Grobner bases for lattices and an algebraic decoding algorithm

Malihe Aliasgari, Mohammad Reza Sadeghi, Daniel Panario

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al.. We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E7 and E8) show that for low SNR the performance of these lattices is near to the lower bounds given in.

Original languageEnglish
Article number6466341
Pages (from-to)1222-1230
Number of pages9
JournalIEEE Transactions on Communications
Volume61
Issue number4
DOIs
StatePublished - 2013

Keywords

  • division algorithm
  • Grobner bases
  • label code
  • lattices

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