发表机构
University of Bristol; Indian Institute of Science（布里斯托大学; 印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究函数校正码中系统性编码的冗余度代价，证明非系统性编码可降低冗余度，并给出最优生成矩阵的构造算法及二值函数冗余度充分条件。
AI 中文摘要
函数校正码（FCCs）保护消息的某个函数 $f$ 的值免受 $t$ 个错误的影响。在 Lenz 等人（2023）的原始表述中，编码是系统性的，且冗余度下界 $2t$ 依赖于这种形式。最近，D. Ho（arXiv，2026）表明，对于线性函数和线性编码，系统性可能带来冗余度代价。我们首先以 OR 函数为例，证明这种代价并不局限于线性情形：非系统性编码可实现冗余度 $1$，而任何系统性编码需要 $2t$。对于线性函数 $f$ 和固定线性码 $C$，设 $d_f$ 表示具有不同函数值的消息的码字之间的最小距离。$C$ 的不同生成矩阵为消息分配不同的码字，从而可能给出不同的 $d_f$ 值。我们研究 $C$ 的哪个生成矩阵能给出最大的 $d_f$。我们给出一个算法，构造最优生成矩阵，并将最优值确定为 $C$ 的贪心基中一个码字的权重。然后，我们利用信息集刻画系统性形式的生成矩阵何时达到该最优值，并给出一个仅需检查 $C$ 的低权重码字即可验证的必要条件。对于二值函数，我们同时放弃线性和系统性。利用单纯序的初始段关于汉明邻域的最小性，我们证明：存在长度为 $n$ 的非系统性 $(f,t)$-FCC 当且仅当某个仅依赖于 $f$ 的较小原像大小的条件成立。对于 $t=1$，当 $k\ge 10$ 时，冗余度 $1$ 对 $\mathbb{F}_2^k$ 上的每个非常数二值函数都充分。对于一般 $t$，当 $k$ 足够大时，冗余度 $1$ 对所有二值函数都充分，并且对于每个 $s<2t$，我们给出阈值 $k$ 的上界，超过该阈值冗余度 $s$ 就充分。
英文摘要
Function-correcting codes (FCCs) protect the value of a function $f$ of the message against $t$ errors. In the original formulation of Lenz et al. (2023), the encoding is systematic, and the lower bound of $2t$ on the redundancy relies on this form. Recently, D. Ho (arXiv, 2026) showed, for linear functions and linear encodings, that systematicity can cost redundancy. We begin with the OR function to show that the cost is not confined to the linear setting: a non-systematic encoding attains redundancy $1$ while every systematic encoding needs $2t$. For a linear function $f$ and a fixed linear code $C$, let $d_f$ denote the minimum distance between codewords of messages with different function values. Different generator matrices of $C$ assign different codewords to the messages and can give different values of $d_f$. We study which generator matrix of $C$ gives the largest $d_f$. We give an algorithm that constructs an optimal generator matrix and determines the optimal value as the weight of a codeword in a greedy basis of $C$. We then characterize, in terms of information sets, when a generator matrix in systematic form attains this optimum, and give a necessary condition that is checked on the low-weight codewords of $C$ alone. For two-valued functions, we drop both linearity and systematicity. Using the minimality of initial segments of the simplicial order with respect to Hamming neighbourhoods, we show that a non-systematic $(f,t)$-FCC of length $n$ exists if and only if a condition depending on $f$ only through the size of its smaller preimage holds. For $t=1$, redundancy $1$ is sufficient for every nonconstant two-valued function on $\mathbb{F}_2^k$ with $k\ge 10$. For general $t$, redundancy $1$ suffices for all two-valued functions once $k$ is large enough, and for each $s<2t$ we give an upper bound on the threshold in $k$ beyond which redundancy $s$ suffices.
