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.
翻译：著名的Barnes-Wall lattices可以通过对Reed-Muller代码链应用建筑 D 来获得。 通过将建筑 $ ⁇ D ⁇ ⁇ (cyc) ⁇ ($$) ⁇ ($) ⁇ ($) ⁇ ($) $($) $($) ) 应用到一个由Reed-Muler代码、Hu 和 Nebe 之间混合的扩展循环代码链(J. London Math. Soc. (2) 101 (2020) 1068-1089), 建造了一套由Barnes-Wall lattices之间混合的、 1068- 1089 的、 普遍非常完美的顶层。 在本文中, 我们首先将其构建扩展至通用 Reed- Muller 代码, 然后为某些特殊案例明确确定这些新的三明治Reed-Muler代码的最小载体。
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