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
$\mathrm{D}^{(\text{cyc})}$
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 explicitly determine the minimum weight codewords of those codes for some special cases.