Non-Random Coding Error Exponent for Lattices

TL;DR

This paper introduces a new upper bound on lattice error probability based on their distance-spectrum, leading to an error-exponent for lattice sequences over AWGN channels, which helps measure their capacity gap.

Contribution

It presents a novel bound for lattice error probability that differs from traditional methods and derives an error-exponent for lattice sequences over AWGN channels.

Findings

New bound resembles Shulman-Feder bound for linear codes

Derived error-exponent for lattice sequences over AWGN

Demonstrated measurement of capacity gap using the new exponent

Abstract

An upper bound on the error probability of specific lattices, based on their distance-spectrum, is constructed. The derivation is accomplished using a simple alternative to the Minkowski-Hlawka mean-value theorem of the geometry of numbers. In many ways, the new bound greatly resembles the Shulman-Feder bound for linear codes. Based on the new bound, an error-exponent is derived for specific lattice sequences (of increasing dimension) over the AWGN channel. Measuring the sequence's gap to capacity, using the new exponent, is demonstrated.