Bounds of fast decodability of space time block codes, skew-Hermitian matrices, and Azumaya algebras

TL;DR

This paper investigates the limits of fast decodability for space-time block codes, establishing necessary conditions for orthogonality, bounds on group decodability using Azumaya algebras, and implications for decoding complexity.

Contribution

It provides new bounds on the maximum number of orthogonal groups in space-time codes using Azumaya algebra theory, and clarifies the limitations of g-group decodability.

Findings

Orthogonality condition is necessary and sufficient for fast decodability.

Maximum number of orthogonal groups is limited by the 2-adic value of n.

Decoding complexity cannot be improved beyond certain bounds for full-rate codes.

Abstract

We study fast lattice decodability of space-time block codes for $n$ transmit and receive antennas, written very generally as a linear combination $\sum_{i=1}^{2l} s_i A_i$, where the $s_i$ are real information symbols and the $A_i$ are $n\times n$ $\mathbb R$-linearly independent complex valued matrices. We show that the mutual orthogonality condition $A_iA_j^* + A_jA_i^*=0$ for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate ($l = n^2$) transmission, the decoding complexity can be no better than $|S|^{n^2+1}$, where $|S|$ is the size of the effective real signal constellation. We also show that for full-rate transmission, $g$-group decodability, as defined in [1], is impossible for any $g \ge 2$. We then use the theory of Azumaya algebras to derive bounds on the maximum number of groups into which the basis matrices can be partitioned so that the matrices in different groups are mutually orthogonal---a key measure of fast decodability. We show that in general, this maximum number is of the order of only the $2$-adic value of $n$. In the case where the matrices $A_i$ arise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only $4$. As a result, the decoding complexity for this case is no better than $|S|^{\lceil l/2 \rceil}$ for any rate $l$.