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A2-frames, negacyclic glue, and extremal strongly $6$-modular lattices

  • Ian Teixeira
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Abstract

The classical A2-frame construction of the Coxeter--Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring S=ℤ[i]/(1+i)3. For odd ℓ, a one-generator cyclic graph code over S lifts from an A2(4)2ℓ-frame to an even strongly $6$-modular lattice Lℓ of rank 4ℓ. In the real frame ordering the same glue is a one-generator negacyclic code. The cases L5 and L7 have minima $6$ and $8$ and give examples for the type-6a entries in dimensions $20$ and $28$, which are listed as open in the Nebe--Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for L7 the key relation is the boundary map of a $7$-cycle. For ℓ=3, L3 has an index-$8$ overlattice isometric to the Coxeter--Todd lattice.

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Cite

BibTeXRIS
BibTeX
@misc{teixeira2026a2frames,
  title = {{$A_2$-frames, negacyclic glue, and extremal strongly $6$-modular lattices}},
  author = {Ian Teixeira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27148},
  eprint = {2609.27148},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27148},
}
RIS
TY  - UNPB
TI  - A_2-frames, negacyclic glue, and extremal strongly $6$-modular lattices
AU  - Ian Teixeira
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.27148
UR  - https://arxiv.org/abs/2609.27148
AB  - The classical A_2-frame construction of the Coxeter--Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring S=ℤ[i]/(1+i)^3. For odd ℓ, a one-generator cyclic graph code over S lifts from an A_2(4)^2ℓ-frame to an even strongly $6$-modular lattice L_ℓ of rank 4ℓ. In the real frame ordering the same glue is a one-generator negacyclic code. The cases L_5 and L_7 have minima $6$ and $8$ and give examples for the type-6a entries in dimensions $20$ and $28$, which are listed as open in the Nebe--Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for L_7 the key relation is the boundary map of a $7$-cycle. For ℓ=3, L_3 has an index-$8$ overlattice isometric to the Coxeter--Todd lattice.
ER  -