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$A_2$-frames, negacyclic glue, and extremal strongly $6$-modular lattices
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 $S=\mathbb{Z}[i]/(1+i)^3$. For odd $\ell$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4)^{2\
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