A non-commutative analogue of the Odlyzko bounds and bounds on performance for space-time lattice codes

TL;DR

This paper develops non-commutative analogues of Odlyzko bounds to limit the coding gain of space-time lattice codes over multiple fading blocks, providing benchmarks for code design and performance analysis.

Contribution

It introduces non-commutative Odlyzko bounds for division algebra-based space-time codes, extending previous bounds from algebraic number fields to a more general setting.

Findings

Derived bounds limit the normalized coding gain for space-time codes.

Bounds serve as benchmarks for practical code design.

Tools for analyzing asymptotic performance as blocks increase.

Abstract

This paper considers space-time coding over several independently Rayleigh faded blocks. In particular we will concentrate on giving upper bounds for the coding gain of lattice space-time codes as the number of blocks grow. This problem was previously considered in the single antenna case by Bayer et al. in 2006. Crucial to their work was Odlyzko's bound on the discriminant of an algebraic number field, as this provides an upper bound for the normalized coding gain of number field codes. In the MIMO context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding (non-commutative) algebra. In this paper we will develop analogues of the Odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space-time codes. These bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases.