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 $\mathcal{O}(M^{15})$.
Highlights
Space-time block codes (STBCs) are used for reliable high rate transmission over wireless digital channels with multiple antennas at both the transmitter and receiver ends
Several different constructions of nonassociative algebras appeared in the literature on fast decodable space-time block code (STBC), cf. for instance Markin and Oggier [1], Srinath and Rajan [2], or [4], [5], [8], [9]
A space-time block code (STBC) for an nt transmit antenna multiple input multiple output (MIMO) system is a set of complex nt × T matrices, called codebook, that satisfies a number of properties which determine how well the code performs
Summary
Space-time block codes (STBCs) are used for reliable high rate transmission over wireless digital channels with multiple antennas at both the transmitter and receiver ends. The family of matrices representing left multiplication in any of the three cases can be used to define a STBC C, which is fully diverse if and only if A is division, and fast-decodable for the right choice of D. The algebras Itn(D, τ, d) and ItnR(D, τ, d) appear when designing fast-decodable asymmetric multiple input double output (MIDO) codes: ItnR(D, τ, d) is implicitly used in [2] but not mentioned there, the algebras Itn(D, τ, d) are canonical generalizations of the ones behind the iterated codes of [1], and are employed in [4] Both times they are used to design fast decodable rate-2 MIDO space-time block codes with n antennas transmitting and 2 antennas receiving the data. The suggested codes have maximal rate in terms of the number of complex symbols per channel use (cspcu)
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