The nonassociative algebras used to build fast-decodable space-time block codes

TL;DR

This paper introduces new nonassociative algebras based on cyclic Galois extensions, used to construct fast-decodable, fully diverse space-time block codes with optimal diversity-multiplexing tradeoff for multiple antenna systems.

Contribution

It presents three families of nonassociative algebras for space-time coding, providing conditions for division algebras and a construction of rate-m codes with low decoding complexity.

Findings

Constructed a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas.

Provided conditions for the algebras to be division algebras.

Achieved ML-decoding complexity at most O(M^{15}) for the proposed codes.

Abstract

Let $K/F$ and $K/L$ be two cyclic Galois field extensions and $D=(K/F,\sigma,c)$ a cyclic algebra. Given an invertible element $d\in D$, we present three families of unital nonassociative algebras over $L\cap F$ defined on the direct sum of $n$ copies of $D$. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors. We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-$m$ for $nm$ transmit and $m$ receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most $\mathcal{O}(M^{15})$.