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PreprintCC BY 4.0arXiv

Twisted group codes of dimension 2 and 3

  • Alvaro Otero Sanchez
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Abstract

It is known that all group codes of dimension at most $3$ are permutation equivalent to abelian group codes. We prove that twisted group codes of dimension at most $3$ are monomially equivalent either to an abelian group code or to an irreducible constacyclic code, and in the second case we describe the code explicitly by means of a trace map. Monomial equivalence cannot be replaced by permutation equivalence, and the second case cannot be avoided: the ternary tetracode is an ideal of 𝔽3[x]/(x4+1) but it is not monomially equivalent to any group code. More generally, we give two infinite families, in dimensions $2$ and $3$, of irreducible constacyclic codes which attain the Griesmer bound, whereas every group code with the same length and dimension has smaller minimum distance.

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Cite

BibTeXRIS
BibTeX
@misc{sanchez2026twisted,
  title = {{Twisted group codes of dimension 2 and 3}},
  author = {Alvaro Otero Sanchez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28297},
  eprint = {2610.28297},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28297},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Twisted group codes of dimension 2 and 3
AU  - Alvaro Otero Sanchez
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.28297
UR  - https://arxiv.org/abs/2610.28297
AB  - It is known that all group codes of dimension at most $3$ are permutation equivalent to abelian group codes. We prove that twisted group codes of dimension at most $3$ are monomially equivalent either to an abelian group code or to an irreducible constacyclic code, and in the second case we describe the code explicitly by means of a trace map. Monomial equivalence cannot be replaced by permutation equivalence, and the second case cannot be avoided: the ternary tetracode is an ideal of 𝔽_3[x]/(x^4+1) but it is not monomially equivalent to any group code. More generally, we give two infinite families, in dimensions $2$ and $3$, of irreducible constacyclic codes which attain the Griesmer bound, whereas every group code with the same length and dimension has smaller minimum distance.
ER  -