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 3nmid 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).
Related
- Solutions to Three Conjectures and an Open Problem on Binary BCH Codes
- External Difference Families Arising from Two or Three Cyclotomic Classes
- Additive codes arising from hypergraphs
- Three Conjectures on Binary Channels for the Doubly Symmetric Binary Source
- DS and AMDS Symbol-Pair Codes from Three Families of Simple Root Cyclic Codes
BibTeX
@misc{wang2026families,
title = {{Six Families of Binary Codes Arising from Ding's Conjectures}},
author = {Xiaoqiang Wang and Shiyan Xiong and Mu yuan and Jing Qiu and Dabin Zheng and Jiawei He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37093},
eprint = {2609.37093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37093},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Six Families of Binary Codes Arising from Ding's Conjectures
AU - Xiaoqiang Wang
AU - Shiyan Xiong
AU - Mu yuan
AU - Jing Qiu
AU - Dabin Zheng
AU - Jiawei He
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37093
UR - https://arxiv.org/abs/2609.37093
AB - 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 3nmid 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).
ER -