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Grobner Bases for Lattices and an Algebraic Decoding Algorithm
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DOI：10.1109/TCOMM.2013.13.120317.png)
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Abstract
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In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grobner bases for binary linear codes were introduced by Borges-Quintana et al. [4]. We extend their work to non-binary group block codes. Then, given a lattice Lambda and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Lambda 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 [16].
Keywords:
Grobner bases
division algorithm
lattices
label code
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