Construction of Barnes-Wall lattices from linear codes over rings
J. Harshan, Emanuele Viterbo, J.-C. Belfiore · 2012
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 BW2min C2mfor any m ≥ 1, we first identify a linear code C2mover the quotient ring Um= F2[u]/umand then propose a mapping ψ : Um→ Z[i] such that the code L2m= ψ (C2m) is a lattice constellation. Further, we show that L2mhas the cubic shaping property when m is even. Finally, we show that BW2mcan be obtained through Construction A' as BW2m= (1 + i)mZ[i]2m⊕ L2m.