Solutions to Three Conjectures and an Open Problem on Binary BCH Codes
Abstract
BCH codes are among the most important classes of cyclic codes and have played a central role in coding theory and its applications. One of the fundamental problems in the study of BCH codes is to determine their exact minimum distances, which directly govern their error-correcting capability. Although the BCH bound provides a general lower bound, determining the exact minimum distance is often difficult, and many parameter families remain unresolved. In this paper, we investigate three conjectures and an open problem on binary BCH codes proposed by Chen, Xie, and Ding in \cite{Chen59}. We settle these conjectures on the exact minimum distances of three families of binary BCH codes by constructing codewords attaining the BCH bound. Beyond these conjectures, we further study the more difficult family codes and determine its minimum distance for some cases. We further study Open Problem 8.4: affirmative answers are obtained for the first two length families, while for the third family a sufficient condition is established and a counterexample shows that the unrestricted assertion does not hold in general.
Related
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BibTeX
@misc{wang2026solutions,
title = {{Solutions to Three Conjectures and an Open Problem on Binary BCH Codes}},
author = {Xiaoqiang Wang and Jiawei He and Boru Yi and Dabin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00532},
eprint = {2609.00532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solutions to Three Conjectures and an Open Problem on Binary BCH Codes
AU - Xiaoqiang Wang
AU - Jiawei He
AU - Boru Yi
AU - Dabin Zheng
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.00532
UR - https://arxiv.org/abs/2609.00532
AB - BCH codes are among the most important classes of cyclic codes and have played a central role in coding theory and its applications. One of the fundamental problems in the study of BCH codes is to determine their exact minimum distances, which directly govern their error-correcting capability. Although the BCH bound provides a general lower bound, determining the exact minimum distance is often difficult, and many parameter families remain unresolved. In this paper, we investigate three conjectures and an open problem on binary BCH codes proposed by Chen, Xie, and Ding in \cite{Chen59}. We settle these conjectures on the exact minimum distances of three families of binary BCH codes by constructing codewords attaining the BCH bound. Beyond these conjectures, we further study the more difficult family codes and determine its minimum distance for some cases. We further study Open Problem 8.4: affirmative answers are obtained for the first two length families, while for the third family a sufficient condition is established and a counterexample shows that the unrestricted assertion does not hold in general.
ER -