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PreprintCC BY-NC-ND 4.0arXiv

Free and projective LCD codes

  • Dominik Krasula
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Abstract

LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the size condition of a code and its dual to the noncommutative setting. This result is applied to prove that group codes are LCD if and only if they are generated by a central idempotent.

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Cite

BibTeXRIS
BibTeX
@misc{krasula2026free,
  title = {{Free and projective LCD codes}},
  author = {Dominik Krasula},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15423},
  eprint = {2609.15423},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15423},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Free and projective LCD codes
AU  - Dominik Krasula
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15423
UR  - https://arxiv.org/abs/2609.15423
AB  - LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the size condition of a code and its dual to the noncommutative setting. This result is applied to prove that group codes are LCD if and only if they are generated by a central idempotent.
ER  -