In this paper, we show that the secrecy function conjecture that states that the maximum of the secrecy function of an l-modular lattice occurs at 1/√l is false, by proving that the four-modular lattice C(4)= ℤ ⊕√2ℤ ⊕ 2ℤ fails to satisfy this conjecture. After this, we indicate how the secrecy function must be modified in the l-modular case to provide a more meaningful comparison for l-modular lattices, and show that this new secrecy function indeed has a maximum at 1/√l for various two-modular lattices and for C(4). We conjecture that this must hold true for all l-modular lattices.

We show that the secrecy function conjecture that states that the maximum of the secrecy function of an l-modular lattice occurs at 1/√(l) is false, by proving that the 4-modular lattice C(4)= ℤ⊕√(2)ℤ ⊕ 2ℤ fails to satisfy this conjecture.

We study fast lattice decodability of space-time block codes for n transmit and receive antennas, written very generally as a linear combination Σi=12lsiAi, where the si are real information symbols and the Aiare n×n ℝ-linearly independent complex-valued matrices. We show that the mutual orthogonality condition AiAj* + AjAi* = 0 for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate (l = n2) transmission, the decoding complexity can be no better than |S|(n2+1), where |S| is the size of the effective real signal constellation. We also show that for full-rate transmission, g-group decodability, as defined by Jithamithra and Rajan, is impossible for any g ≥ 2. We then use the theory of Azumaya algebras to derive bounds on the maximum number of groups into which the basis matrices can be partitioned so that the matrices in different groups are mutually orthogonal-a key measure of fast decodability. We show that in general, this maximum number is of the order of only the 2-adic value of n. In the case where the matrices Aiarise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only 4. As a result, the decoding complexity for this case is no better than |S|⌈l/2⌉ for any rate l.

We study fast decodability for general rate space-time block codes. We derive bounds on the number of generators that can be in mutually orthogonal groups and use these to show that that the ML decoding complexity of a full-rate n × n space-time block code is unfortunately quite high: at least of the order of |S|n2+1, where S is the effective real constellation. We also show that g-group decodability is not possible for full-rate codes.

For the quasi-static, Rayleigh-fading multiple-input multiple-output (MIMO) channel with$n_{t}$transmit and$n_{r}$receive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and spatial-multiplexing gains, referred to as the diversity–multiplexing gain (D-MG) tradeoff. Subsequently, El Gamal, Caire, and Damen considered signaling across the same channel using an$L$-round automatic retransmission request (ARQ) protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information per use of the feedback channel. They showed that given a fixed number$L$of ARQ rounds and no power control, there is a tradeoff between diversity and multiplexing gains, termed the diversity–multiplexing–delay (DMD) tradeoff. This tradeoff indicates that the diversity gain under the ARQ scheme for a particular information rate is considerably larger than that obtainable in the absence of feedback.

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.

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.

For the quasi-static, Rayleigh-fading MIMO channel with nttransmit and nrreceive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and multiplexing gains, referred to as the diversity-multiplexing gain (D-MG) tradeoff. Explicit constructions for D-MG optimal ST codes are now available. In a subsequent paper, El Gamal, Caire and Damen considered signaling across the quasi-static ST channel using an L-round ARQ protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information (ACK or NACK) per use of the feedback channel. They showed that given a fixed number of ARQ rounds L, there is a tradeoff between diversity and multiplexing gains under which the optimum diversity gain of the ARQ channel transmitting R = r log(SNR) bits per channel use, is that of the quasi-static channel without feedback investigated by Zheng and Tse, transmitting at (1/L)th the information rate. This tradeoff, which now is a function of the number L of ARQ rounds, is termed the diversity-multiplexing gain-delay (DMD) tradeoff. In the current paper, a sufficient condition under which a ST code will achieve the DMD tradeoff is presented for the case nrges nt. The cyclic-division-algebra-based constructions of DMG optimal ST codes by Elia et. al. are then modified to yield codes which meet this sufficient criterion and are thereby DMD-optimal. This modification requires that either nt|L or L|nt

We construct full-rank, rate-n space-time block codes (STBC), over any a priori specified signal set for n-transmit antennas using crossed-product division algebras and give a sufficient condition for these STBCs to be information lossless. A class of division algebras for which this sufficient condition is satisfied is identified. Simulation results are presented to show that STBCs constructed in this paper perform better than the best known codes, including those constructed from cyclic division algebras and also to show that they are very close to the capacity of the channel with QAM input.

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.

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 their's.

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. Alamouti's (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.

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 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 /spl Copf/ as a maximal subfield.

An n/spl times/l (l/spl ges/n) space time block code (STBC) C over a complex signal set S consists of a finite number of n/spl times/l matrices with elements from 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 matrices, called the rank of the code. C is of full rank if its rank is n and is of minimum delay if l=n. The rate R, in bits per second per Hertz, of a full rank minimum delay code over S is upper bounded by log/sub 2/|S| and those meeting this bound are referred to as full rate codes. A full rank, full rate, minimum delay, space time block code over S is said to be rate-optimal. We present some general techniques for constructing rate-optimal codes from field extensions embedded in matrix rings. Working mostly with cyclotomic fields, we construct rate-optimal n/spl times/n STBCs over m-PSK signal sets for arbitrary values of m and a large set of values of n.

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.