Six Families of Binary Codes Arising from Ding's Conjectures

Abstract

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 $(b)$.