发表机构
The University of Tokyo; RIKEN Center for Advanced Intelligence Project（东京大学; 理化学研究所先进智能项目中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对二元纠错码打包问题，给出三组显式二元码，改进了三个通用二元码下界，其中首次66年突破字典序构造的下界，还发布码字与验证器供独立验证。
AI 中文摘要
纠错码是在通信和数据存储中，用于鲁棒表示信息以抵御噪声的基础技术。二元码打包问题旨在固定码长和最小距离的条件下最大化码字数量，是编码理论中的基础问题，对许多参数选择仍未解决。我们给出参数为$(n,M,d)=(22,84,9)$、$(24,196,9)$和$(25,65,11)$的显式二元码，其中$n$表示码长，$M$表示码字数量，$d$表示最小距离。这些结果确立了下界$A_2(22,9)\ge84$、$A_2(24,9)\ge196$和$A_2(25,11)\ge65$，其中$A_2(n,d)$表示最大可能的码字数量，分别将通用二元码公开表格中的对应下界提高了80、192和64。最值得注意的是，第三个结果66年来首次改进了字典序构造提供的下界，而前两个结果改进了已存在21年的下界。新码在不改变码长或纠错保证的情况下可表示更多消息，增益在各码块间呈乘法累积。我们发布所有码字和详尽验证器以支持独立验证。
英文摘要
Error-correcting codes are a foundational technology for representing information robustly against noise in communication and data storage. The binary code packing problem, which seeks to maximize the number of codewords at a fixed code length and minimum distance, is a fundamental problem in coding theory that remains open for many parameter choices. We give explicit binary codes with parameters $(n,M,d)=(22,84,9)$, $(24,196,9)$, and $(25,65,11)$, where $n$, $M$, and $d$ denote the code length, number of codewords, and minimum distance, respectively. These establish the lower bounds $A_2(22,9)\ge84$, $A_2(24,9)\ge196$, and $A_2(25,11)\ge65$, where $A_2(n,d)$ denotes the maximum possible number of codewords, improving the respective lower bounds 80, 192, and 64 in the public table of general binary codes. Most notably, the third result improves the lower bound supplied by a lexicographic construction for the first time in 66 years, while the first two improve bounds that have stood for 21 years. The new codes represent more messages without changing the code length or error-correction guarantees, with gains compounding multiplicatively across blocks. We release all codewords and an exhaustive verifier to enable independent verification.
