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.

翻译：函数纠错码在数据保护功能中同时以不同的纠错级别保护数据及其函数。当函数获得比数据更强的保护时，这类码称为具有数据保护功能的严格函数纠错码。已有研究表明完美码和MDS码无法作为严格函数纠错码，但哪些码能胜任此角色以及如何构造它们仍是未解问题。本文通过三项主要贡献论述了线性码作为严格函数纠错码的存在性与构造问题。首先，利用我们先前工作中的α-距离图框架，建立了码可作为严格函数纠错码的图论存在性条件。对于线性码，我们证明该距离图同构于Cayley图，这意味着其连通分支是由低权重码字生成的子码的陪集，从而将存在性问题转化为子码生成问题。其次，Simonis的经典结论表明任何线性码可转换为具有相同参数但基向量全由最小权重码字构成的新码。我们发展了其逆构造：在权重分布的特定条件下，线性码可转换为具有相同参数但独立最小权重码字数量更少的新码，从而生成适用于严格函数纠错码的码族。为提供满足这些条件的码源，我们引入了链式码——一类由最小权重码字生成的无限线性码族。第三，我们通过证明设计距离为3的窄意义BCH码的最小权重码字包含于真子码中，独立构造了此类码的严格函数纠错码。