Closest point search in lattices

Abstract

In this semi-tutorial paper, a comprehensive survey of closest-point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest-point search algorithm, based on the Schnorr-Euchner variation of the Pohst method, is implemented. Given an arbitrary point x ∈ R m and a generator matrix for a lattice Λ, the algorithm computes the point of Λ that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent Viterbo-Boutros decoder. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, computing the Voronoi-relevant vectors, and finding a Korkine-Zolotareff reduced basis.

An Improved Soft Decision Based MIMO Detection Using Lattice Reduction

“…This MIMO detection problem is also known as closest point problem in [14] that represents a search through the set of all possible lattice points. Each antenna provides a 2-level search: one for detecting real part and the other for imaginary part of transmitted symbol of that antenna.…”

VLSI Implementation of Hard- and Soft-Output Sphere Decoding for Wide-Band MIMO Systems

“…Sphere-Decoding Algorithm The SD algorithm [5][6][7][8][9] starts with the QR decomposition (QRD) of the channel matrix H = QR, where the M R × M T matrix Q satisfies Q H Q = I MT , and the M T ×M T matrix R is upper-triangular. The QRD enables us to rewrite the ML-detection problem (2) as follows:…”

“…The ML solution (3) corresponds to the path through the tree starting by the root and leading to the leaf associated with the smallest PED. The basic ideas underlying the SD algorithm, as described in [9,12], are briefly summarized as follows: The search in the tree is constrained to nodes which lie within a radius r aroundỹ (and hence, nodes from the tree are pruned for which d i s (i) is larger than r) and tree traversal is performed depth-first, visiting the children of a given node in ascending order of their PEDs. The method using this enumeration scheme is also known as the Schnorr-Euchner (SE) SD algorithm [6].…”

“…We emphasize that the LLRs in (9) are normalized with respect to the noise variance N o in order to get rid of the factor 1/N 0 on the right hand side (RHS) of (9). This normalization simplifies the exposition and does not degrade the error rate performance with max-log-based channel decoders (see [11] for the details).…”

“…Consequently, to meet the practically important requirement of a fixed throughput, the algorithm's run-time must be constrained. This, in turn, leads to a constraint on the maximum detection effort or, equivalently, to a constraint on the maximum number of nodes that the SD is allowed to visit, which will clearly prevent the detector from achieving ML performance or to be able to deliver the exact max-log LLRs in (9). It is therefore of paramount importance to find a way of imposing run-time constraints while keeping the resulting performance degradation at a minimum.…”