# Shortest vectors in tensor product and maximal lattices in tensor product

$$\mathcal L$$ and $$\mathcal L'$$ be full rank lattices in $$\mathbb R^n$$ with shortest vectors $$v_1,\dots,v_n$$ and $$v_1',\dots,v_n'$$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\leq\dots\leq\|v_n'\|_2$$ hold and $$\mathcal L'=Span(v_1,\dots,v_n)$$ and $$\mathcal L=Span(v_1',\dots,v_n')$$.

1. Do the shortest vectors of $$\mathcal L\otimes\mathcal L'$$ have any relation to shortest vectors of $$\mathcal L$$ and $$\mathcal L'$$ and what is the span of $$v_i\otimes v_j'$$?

2. Do maximal lattices in $$\mathcal L\otimes\mathcal L'$$ have any relation to $$\mathcal L$$ and $$\mathcal L'$$ and $$v_1,\dots,v_n$$ and $$v_1',\dots,v_n'$$?