Embedding linear codes over Z4 into self-orthogonal codes
Abstract
The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.
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BibTeX
@misc{an2026embedding,
title = {{Embedding linear codes over Z4 into self-orthogonal codes}},
author = {Junmin An and Jon-Lark Kim and San Ling},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.08964},
eprint = {2606.08964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08964},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Embedding linear codes over Z4 into self-orthogonal codes
AU - Junmin An
AU - Jon-Lark Kim
AU - San Ling
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2606.08964
UR - https://arxiv.org/abs/2606.08964
AB - The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.
ER -