= Tetracode length-isospectral lattice construction
{title2=$L_\pm=(\pm3I+S)\mathbb Z^4$}
For the skew matrix $S=\left(\begin{smallmatrix}0&1&1&1\\-1&0&-1&1\\-1&1&0&-1\\-1&-1&1&0\end{smallmatrix}\right)$, 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.
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