Grobner Bases for Lattices and an Algebraic Decoding Algorithm

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 associ...

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Vydáno v:IEEE transactions on communications Ročník 61; číslo 4; s. 1222 - 1230
Hlavní autoři: Aliasgari, M., Sadeghi, M., Panario, D.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York, NY IEEE 01.04.2013
Institute of Electrical and Electronics Engineers
Témata:
Applied sciences
Coding, codes
division algorithm
Exact sciences and technology
Generators
Grobner bases
Information, signal and communications theory
label code
Lattices
Maximum likelihood decoding
Polynomials
Signal and communications theory
Telecommunications and information theory
Vectors
Zinc
Performance evaluation
Gröbner bases
Lower bound
Block code
Gröbner basis
Trellis code
Decoding
Division algorithm
Binary code
Polynomial time
Linear code
label code
Integer lattice
lattices
Time complexity
Signal to noise ratio
ISSN:0090-6778
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Popis
Shrnutí: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 (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .
ISSN:0090-6778
DOI:10.1109/TCOMM.2013.13.120317