Almost universal codes achieving ergodic MIMO capacity within a constant gap

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. A design criterion based on the normalized minimum determinant is proposed for division algebra multiblock space-time codes over fading channels; this plays a similar role to the Hermite invariant for Gaussian channels. It is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for slow fading channels and ergodic fading channels. This performance is achieved both under maximum likelihood decoding and naive lattice decoding. In the case of independent identically distributed Rayleigh fading, it is also shown that the error probability vanishes exponentially fast. 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. 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.

Polar Codes and Polar Lattices for Independent Fading Channels

“…A more recent work showed that number field lattices are able to achieve the Gaussian and the Rayleigh channel capacity within a constant gap [33]. This scheme is universal and extended to the multi-input and multi-output (MIMO) context [34].…”

Lattice Coding and Decoding for Multiple-Antenna Ergodic Fading Channels

“…Furthermore, for the special case of SISO channels the gap to capacity is characterized for all SNR values and over a wide range of fading distributions. Unlike [23], the proposed scheme achieves positive rates at low SNR where the gap to capacity vanishes. At moderate and high SNR, the gap to capacity is bounded by a constant that is independent of SNR and only depends on the fading distribution.…”

“…Vituri [22] studied the performance of lattice codes with unbounded power constraint under regular fading channels. Recently, Luzzi and Vehkalahti [23] showed that a class of lattices belonging to a family of division algebra codes achieve rates within a constant gap to the ergodic capacity at all SNR, where the gap depends on the algebraic properties of the code as well as the antenna configuration. Unfortunately, the constant gap in [23] can be shown to be quite large at many useful antenna configurations, in addition to requiring substantial transmit power to guarantee any positive rate.…”

On the Separability of Ergodic Fading MIMO Channels: A Lattice Coding Approach

“…In [25,Section 4.5] Vituri analyzed the performance of lattice codes under fading channels without power constraint. Under ergodic fading and CSIR only, Luzzi and Vehkalahti [26] recently showed that a class of lattices belonging to a family of division algebra codes achieve rates within a constant gap to capacity, however, this gap can be large. In [27], a lattice coding scheme was proposed whose decoder does not depend on the fading realizations, achieving rates within a constant gap to capacity.…”