An algebraic description of orthogonal designs and the uniqueness of the Alamouti code

An n × l (l ≥ n) space time block code (STBC) C consists of a finite number |C| of n × l matrices with entries from the complex field. If the entries of the codeword matrices are from a complex signal set S or complex linear combinations of elements of S then the code is said to be over S. For quasi-static, flat fading channels a primary performance index of C is the minimum of the rank of the difference of any two codeword matrices, called the rank of the code. C is of full-rank if its rank is n and is of minimal-delay if l = n. The rate of the code R in symbols per channel use is given by 1/l log_|S|(|C|). It is well known that orthogonal designs provide rate 1, mimial-delay, full-rank STBCs with linear decodability, but exist only for n = 2 (Alamouti code) for complex constellations and for n = 2, 4 and 8 only for real constellations. In this paper, we present some general techniques for constructing rate 1, full-rank, minimal-delay STBCs over S using non-commutative division algebras of the rational field Q embedded in matrix rings. Using two ways of embedding non-commutative division algebras into matrices, namely, the left regular representation and representation over maximal cyclic subfields, we observe that (i) the Alamouti´s (1998) code and real orthogonal designs for n = 2 and 4 are just special cases of our construction and (ii) algebraically, the uniqueness of the Alamouti code is due to the fact: Hamilton´s quaternions H is the only non-commutative division algebra which has C as a maximal subfield.