B. A. Sethuraman

Full-diversity, high-rate space-time block codes from division algebras

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 /spl Qopf/ embedded in matrix rings. The first half of the paper deals with constructions using field extensions of /spl Qopf/. 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/spl times/4 real orthogonal design is obtained by the left regular representation of quaternions. Alamoutis (1998) 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 nth 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 (1933) is discussed and several examples of STBCs derived from each of these constructions are presented.

Perfect Space–Time Codes for Any Number of Antennas

In a recent paper, perfect (n times n) space-time codes were introduced as the class of linear dispersion space-time (ST) codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions, and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the Zheng-Tse diversity-multiplexing gain tradeoff (DMT), as well as excellent low signal-to-noise ratio (SNR) performance. Yet perfect space-time codes have been constructed only for two, three, four, and six transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of ST codes, and extend the notion of a perfect code to the rectangular case.

Perfect space-time codes with minimum and non-minimum delay for any number of antennas

We here introduce explicit constructions of minimum-delay perfect space-time codes for any number n/sub t/ of transmit antennas and any number n/sub t/ of receive antennas. We also proceed to construct non-minimal delay perfect space-time codes for any n/sub t/, n/sub r/ and any block length T /spl ges/ n/sub t/. Perfect space-time codes were first introduced in F. Oggier et al. (2004) for dimensions of 2 /spl times/ 2, 3 /spl times/ 3, 4 /spl times/ 4 and 6 /spl times/ 6, 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. The term perfect corresponds to the fact that the code simultaneously satisfies all the above mentioned important criteria. As a result, perfect codes have proven to have extraordinary performance. Finally, we point out that the set of criteria in F. Oggier et al. (2004) of non-vanishing determinant, full diversity, and full rate, is a subset of the more general and more strict set of criteria for optimally in the diversity-multiplexing gain (D-MG) tradeoff [Elias, P. et al., 2004], [Kumar, K.R. et al., 2005], an approach that takes special significance when constructing non-minimal delay perfect codes. Both minimum and non-minimum delay perfect codes are shown to be D-MG optimal.

Information-Lossless Space–Time Block Codes From Crossed-Product Algebras

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 proposed a design for two transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting space-time block code (STBC) when used with a signal set an information-lossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori 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 full rank. 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 quadrature amplitude modulation (QAM) symbols as input

Commuting pairs and triples of matrices and related varieties

Abstract In this note, we show that the set of all commuting d -tuples of commuting n×n matrices that are contained in an n -dimensional commutative algebra is a closed set, and therefore, Gerstenhabers theorem on commuting pairs of matrices is a consequence of the irreduciblity 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 .

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

An n /spl times/ l (l /spl ges/ n) space time block code (STBC) C consists of a finite number |C| of n /spl times/ 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/sub |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 /spl Qopf/ 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 Alamoutis (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: Hamiltons quaternions H is the only non-commutative division algebra which has /spl Copf/ as a maximal subfield.

STBCs using capacity achieving designs from cyclic division algebras

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that only for the case of two transmit and one receive antennas. M.O. Damen et al. (see IEEE Trans. Inform. Theory, vol.48, no.3, p.753-60, 2002) gave a design for 2 transmit antennas, which achieves capacity for any number of receive antennas, calling it an information lossless STBC. We construct capacity achieving designs using cyclic division algebras for an arbitrary number of transmit and receive antennas. For the STBCs obtained using these designs, we present simulation results for those numbers of transmit and receive antennas for which Damen et al. also gave results, and show that our STBCs perform better than theirs.

Information-lossless STBCs from crossed-product algebras

This work presents the construction of STBCs, using crossed-product algebras, for arbitrary number of transmit antennas over an a priori specified signal set. It obtains a condition under which these STBCs from arbitrary crossed-product algebras are information-lossless.

STBC from field extensions of the rational field

The construction of space time block code (STBCS) over symmetric m-PSK (m-arbitrary) signal sets for a restricted set of number of antennas n is presented by Sethuraman and Sundar Rajan (see ICC 2002, New York, 2002), using cyclotomic field extensions of the field of rationals /spl Qopf/. We extend the method of Sethuraman et al. and achieve construction of minimal-delay, full-rank, rate-1 STBCs over (i) certain asymmetric m-PSK (m-arbitrary) signal sets, QAM and cross-constellations, (ii) certain rotationally invariant signal sets, for arbitrary number of antennas, using transcendental extensions of /spl Qopf/ and (iii) using non-cyclotomic field extensions of /spl Qopf/ for arbitrary number of antennas.

Full-rank, full-rate STBCs from division algebras

Construction of rate-optimal full-diversity space-time block codes (STBC) over symmetric PSK signal sets using cyclotomic field extensions of the field of rational /spl Qopf/ was reported previously, and for a variety of other signal sets using non-cyclotomic field extensions. Fields are commutative division algebras. Construction of full-rate STBCs with full-diversity using a class of non-commutative division algebras (cyclic division algebras) was also reported previously and the Alamouti code was shown to be a special case with an algebraic uniqueness property. In this paper, we present the basic principle behind these constructions and also obtain full-diversity, full-rate STBCs using a different class of non-commutaive division algebra constructed by Brauer. The interrelationship between codes constructed from division algebras, linear dispersion codes and codes from orthogonal designs is discussed.