Six conjectures on binary codes receive new analysis and counterexamples
Six Families of Binary Codes Arising from Ding's Conjectures
Information Theory
Summary
Binary codes are ways to represent data for computers and communication. A researcher named Ding proposed ten ideas about these binary codes, some of which were already proved or disproved by others. This paper looks at six remaining ideas, confirming some cases and providing examples where original guesses were not quite right. The authors also completely solve one last tricky case by using earlier work. Their results give a clearer picture of how these special binary codes can behave.
What this means in practice
- •For error correction engineers: Use refined weight distributions of binary codes to design better error-correcting algorithms with predictable performance.
- •For communications systems developers: Incorporate insights from new code families to improve data transmission reliability in digital communication systems.
A theory result. No direct application yet.
Authors
Xiaoqiang Wang, Shiyan Xiong, Mu yuan, Jing Qiu, Dabin Zheng, Jiawei He
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 $(b)$.