Tetracode length-isospectral lattice construction ($L_{\pm }=(\pm 3I+S){\mathbb{Z}}^4$)

For the skew matrix $S=\left(\begin{array}{cccc}0& 1& 1& 1\\ -1& 0& -1& 1\\ -1& 1& 0& -1\\ -1& -1& 1& 0\end{array}\right)$, both Euclidean lattices reduce to the tetracode modulo $3$. A reflection depending on the reduction coset matches their vectors coordinatewise in absolute value. Their weighted length spectra agree for all positive diagonal axis weights. Rationally independent weights force any possible linear isometry to be a coordinate sign change; the code then forces an overall sign, which does not identify the two lattices.

Choose positive axis weights linearly independent over the rationals, and suppose both integer Euclidean lattices contain $q{\mathbb{Z}}^d$ for some positive integer $q$. A length-preserving linear map sends each $q{e}_i$ to an integer vector with exactly that absolute-coordinate pattern, by the displayed implication. Thus the map must be a diagonal sign change. For the four-coordinate tetracode, preserving the code forces all four signs to agree, because each of its four nonzero projective words has support on a different triple of coordinates. A global sign cannot exchange the two odd-vector parity conditions, proving nonisometry for these weights.

Add respectively the conditions ${\sum }_i{x}_i\equiv 0(\mathrm{mod}4)$ and ${\sum }_i{x}_i-2x_1\equiv 0(\mathrm{mod}4)$ to the displayed common conditions. These define the two integer Euclidean lattices underlying the tetracode length-isospectral lattice construction. They coincide on their all-even vectors and differ on all-odd vectors. Reflecting a coordinate divisible by three preserves the code condition; on odd vectors it changes the coordinate sum by two modulo four, while on even vectors both conditions remain satisfied. This yields the coordinate reflection bijection for tetracode lattices.

For the tetracode length-isospectral lattice construction $L_{\pm }=(\pm 3I+S){\mathbb{Z}}^4$, choose the first coordinate $i$ divisible by three and reverse its sign. Such a coordinate always exists because each nonzero tetracode word has one zero coordinate. The reflection preserves the reduction modulo three and switches the two lattice membership conditions modulo four. It is an involution between $L_{+}$ and $L_{-}$ preserving every absolute coordinate, so arbitrary positive diagonal weights preserve equality of their vector-length multisets.