Estimates of lengths of shortest nonzero vectors in some lattices. II (Q2081017)

The paper considers the minimum norm of 5-dimensional lattices \(L(a,W)\) spanned by the rows of \[\left( \begin{array}{ccccc} W & 0 & 0 & \ldots & 0 \\ a & -1 & 0 & \ldots & 0 \\ a^2 & 0 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ a^4 & 0 & \ldots & 0 & =1 \end{array} \right) \] where \(W >0\), \(a \in [0,\ldots,W] \) are given integers. It is shown that for any sufficiently large number \(W\) and any positive \(\varepsilon < 1/1157\) and all but \(W^{1-\varepsilon} \) values of \(a\) the mininum is \(\min (L(a,W)) \geq W^{(1/5-\varepsilon)}\). For Part I see [the author, ibid. 32, No. 3, 207--218 (2022; Zbl 07596775); translation from Diskretn. Mat. 33, No. 1, 31--46 (2021)].