Division algebra codes achieve MIMO block fading channel capacity within a constant gap

TL;DR

This paper demonstrates that division algebra-based lattice codes can approach the capacity of multi-antenna block fading channels within a constant gap, using number theory for code construction.

Contribution

It introduces a novel multiblock code construction based on division algebras that achieves near-capacity performance, contrasting with prior random lattice ensemble methods.

Findings

Codes achieve rates within a constant gap of capacity

Discriminant of division algebra influences the gap size

Number theory enables explicit code constructions with small discriminants

Abstract

This work addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. In contrast to the standard approach in the literature which employs random lattice ensembles, the existence results in this paper are derived from number theory. It is shown that a multiblock construction based on division algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. First the gap to capacity is shown to depend on the discriminant of the chosen division algebra; then class field theory is applied to build families of algebras with small discriminants. The key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants.