Lattices from abelian extensions and error-correcting codes

• Autor(a) principal: Interlando, J. Carmelo
• Data de Publicação: 2021
• Outros Autores: da Nóbrega Neto, Trajano Pires [UNESP], Nunes, José Valter Lopes, Lopes, José Othon Dantas
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://dx.doi.org/10.1216/rmj.2021.51.903; http://hdl.handle.net/11449/233424
• topic: Abelian extensions; Cyclotomic fields; Error-correcting codes; Lattice packing; Quadratic forms

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.