Universal codes and unimodular lattices

Bonnecaze, Calderbank and Sole (see IEEE Trans. Inform. Theory, vol.41, p.366-77, 1995) introduced for primes p≡±(mod8), codes 𝒬 and 𝒩, the universal extended quadratic residue codes, of length p+1 over the 2-adic integers Z_2∞. For positive integers s they consider their reductions 𝒬_2s and 𝒩_2s modulo 2s; 𝒬₂ and 𝒩₂ are just the standard binary extended quadratic residue codes, while 𝒬₄ and 𝒩₄ are the quaternary quadratic residue codes. Given a code C of length n over Z₄ define Λ(C) as the set of vectors in Zⁿ which reduce modulo 4 to elements of C. We show, by means of an explicit isomorphism, that if p≡-1(mod 8) and p⩽31 then ½Λ(𝒬₄) is isometric to lattice L(𝒬 ₂) constructed from the binary quadratic residue code in a manner (construction B plus density doubling) generalizing the original construction of the Leech lattice