Universal Lattice Codes for MIMO Channels

Abstract

We propose a coding scheme that achieves the capacity of the compound MIMO channel with algebraic lattices. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned channel realizations. To shape the constellation, a discrete Gaussian distribution over the lattice points is applied. These techniques, along with algebraic properties of the proposed lattices, are then used to construct a sub-optimal de-coupled coding schemes that achieves a gap to compound capacity by decoding in a lattice that does not depend of the channel realization. The gap is characterized in terms of algebraic invariants of the codes, and shown to be significantly smaller than previous schemes in the literature. We also exhibit alternative algebraic constructions that achieve the capacity of ergodic fading channels.

Semantically Secure Lattice Codes for Compound MIMO Channels

“…Proof. The proof is analogous to the quantization argument for the probability of error in [29], which, in turn follows [36]. (i) Fixed H e .…”

“…The key method is discrete Gaussian shaping and a "direct" proof of the universal flatness of the eavesdropper's lattice. This method is similar to that used in [29] to approach the capacity of compound MIMO channels so that the present paper can be considered a companion paper of [29] for wiretap channels. Note that [28] used an "indirect" proof, which was based on an upper bound on the smoothing parameter in terms of the minimum distance of the dual lattice.…”

“…It was shown in [29] that if X ∼ D Λ T b ,σs , then the maximum-a-posteriori (MAP) decoder for the signal Y b is equivalent to lattice decoding of F b Y b , where F b is the MMSE-GDFE matrix to be defined in the sequel. We cannot claim directly that X ∼ D Λ T b ,σs , since the message distribution in M T need not be uniform.…”

“…bounded and if we choose a universally good lattice, the probability vanishes for all possible R b . This is possible [29] provided that…”

“…More refined "direct" constructions can be obtained by using number theory and prime ideals of Z[i]. For instance, if Λ base is the embedding of the ring of integers of a number field and ψ is the reduction modulo a prime ideal we can recover the constructions in [29]. Notice that, for this construction, if C 1 ⊂ C 2 , we obtain two nested lattices Λ(C 1 ) ⊂ Λ(C 2 ).…”