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Double circulant LCD codes over 𝔽2+u𝔽2 and constructions for binary LCD codes

  • Xingxing Xu
  • Minjia Shi
  • Jon-Lark Kim
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Abstract

Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring R=𝔽2+u𝔽2, where u2=0. Using the Chinese Remainder Theorem and the factorization of xnβˆ’1, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to xβˆ’1, which is not obtained by a formal specialization of the previously known counting formula over 𝔽q[u]/(uk). Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate 1/2 whose relative distance can approach H2βˆ’1(1/4) from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.

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Cite

BibTeXRIS
BibTeX
@misc{xu2026double,
  title = {{Double circulant LCD codes over $\mathbbF_2+u\mathbbF_2$ and constructions for binary LCD codes}},
  author = {Xingxing Xu and Minjia Shi and Jon-Lark Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16121},
  eprint = {2610.16121},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16121},
}
RIS
TY  - UNPB
TI  - Double circulant LCD codes over 𝔽_2+u𝔽_2 and constructions for binary LCD codes
AU  - Xingxing Xu
AU  - Minjia Shi
AU  - Jon-Lark Kim
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16121
UR  - https://arxiv.org/abs/2610.16121
AB  - Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring R=𝔽_2+u𝔽_2, where u^2=0. Using the Chinese Remainder Theorem and the factorization of x^nβˆ’1, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to xβˆ’1, which is not obtained by a formal specialization of the previously known counting formula over 𝔽_q[u]/(u^k). Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate 1/2 whose relative distance can approach H_2^βˆ’1(1/4) from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.
ER  -