Linear-time nearest point algorithms for Coxeter lattices

Details Linear-time nearest point algorithms for Coxeter lattices Linear-time nearest point algorithms for Coxeter lattices

2009-03-04

arxiv.org/abs/0903.0673v1

Computer Science

Information Theory

submitted to IEEE Transactions on Information Theory

Scientific paper

The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices containing many of the important lattices in low dimensions. This includes $A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexity $O(n\log{n})$ and the other with worst case complexity O(n) where $n$ is the dimension of the lattice. We show that for the particular lattices $A_n$ and $A_n^*$ the algorithms reduce to simple nearest point algorithms that already exist in the literature.

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