Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes
Abstract
Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram [3,3,3,5] with rank distance 3. We construct an optimal [[3,3,3,5],6,3]q Ferrers diagram rank-metric code for every prime power q, thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a [3×4,4,3]q MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a [3×3,3,3]q MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the [[3,3,3,5],6,3]q construction and holds for every prime power q, while the second is binary and obtained directly by the puncturing-inclusion construction.
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BibTeX
@misc{xu2026puncturinginclusion,
title = {{Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes}},
author = {Dengming Xu and Chengming Duan and Yong Jin and Yuhao Ning},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17570},
eprint = {2610.17570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17570},
}
RIS
TY - UNPB
TI - Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes
AU - Dengming Xu
AU - Chengming Duan
AU - Yong Jin
AU - Yuhao Ning
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17570
UR - https://arxiv.org/abs/2610.17570
AB - Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram [3,3,3,5] with rank distance 3. We construct an optimal [[3,3,3,5],6,3]_q Ferrers diagram rank-metric code for every prime power q, thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a [3×4,4,3]_q MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a [3×3,3,3]_q MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the [[3,3,3,5],6,3]_q construction and holds for every prime power q, while the second is binary and obtained directly by the puncturing-inclusion construction.
ER -