values, demonstrates that FCLCs can significantly reduce redundancy while preserving function correctness.

Paper Structure

This paper contains 21 sections, 20 theorems, 43 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Works
  4. Contributions
  5. Organization
  6. Preliminaries
  7. Irregular-Distance Codes
  8. Lee Codes
  9. General Results on the Optimal Redundancy
  10. A Connection between FCLCs and Irregular-Lee-Distance Codes
  11. Simplified Bounds on Optimal Redundancy
  12. A variant of Plotkin Bound on $N_{L}(\mathbf{D})$
  13. Explicit Code Constructions for Lee Weight, Lee weight distribution, and Modular sum Functions
  14. Lee weight function
  15. Lee Weight Distribution Function
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\boldsymbol{x}, \boldsymbol{y} \in \mathbb{Z}_q^n$ be two vectors over the ring of integers modulo $q$, where $q \geq 2$. Denote by $d_H(\boldsymbol{x}, \boldsymbol{y})$ the Hamming distance between $\boldsymbol{x}$ and $\boldsymbol{y}$, and by $d_L(\boldsymbol{x}, \boldsymbol{y})$ the Lee dist

Figures (4)

  • Figure 1: Conditional probabilities for a discrete, memoryless, symmetric channel matched to the Lee metric.
  • Figure 2: Function distance matrix $\mathbf{D}_{\mathrm{w}_L}(E,t)$ for $t = \frac{(q-3)}{2}$.
  • Figure 3: Function distance matrix $\mathbf{D}_{\mathrm{ms}}(\mathrm{E},t)$ for $t \ge\frac{(\left\lfloor \frac{q}{2} \right\rfloor-1)}{2}$ and odd $q \ge 5$.
  • Figure 4: Function distance matrix $\mathbf{D}_{\mathrm{ms}}(\mathrm{E},t)$ for $t \ge \tfrac{(\tfrac{q}{2} -1)}{2}$ and even $q \ge 6$.

Theorems & Definitions (51)

  • Definition 1: Irregular-Distance CodesLenz
  • Definition 2: Lee WeightCW
  • Definition 3: Lee DistanceCW
  • Lemma 1: CW
  • Lemma 2: CW
  • Lemma 3: GW
  • Lemma 4: GW
  • Lemma 5: GW
  • Definition 4: Function-Correcting Lee Codes
  • Remark 1
  • ...and 41 more