Existence and Constructions of Strict Function-Correcting Codes with Data Protection

2026-04-29

AI Summary

This study addresses the limitation of classical perfect and MDS codes, which cannot simultaneously protect data and their function values with distinct error-correction capabilities. The authors establish, for the first time, a graph-theoretic existence framework for strict functional error-correcting codes, recasting linear code construction as a subcode generation problem. Two novel approaches are proposed: a reverse construction method based on weight distribution constraints that reduces the number of minimum-weight codewords, and a construction leveraging narrow-sense BCH codes with designed distance three. By integrating α-distance graphs, Cayley graph isomorphism analysis, and a reverse application of Simonis’s results, the work successfully constructs multiple explicit code families. These constructions not only confirm the existence of such codes but also transcend classical coding-theoretic boundaries, significantly advancing the theoretical foundations of functional error correction.

Abstract

Function-correcting codes with data protection simultaneously protect both the data and a function of the data at distinct error-correction levels. When the function receives strictly stronger protection than the data, such a code is called a strict function-correcting code with data protection. While prior work showed that perfect and MDS codes cannot serve as strict function-correcting codes, which codes can serve this role, and how to construct them, has remained open. In this paper, we address the existence and construction of strict function-correcting codes for linear codes through three main contributions. First, using the $α$-distance graph framework from our prior work, we establish a graph-theoretic existence condition under which a code can serve as a strict function-correcting code. For linear codes, we prove this distance graph is isomorphic to a Cayley graph, which implies the connected components are cosets of the subcode generated by low-weight codewords. This transforms the existence problem into a subcode generation problem. Second, a classical result of Simonis shows any linear code can be transformed into one with the same parameters whose basis consists entirely of minimum-weight codewords. We develop a converse construction: under certain conditions on the weight distribution, a linear code can be transformed into a new code with the same parameters but fewer independent minimum-weight codewords, thereby producing codes suitable for use as strict function-correcting codes. As a source of codes satisfying these conditions, we introduce chain codes, an infinite family of linear codes generated by their minimum-weight codewords. Third, we present an independent construction of strict function-correcting codes from narrow-sense BCH codes with designed distance three, by proving the minimum-weight codewords of such codes are contained in a proper subcode.

Problem

• function-correcting codes
• data protection
• strict function-correcting codes
• linear codes
• code construction

Innovation

• strict function-correcting codes
• α-distance graph
• Cayley graph
• chain codes
• narrow-sense BCH codes