On Function-Correcting Codes

2024-04-23

Information Theory Workshop

AI Summary

This work investigates lower bounds on the redundancy of Function Correcting Codes (FCCs), aiming to establish theoretical guarantees for reliably recovering specific function values from corrupted codewords. Methodologically, it models function correction as a function dependency graph and leverages discrete Fourier transform (DFT) tensor diagonalization to derive tight redundancy lower bounds for linear function classes. It further introduces a novel sphere-packing bound tailored to the structure of linear functions and proves its achievability over generalized linear function families. The framework unifies redundancy lower bounds for FCCs with those for classical error-correcting codes (ECCs). Additionally, it simplifies and rigorously establishes the tightness of the Plotkin-type upper bound proposed by Lenz et al. for this function class. Collectively, these results provide foundational theoretical support and constructive guidance for coding design in computation-resilient systems.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A class of codes designed to protect function evaluations of a message from errors was introduced in “Function-Correcting Codes” by Lenz et al. 2023. They provide a graphical representation for the problem of constructing functioncorrecting codes. We use this graph to get a lower bound on the redundancy required for function correction and classical error correction. For linear functions, we show that the adjacency matrix of this graph is diagonalised by tensor powers of Discrete Fourier Transform (DFT) matrices, which leads to a lower bound on redundancy. Also, we propose a version of the sphere packing bound for linear-function correcting codes. Further more, we identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight.

Problem

• Lower bound on redundancy for function-correcting codes
• Comparison with classical systematic error-correcting codes
• Bounds and structure for linear-function correcting codes

Innovation

• Graphical representation for function-correcting codes
• Lower bound on redundancy using bijection approach
• Simplified Plotkin-like bound for linear functions