Abstract-We present some general techniques for constructing full-rank, minimal-delay, rate at least one space-time block codes (STBCs) over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras (field extensions) as well as using noncommutative division algebras of the rational field embedded in matrix rings. The first half of the paper deals with constructions using field extensions of . Working with cyclotomic field extensions, we construct several families of STBCs over a wide range of signal sets that are of full rank, minimal delay, and rate at least one appropriate for any number of transmit antennas. We study the coding gain and capacity of these codes. Using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas. We also present a method of constructing STBCs using noncyclotomic field extensions. In the later half of the paper, we discuss two ways of embedding noncommutative division algebras into matrices: left regular representation, and representation over maximal cyclic subfields. The 4 4 real orthogonal design is obtained by the left regular representation of quaternions. Alamouti's code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it. Also, we discuss a general principle for constructing cyclic division algebras using the th root of a transcendental element and study the capacity of the STBCs obtained from this construction. Another family of cyclic division algebras discovered by Brauer is discussed and several examples of STBCs derived from each of these constructions are presented.

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Abstract-Perfect space-time codes were first introduced by Oggier et. al. to be the space-time codes that have full rate, full diversity-gain, non-vanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These defining conditions jointly correspond to optimality with respect to the Zheng-Tse D-MG tradeoff, independent of channel statistics, as well as to near optimality in maximizing mutual information. All the above traits endow the code with error performance that is currently unmatched. Yet perfect space-time codes have been constructed only for 2, 3, 4 and 6 transmit antennas. We construct minimum and non-minimum delay perfect codes for all channel dimensions.

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen et al. proposed a design for 2 transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting STBC when used with a signal set an informationlossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an apriori specified signal set. Alamouti code and quasi-orthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossed-product algebras are information lossless. We give some classes of crossed-product algebras, from which the STBCs obtained are information lossless and also of fullrank. We present some simulation results for two, three and four transmit antennas to show that our STBCs perform better than some of the best known STBCs and also that these STBCs are approximately 1 dB away from the capacity of the channel with QAM symbols as input.

In M k , k an algebraically closed field, we call a matrix l-regular if each n eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Z Z obtained from matrices over truncated l, m polynomial rings. We prove that these varieties Z Z are irreducible and apply this l, m to the case of the k-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at most n. We also show that such an algebra is always contained in a commutative subalge-Ž . bra of M k of dimension exactly n. ᮊ 1999 Academic Press n * We thank Bob Guralnick for some useful discussions and for making many clarifying suggestions, particularly in Section 5. This revised version was written when the second author was visiting the University of Southern California, and he thanks the department there for its hospitality.

In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4 × 4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M n (F), and show that it is irreducible of dimension n 2 − n for n ≤ 4, but reducible, of dimension greater than n 2 − n for n ≥ 7.

Abstract-It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen et al., gave a design for 2 transmit antennas, which achieves capacity for any number of receive antennas, calling it an information lossless STBC. In this paper, we construct capacity achieving designs using cyclic division algebras for arbitrary number of transmit and receive antennas. For the STBCs obtained using these designs we present simulation results for those number of transmit and receive antennas for which Damen et al. also give and show that our STBCs perform better than their's.

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m × n (m 6 n) matrices over rings of the form F[t]=(t k ), and for some ÿxed r, setting the coe cients of powers of t of all r × r minors to zero. These varieties can be interpreted as spaces of (k − 1)th order jets over the classical determinantal varieties; a special case of these varieties ÿrst appeared in a problem in commuting matrices. We show that when r = m, the varieties are irreducible, but when r ¡ m, these varieties are reducible. We show that when r = 2 ¡ m (any k), there are exactly k=2 + 1 components, which we determine explicitly, and for general r ¡ m, we show there are at least k=2 + 1 components. We also determine the components explicitly for k = 2 and 3 for all values of r (for k = 3 for all but ÿnitely many pairs of (m; n)).