MDS codes containing a fixed subcode over finite field extensions
Abstract
Let 2≤ k≤ n and let U≤𝔽qn be a fixed (k−1)-dimensional linear code. We study the [n,k] maximum distance separable (MDS) codes over finite extensions of 𝔽q that contain the scalar extension of U. Our approach constructs an auxiliary linear code from U. Its full-weight words characterize the desired MDS completions. We prove that U admits an [n,k] MDS completion over some finite extension of 𝔽q if and only if d(U)≥ n−k+1. When this condition holds, the number of these MDS codes over 𝔽qm is #{W≤𝔽qmn:U⊗𝔽q𝔽qm⊆ W, dim𝔽qm W=k, W is MDS}=χδGM(G)(qm)/(qm−1), where G is a generator matrix of U, δGM(G) is its represented derived matroid, and χδGM(G) is its characteristic polynomial. In particular, the formula determines existence over each extension field and the least extension degree for which such an MDS code exists. Examples show that these counts may distinguish different representations of the same column matroid, while monomially inequivalent codes may have identical counts over every finite extension.
Related
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BibTeX
@misc{li2026codes,
title = {{MDS codes containing a fixed subcode over finite field extensions}},
author = {Yangcheng Li and Pingzhi Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34016},
eprint = {2610.34016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34016},
}
RIS
TY - UNPB
TI - MDS codes containing a fixed subcode over finite field extensions
AU - Yangcheng Li
AU - Pingzhi Yuan
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34016
UR - https://arxiv.org/abs/2610.34016
AB - Let 2≤ k≤ n and let U≤𝔽_q^n be a fixed (k−1)-dimensional linear code. We study the [n,k] maximum distance separable (MDS) codes over finite extensions of 𝔽_q that contain the scalar extension of U. Our approach constructs an auxiliary linear code from U. Its full-weight words characterize the desired MDS completions. We prove that U admits an [n,k] MDS completion over some finite extension of 𝔽_q if and only if d(U)≥ n−k+1. When this condition holds, the number of these MDS codes over 𝔽_q^m is #{W≤𝔽_q^m^n:U⊗_𝔽_q𝔽_q^m⊆ W, dim_𝔽_q^m W=k, W is MDS}=χ_δ_GM(G)(q^m)/(q^m−1), where G is a generator matrix of U, δ_GM(G) is its represented derived matroid, and χ_δ_GM(G) is its characteristic polynomial. In particular, the formula determines existence over each extension field and the least extension degree for which such an MDS code exists. Examples show that these counts may distinguish different representations of the same column matroid, while monomially inequivalent codes may have identical counts over every finite extension.
ER -