Construction of Barnes-Wall lattices from linear codes over rings

Dense lattice packings can be obtained via the well-known Construction A from binary linear codes. In this paper, we use an extension of Construction A called Construction A´ to obtain Barnes-Wall lattices from linear codes over polynomials rings. To obtain the Barnes-Wall lattice BW_2m in C^2m for any m ≥ 1, we first identify a linear code C_2m over the quotient ring U_m = F₂[u]/u^m and then propose a mapping ψ : U_m → Z[i] such that the code L_2m = ψ (C_2m) is a lattice constellation. Further, we show that L_2m has the cubic shaping property when m is even. Finally, we show that BW_2m can be obtained through Construction A´ as BW_2m = (1 + i)^m Z[i]^2m ⊕ L_2m.