Plotkin-type Bound on Function-Correcting Codes for Monomial Mappings

2026-10-03

AI Summary

This study addresses the long-standing absence of unified theoretical bounds on the redundancy of function-correcting codes (FCCs). By leveraging algebraic analysis over finite fields and coding theory under the Hamming metric, we derive generalized Plotkin-type redundancy bounds for arbitrary functions. The proposed framework transcends the limitations of traditional function-specific analyses by uniformly accommodating diverse mapping scenarios, including linear functions, weight-threshold functions, and monomials, while naturally reducing to classical error-correcting code results as special cases. Furthermore, we obtain explicit redundancy bounds for these mappings, thereby establishing a rigorous theoretical foundation for FCCs and substantially refining their overall theoretical framework.

Abstract

Function-correcting codes, recently introduced by Lenz et al.\ (2023), provide a framework for the receiver to provide a higher level of protection against channel errors than the level of protection for messages. In this paper, we derive a generalised Plotkin-type bound on the redundancy of FCCs under the Hamming metric for arbitrary functions over finite fields. The proposed bound recovers the classical Plotkin bound for classical error-correcting codes as a special case. We further obtain explicit Plotkin-type bounds for linear functions, Hamming-weight functions, and Hamming-weight threshold functions. In addition, we study FCCs for monomial mappings $f(x)=x^n$ over $\mathbb{F}_{q^k}$, which are widely utilised in coding theory, cryptography, and finite geometry, and derive an explicit Plotkin-type redundancy bound.

Problem

• Function-correcting codes
• Plotkin-type bound
• Redundancy
• Monomial mappings
• Finite fields

Innovation

• Function-correcting codes
• Plotkin-type bound
• Monomial mappings
• Hamming metric
• Redundancy bound