Function-Correcting Codes for Sum-Rank Metric

2026-07-04

AI Summary

This study addresses the high redundancy overhead of traditional error-correcting codes by investigating low-redundancy fault-tolerant mechanisms tailored for reliable computation of specific functions. It extends functional error-correcting codes to the sum-rank metric, establishes a Plotkin-type upper bound applicable to irregular-distance codes, and derives general upper and lower bounds on redundancy under this metric. Building on these theoretical foundations, the work presents an explicit coding construction for local binary functions that achieves optimal redundancy while guaranteeing correctness of function outputs, thereby significantly reducing communication and storage overhead.

Abstract

Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. In this paper, we study function-correcting codes under the sum-rank metric, which is a natural generalization of both the Hamming metric and the rank-metric and also we derive general upper and lower bounds on the optimal redundancy of FCCs in the sum-rank metric. In particular, we establish a Plotkin-like bound for irregular-distance codes in sum-rank metric. Furthermore, we present explicit construction of function-correcting sum-rank metric codes (FCSRCs) for locally binary functions with optimal redundancy.

Problem

• Function-Correcting Codes
• Sum-Rank Metric
• Error Correction
• Redundancy
• Channel Errors

Innovation

• function-correcting codes
• sum-rank metric
• optimal redundancy
• Plotkin-like bound
• explicit construction