We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.

翻译：我们利用arXiv:2308.15275中的结果，证明了数域上模格中最短向量的紧概率界。此外，通过建立代数整数元素且欧几里得长度有界的固定秩矩阵计数的渐近公式，我们证明了从代数编码提升得到的模格离散集的近似罗杰斯积分公式。这进而表明，arXiv:2308.15275中的矩估计以及上述关于最短向量的界对于足够大的模格离散集同样成立。