  The famous Barnes-Wall lattices can be obtained by applying Construction D to
a chain of Reed-Muller codes. By applying Construction ${{D}}^{{(cyc)}}$ to a
chain of extended cyclic codes sandwiched between Reed-Muller codes, Hu and
Nebe (J. London Math. Soc. (2) 101 (2020) 1068-1089) constructed new series of
universally strongly perfect lattices sandwiched between Barnes-Wall lattices.
In this paper, we first extend their construction to generalized Reed-Muller
codes, and then explicitly determine the minimum vectors of those new
sandwiched Reed-Muller codes for some special cases.
