Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq
Abstract
This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over π½q+uπ½q. By decomposing Gram matrices over π½q+uπ½q into pairs of symmetric matrices over the finite field π½q, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over π½q+uπ½q with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over π½q.
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BibTeX
@misc{an2026shortest,
title = {{Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq}},
author = {Junmin An and Jon-Lark Kim},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28222},
eprint = {2608.28222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28222},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq
AU - Junmin An
AU - Jon-Lark Kim
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28222
UR - https://arxiv.org/abs/2608.28222
AB - This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over π½_q+uπ½_q. By decomposing Gram matrices over π½_q+uπ½_q into pairs of symmetric matrices over the finite field π½_q, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over π½_q+uπ½_q with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over π½_q.
ER -