Universal bound on the performance of lattice codes

Abstract

We present a lower bound on the probability of symbol error for maximum-likelihood decoding of lattices and lattice codes on a Gaussian channel. The bound is tight for error probabilities and signal-to-noise ratios of practical interest, as opposed to most existing bounds that become tight asymptotically for high signal-to-noise ratios. The bound is also universal; it provides a limit on the highest possible coding gain that may be achieved, at specific symbol erro. r probabilities, using any lattice or lattice code in n dimensions. In particular, it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains. The asymptotic (as n ! 1) behavior of the new bound is shown to coincide with the Shannon limit for Gaussian channels.

Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward

“…It is well known that the above probability expression depends on the shape of the Voronoi region. The bounds of such probability of error can be found in [16].…”

Sphere Lower Bound for Rotated Lattice Constellations in Fading Channels

“…In this Section, we recall the basics of the SLB for infinite lattices S = Λ [7], [8] and we apply it to bound P f (ρ). The first simplification stems from the geometrical uniformity of lattices, which implies that [7], [8] …”

“…Due to the circular symmetry of the Gaussian noise, replacing V Λ (h) by an N-dimensional sphere B(h) of the same volume and radius R(h) [6], yields the corresponding SLB on the lattice performance [7], [8] …”

“…The first simplification stems from the geometrical uniformity of lattices, which implies that [7], [8] …”

“…The SLB dates back to Shannon's work [6], and gives a lower bound to the error probability of spherical codes with a given length in the additive white Gaussian noise (AWGN) channel. The application of the SLB to infinite lattices and lattice codes was studied in [7], [8] for the AWGN channel. This SLB yields a lower bound to the error probability of infinite lattices regardless of the lattice structure.…”