$A_2$-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 $S=\mathbb{Z}[i]\mathrm{/}(1+i{)}^3$. For odd $\mathrm{\ell }$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4{)}^{2\mathrm{\ell }}$-frame to an even strongly $6$-modular lattice $L_{\mathrm{\ell }}$ of rank $4\mathrm{\ell }$. 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-$6\mathrm{a}$ 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 $\mathrm{\ell }=3$, $L_3$ has an index-$8$ overlattice isometric to the Coxeter–Todd lattice.