Lattices from abelian extensions and error-correcting codes

Abstract

A construction of laminated lattices of full diversity in odd dimensions d with 3 ≤ d ≤ 15 is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field F of degree d and prime conductor p, where p is a prime congruent to 1 modulo d, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a Z-submodule of OF, the ring of integers of F. The submodule is defined by the parity-check matrices of a Reed–Solomon code over GF(p) and a suitably chosen linear code, typically either binary or over Z/4Z, the ring of integers modulo 4.