Space-Time Codes from Crossed Product Algebras of Degree 4
We study crossed product algebras of degree 4, and present a new space-time code construction based on a particular crossed product division algebra which exhibits very good performance.
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chool of Mathematics University of Southampton, UK [email protected] 2 Department of Electrical Engineering California Institute of Technology, USA [email protected]
Abstract. We study crossed product algebras of degree 4, and present a new space-time code construction based on a particular crossed product division algebra which exhibits very good performance.
1
Introduction
Wireless systems are nowadays part of every day life. However, to answer the need of higher and higher data rate, researchers have started to investigate wireless systems where both the transmitter and receiver end are equipped with multiple antennas. This new kind of channel required new coding techniques, namely space-time coding [10]. Unlike classical coding, space-time coding involves the design of families of matrices, with the property, called full diversity, that the difference of any two distinct matrices is full rank. Following the seminal work of Sethuraman et al. [7,8], codes based on division algebras have been investigated. This algebraic approach has generated a lot of interest, since division algebras naturally provide linear codes with full diversity. Quaternion algebras [1] and their maximal orders [3], cyclic algebras [8,4], Clifford algebras [9] and crossed product algebras [6] have been studied. In this paper, we study crossed product algebras of degree 4, and, unlike in [6], we focus on the case where the Galois group is not cyclic. For this scenario, we derive conditions for crossed product algebras to be division algebras, which yields the full diversity property, and optimize the code design.
2
Crossed Product Algebras of Degree 4
Let L/K be a Galois extension. A central simple K-algebra is called a crossed product algebra over L/K if it contains L as a maximal commutative subfield. A crossed product algebra can be described nicely in terms of generators and
This work was partly supported by the Nuffield Newly Appointed Lecturers Scheme 2006 NAL/32706, F. Oggier is now visiting RCIS, AIST, Tokyo, Japan.
S. Bozta¸s and H.F. Lu (Eds.): AAECC 2007, LNCS 4851, pp. 90–99, 2007. c Springer-Verlag Berlin Heidelberg 2007
Space-Time Codes from Crossed Product Algebras of Degree 4
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√ √ L = K( d, d )
√
2
K( d)
HH τ
H H
2 HH H
K
σ
√ K( d )
Fig. 1. A biquadratic extension of K
relations, and when L/K has cyclic Galois group, we recover the concept of cyclic algebra. Since in degree 2 and 3 Galois extensions have necessarily cyclic groups, the first interesting example of crossed product algebra arises in degree 4. This is the case we focus on in this work. For definitions and basic facts on crossed product algebras, the reader may refer to [2]. 2.1
Definition and Examples
Consider a Galois extension L/K of degree 4. Its Galois group is either cyclic of order 4 or a product of two cyclic groups of order 2. We focus on the latter, and consider the case where L/K is a biquadratic extension (see Fig. 1), namely √ √ L = K( d, d ). We set G = Gal(L/K) = {1, σ, τ, στ }, where σ, τ are d
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