Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 3, pp 2413–2427 | Cite as

Bounds for the \(l_1\)-distance of q-ary lattices obtained via Constructions D, D\(^{'}\) and \(\overline{\text{ D }}\)

  • Eleonesio Strey
  • Sueli I. R. Costa
Article
  • 54 Downloads

Abstract

Lattices have been used in several problems in coding theory and cryptography. In this paper, we approach q-ary lattices obtained via Constructions D, \(\text{ D }'\) and \(\overline{\text{ D }}\). It is shown connections between Constructions D and \(\text{ D }'\). Bounds for the minimum \(l_1\)-distance of lattices \({\varLambda }_{D}\), \({\varLambda }_{D'}\) and \({\varLambda }_{\overline{D}}\) and, under certain conditions, a generator matrix for \({\varLambda }_{D'}\) are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum \(l_1\)-distances of the lattices \({\varLambda }_{D}\) and \({\varLambda }_{\overline{D}}\) attached to the distances of the codes used in these constructions.

Keywords

Lattices Lattices from q-ary codes Codes over rings \(l_1\)-distance Constructions D, D\('\) and \({\overline{\mathrm{D}}}\) 

Mathematics Subject Classification

94B05 06B99 52C99 

Notes

Acknowledgements

The authors are thankful to the reviewers for their comments and suggestions.

References

  1. Barnes ES, Sloane NJA (1983) New lattice packings of spheres. Can J Math 35:117–130MathSciNetCrossRefMATHGoogle Scholar
  2. Campello A, Jorge GC, Strapasson JE, Costa SRI (2016) Perfect codes in the \(l_p\) metric. Eur J Comb 53:72–85MathSciNetCrossRefMATHGoogle Scholar
  3. Cohn H, Kumar A (2009) Optimality and uniqueness of the Leech lattice among lattices. Ann Math 170(2009):1003–1050. doi: 10.4007/annals.2009.170.1003
  4. Conway JH, Sloane NJA (1998) Sphere packings, lattices and groups, 3rd edn. Springer, New YorkGoogle Scholar
  5. da Silva PRB, Silva D (2014) Design of lattice network codes based on Construction D. International telecommunications symposiumGoogle Scholar
  6. Costa SIR, Campello A, Jorge GC, Strapasson JE, Qureshi C (2014) Codes and lattices in the \(l_p\) metric. IEEE information theory and applications workshop, pp 1–4Google Scholar
  7. Etzion T, Vardy A, Yaakobi E (2010) Dense error-correcting codes in the Lee metric. IEEE information theory workshop, Dublin, IrelandGoogle Scholar
  8. Etzion T, Vardy A, Yaakobi E (2013) Coding for the Lee and Manhattan metrics with weighing matrices. IEEE Trans Inf Theory 59(10):6712–6723MathSciNetCrossRefMATHGoogle Scholar
  9. Forney GD (1988a) Coset codes-part I: Introduction and geometrical classification. IEEE Trans Inf Theory 34(5):1123–1151CrossRefMATHGoogle Scholar
  10. Forney GD (1988b) Coset codes-part II: Binary lattices and related codes. IEEE Trans Inf Theory 34(5):1152–1187CrossRefMATHGoogle Scholar
  11. Jorge GC, Campello AC, Costa SIR (2013) \(q\)-ary lattices in the \(l_p\) norm and a generalization of the Lee metric. International workshop on coding and cryptography, Bergen, NorwayGoogle Scholar
  12. Khodaiemehr H, Sadeghi M-R, Sakzad A (2017) Practical encoder and decoder for power constrained QC LDPC-lattice codes. IEEE Trans Commun 65(2):486–500CrossRefGoogle Scholar
  13. Lee CY (1958) Some properties of nonbinary error-correcting code. IRE Trans Inf Theory IT-4:72–82Google Scholar
  14. Liu S, Hong Y, Viterbo E (2014) Unshared secret key cryptography. IEEE Trans Wirel Commun 13(12):6670–6683CrossRefGoogle Scholar
  15. Micciancio D, Regev O (2009) Lattice-based cryptography in post quantum cryptography In: Bernstein DJ, Buchmann J, Dahmen E (eds) Post-Quantum Cryptography, Springer, pp 147–191Google Scholar
  16. Minkowski H (1904) Dichteste gitterformige lagerung kongruenter korper. Nachrichten Ges. Wiss. Göttingen, pp 311–355Google Scholar
  17. Rush JA, Sloane NJA (1987) An improvement to the Minkowski–Hlawka bound for packing superballs. Mathematika 34:8–18MathSciNetCrossRefMATHGoogle Scholar
  18. Sadeghi M-R (2010) Lattice and construction of high coding In: Woungang I, Misra S, Chandra Misra S (eds) Selected topics in information and coding theory. Series on coding theory and cryptology, vol 7, chapter 2, pp 41–76. ISBN: 978-981-283-716-5Google Scholar
  19. Sadeghi M-R, Banihashemi AH, Panario D (2006) Low-density parity-check lattices: construction and decoding analysis. IEEE Trans Inf Theory 52(10):4481–4495MathSciNetCrossRefMATHGoogle Scholar
  20. Sakzad A, Sadeghi M-R, Panario D (2012) Turbo lattices: construction and error decoding performance. Available on arXiv:1108.1873v3
  21. Strey E, Costa SIR (2016a) Lattices from codes over \({\mathbb{Z}}_q\): generalization of Constructions D, \(\text{D}^{\prime }\) and \(\overline{\text{ D }}\). doi: 10.1007/s10623-016-0289-1 (to appear in Designs, codes and cryptography)
  22. Strey E, Costa SIR (2016b) Bounds for the \(l_1\)-distance of lattices obtained via Construction D and its variations (in Portuguese). In: 34th Brazilian telecommunications and signal processing symposium, Santarém-PA, BrazilGoogle Scholar
  23. Strey E, Costa SIR (2017) Lattices from codes over finite rings: connections between Constructions D, D\(^{\prime }\) and \(\overline{\text{ D }}\) (in Portuguese). Proc Ser Braz Soc Appl Comput Math 5(1). doi: 10.5540/03.2017.005.01.0228
  24. Ulrich W (1957) Non-binary error correction codes. Bell Syst J 36:1341–1387CrossRefMATHGoogle Scholar
  25. Zamir R (2009) Lattices are everywhere. Information theory and applications workshop, San Diego-CA, pp 392–421Google Scholar
  26. Zamir R (2014) Lattice coding of signals and networks: a structured coding approach to quantization, modulation and multi-user information theory. Cambridge University Press, Cambridge, UKCrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Department of Pure and Applied MathematicsFederal University of Espirito SantoAlegreBrazil
  2. 2.Institute of MathematicsUniversity of CampinasCampinasBrazil

Personalised recommendations