Extended Cyclic Codes Sandwiched Between Reed–Muller Codes

The famous Barnes–Wall lattices can be obtained by applying Construction D to a chain of Reed–Muller codes. By applying Construction
${\text {D}}^{\text {(cyc)}}$
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.