Tail Behavior of Sphere-Decoding Complexity in Random Lattices

TL;DR

This paper investigates the tail behavior of sphere-decoding complexity in random lattices, revealing that it is primarily influenced by the inverse volume of the fundamental region, with implications for Gaussian lattices and lattice reduction techniques.

Contribution

It establishes a general relationship between sphere-decoding complexity tail behavior and lattice fundamental volume, and shows lattice reduction does not improve the tail exponent.

Findings

Complexity tail behavior is determined by inverse fundamental volume.

For Gaussian lattices, the tail follows a Pareto distribution with specific exponent.

Lattice reduction does not improve the tail exponent.

Abstract

We analyze the (computational) complexity distribution of sphere-decoding (SD) for random infinite lattices. In particular, we show that under fairly general assumptions on the statistics of the lattice basis matrix, the tail behavior of the SD complexity distribution is solely determined by the inverse volume of a fundamental region of the underlying lattice. Particularizing this result to NxM, N>=M, i.i.d. Gaussian lattice basis matrices, we find that the corresponding complexity distribution is of Pareto-type with tail exponent given by N-M+1. We furthermore show that this tail exponent is not improved by lattice-reduction, which includes layer-sorting as a special case.