源自Ding猜想的六族二元码

中文摘要 AI 辅助

Ding \cite{Ding2016} 提出了关于由布尔函数产生的二元线性码的十个猜想。其中四个，即猜想38至41，随后由Göloğlu和Krasnayová \cite{GologluKrasnayova2019} 证明。本文研究其余六个猜想，即猜想19、27、30、33、34和37。对于猜想19和27，我们获得了共同的权重限制和几个无限五权重族。对于猜想30，我们证明每个可容许码具有三个、四个或五个非零权重，并且一个显式的四权重例子否证了原始的“三个或五个权重”断言。对于猜想33，获得了一个无限五权重族。对于猜想34，我们获得了一个一般的$(2h+1)$权重上界，并给出了一个显式的六权重反例，表明原始的“三个或五个权重”断言在一般情况下不成立，其中$h$为正整数。$h=3$且$3\nmid m$的情况也被完全确定。最后，通过结合Ahmadi和Shafaeiabr \cite{AhmadiShafaeiabr2023}的已知结果以及对剩余两类$(a)$和$(b)$的处理，猜想37被完全解决。

英文摘要

Ding \cite{Ding2016} proposed ten conjectures on binary linear codes arising from Boolean functions. Four of them, namely Conjectures 38--41, were subsequently proved by Göloğlu and Krasnayová \cite{GologluKrasnayova2019}. In this paper, we investigate the remaining six conjectures, namely Conjectures 19, 27, 30, 33, 34, and 37. For Conjectures~19 and~27, we obtain common weight restrictions and several infinite five-weight families. For Conjecture~30, we prove that every admissible code has three, four, or five nonzero weights, and an explicit four-weight example disproves the original ``three or five weights'' assertion. For Conjecture~33, an infinite five-weight family is obtained. For Conjecture~34, we obtain a general $(2h+1)$-weight upper bound and give an explicit six-weight counterexample, showing that the original ``three or five weights'' assertion is false in general, where $h$ is a positive integer. The case $h=3$ with $3\nmid m$ is also completely determined. Finally, Conjecture~37 is completely resolved by combining the known results of Ahmadi and Shafaeiabr \cite{AhmadiShafaeiabr2023} with the treatment of the two remaining classes $(a)$ and $(b)$.

发表机构

• Hubei University（湖北大学）
• Nanchang Hangkong University（南昌航空大学）

机构由 AI 辅助整理，请以论文原文为准。