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In this paper, the need for the construction of asymmetric and multiblock space-time codes is discussed. Above the trivial puncturing method, i.e., switching off the extra layers in the symmetric multiple-input multiple-output (MIMO) setting, two more sophisticated asymmetric construction methods are proposed. The first method, called the block diagonal method (BDM), can be converted to produce multiblock space--time codes that achieve the diversity--multiplexing tradeoff (DMT). It is also shown that maximizing the density of the newly proposed block diagonal asymmetric space--time (AST) codes is equivalent to minimizing the discriminant of a certain order, a result that also holds as such for the multiblock codes. An implicit lower bound for the density is provided and made explicit for an important special case that contains e.g., the systems equipped with 4Tx + 2Rx antennas. Further, an explicit scheme achieving the bound is given. Another method proposed here is the Smart Puncturing Method (SPM) that generalizes the subfield construction method proposed in earlier work by Hollanti and Ranto and applies to any number of transmitting and lesser receiving antennas. The use of the general methods is demonstrated by building explicit, sphere decodable codes using different cyclic division algebras (CDAs). Computer simulations verify that the newly proposed methods can compete with the trivial puncturing method, and in some cases clearly outperform it. The conquering construction exploiting maximal orders improves upon the punctured perfect code and the DjABBA code as well as the Icosian code. Also extensive DMT analysis is provided. Index Terms--Asymmetric space--time block codes (ASTBCs), cyclic division algebras (CDAs), dense lattices, discriminants, diversity--multiplexing tradeoff (DMT), maximal orders, multiblock codes, multiple-input multiple-output (MIMO) channels, normalized minimum determinant.

A new construction of good, easily encodable, and soft-decodable codes is proposed in this paper. The construction is based on serially concatenating several simple 1+D convolutional codes as the outer code, and a rate-1 1/(1 + D) accumulate code as the inner code. These codes have very low encoding complexity and require only one shift-forward register for each encoding branch. The input-output weight enumerators of these codes are also derived. Divsalar's simple bound technique is applied to analyze the bit error rate performance, and to assess the minimal required signal-to-noise ratio (SNR) for these codes to achieve reliable communication under AWGN channel. Simulation results show that the proposed codes can provide good performance under iterative decoding. Index Terms--Low-density parity-check (LDPC) codes, accumulate codes, convolutional codes.

In this paper, we construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, nonvanishing determinants (NVDs) for four transmit antenna multiple-input-single-output (MISO) space-time (ST) applications. The constructions are based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions and can be extended to a larger number of Tx antennas. The usage of ideals guarantees an NVD larger than one and an easy way to present the exact proofs for the minimum determinants. The idea of finding denser sublattices within a given division algebra is then generalized to a multiple-input-multiple-output (MIMO) case with an arbitrary number of Tx antennas by using the theory of cyclic division algebras (CDAs) and maximal orders. It is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding. Related to the decoding complexity, the notion of sensitivity is introduced, and experimental evidence indicating a connection between sensitivity, decoding complexity, and performance is provided. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi-orthogonal ABBA lattice as well as the diagonal algebraic space-time (DAST) lattice. We also show that our quaternionic lattice is better than the DAST lattice in terms of the diversity-multiplexing gain tradeoff (DMT). Index Terms--Cyclic division algebras (CDAs), dense lattices, maximal orders, multiple-input-multiple-output (MIMO) channels, multiple-input-single-output (MISO) channels, number fields, quaternions, space-time block codes (STBCs), sphere decoding.

Constructions of multiblock space-time coding schemes that are optimal with respect to diversity-multiplexing (D-M) tradeoff when coding is applied over any number of fading blocks are presented in this correspondence. The constructions are based on a left-regular representation of elements in some cyclic division algebra. In particular, the main construction applies to the case when the quasi-static fading interval equals the number of transmit antennas, hence the resulting scheme is termed a minimal delay multiblock space-time coding scheme. Constructions corresponding to the cases of nonminimal delay are also provided. As the number of coded blocks approaches infinity, coding schemes derived from the proposed constructions can be used to provide a reliable multiple-input multiple-output (MIMO) communication with vanishing error probability. Index Terms--Cyclic-division algebras, diversity-multiplexing (D-M) tradeoff, fading channels, multiblock space-time codes, multiple-input multiple-output (MIMO) channels, number fields, space-time codes.

A new construction of optical orthogonal codes is provided in this correspondence which is a generalization of the well-known construction of distinct difference set (DDS) by Bose and Chowla. This construction is optimal with respect to the Johnson bound and has parameters n = [q.sup.a] - 1. [omega] = q, and [lambda] = 1. Index Terms--Distinct difference set (DDS), optical code-division multiple access (OCDMA), optical orthogonal code (OOC), optical CDMA.

Constructions of space-frequency (SF) codes for multiple-input multiple-output (MIMO)-orthogonal frequency-division multiplexing (OFDM) systems with [n.sub.t] transmit antennas and Q subcarriers are considered in this paper. Following the pairwise-error-probability analysis, it is known that in addition to the conventional rank distance criterion, the minimum column distance of ([n.sub.t] x Q) SF codes serves as another benchmark in code design. SF codes with larger minimum column distance are expected to have better performance. Following this principle, the rate--diversity tradeoff for the MIMO-OFDM channels as well as two SF code constructions are presented. The first construction is obtained by right-multiplying the code matrices in a maximal rank-distance (MRD) code by a fixed (Q x Q) nonsingular matrix. Codes obtained from this construction are called linearly transformed MRD (LT-MRD) codes. Minimum column distance of the LT-MRD codes, when averaged over all code ensembles, is shown to meet the Gilbert-Varshamov bound. For the case of constructing the (2 x 256) quadrature phase-shift keying (QPSK)-modulated SF codes, it is shown that the LT-MRD codes can provide a much larger minimum column distance at the value of [greater than or equal to] 50, compared to the values of 3, 5, or 6 obtained by other available constructions. The second code construction, termed cyclotomic construction, is reminiscent of the construction of the Reed-Solomon codes except that the code polynomials are now selected according to the cyclotomic cosets of the underlying field. Exact minimum rank distances of the resultant codes are presented. It is shown that this newly constructed code is asymptotically optimal in terms of rate-diversity tradeoff. Bounds on the minimum column distance of these codes are also given. Index Terms--Algebraic code designs, cyclotomic cosets, linearly transformed maximal rank distance (LT-MRD) codes, multiple-input multiple-output (MIMO), orthogonal frequencydivision multiplexing (OFDM), space-frequency codes, space-time (ST) codes.

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space--time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for T [greater than or equal to] [n.sub.t] + [n.sub.r], - 1 where T is the number of time slots over which coding takes place and [n.sub.t], [n.sub.r] are the number of transmit and receive antennas, respectively. For T < [n.sub.t] + [n.sub.r], - 1, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay ease when T = [n.sub.t] = n, cyclic-division-algebra (CDA)-based ST codes having the nonvanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. For the rectangular, T > [n.sub.t] ease, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all T [greater than or equal to] [n.sub.t] is the same as that previously known to hold for T [greater than or equal to] [n.sub.t] + [n.sub.r] - 1. Index Terms--Cyclic division algebra, diversity-multiplexing gain tradeoff, explicit construction, space-time codes.

Proofs to the conjectures made by Jing and Hassibi on having fully diverse (3 x 3) SU(3) and AB unitary space--time codes are presented in this correspondence. We first prove that the SU(3) codes are fully diverse if and only if the design parameters P, Q, R, and S are all odd integers, and in addition, are relatively prime. For the type I AB codes, it is shown that full diversity can be achieved if and only if the integers P, Q, R, and S are relatively prime. Finally, we show that such condition is also sufficient for having fully diverse type II AB codes. Index Terms--Algebraic number theory, cyclotomic number field, Lie group, multiple-antenna system, unitary space--time code.

Constructions of space--time codes having amplitude-modulated phase-shift keying (AM-PSK) constellations are presented in this paper. The first construction, termed [??]-radii construction, is obtained by extending Hammons' dyadic dual-radii construction to the cases when the size of the constellation is a power of a prime [??], [greater than or equal to] 2. The resultant code is optimal with respect to the rate--diversity tradeoff and has an AM-PSK constellation with signal points distributed over [??]-concentric circles in the complex plane, i.e., there are [??] radii. Also contained in this paper is the identification of rich classes of nontrivial subset-subcodes of the newly constructed space--time codes and it is shown that these subset-subcodes are again, all optimal. Finally, a new generalization of the super-unified construction by Hammons is presented. It is shown that codes obtained from several previously known constructions are subset-subcodes of the one derived from this generalized construction. Index Terms--Algebraic code designs, algebraic integers, amplitude-modulated phase-shift keying (AM-PSK) constellation, Dobinski-type summations, multiple-input multiple-output (MIMO), space--time codes, subset-subcodes.

This correspondence presents a new signal mapper that maps the maximal rank distance codes to space--time (ST) codes with amplitude modulation phase-shift keying (AM-PSK) constellations. It is shown that this new mapper is rank-distance preserving. Comparing to the multi-radii construction proposed by Hammons, this new mapper has linear increase in the radii and the resulting signal constellations have larger minimum distance and lower peak-to-average power ratio. Variations of this new mapper are also given to provide ST codes with rotated AM-PSK constellations. Index Terms--Amplitude modulation phase-shift keying (AM-PSK) constellation, diversity gain advantage, multiple antennas, multiple input multiple output (MIMO), rate-diversity tradeoff, space--time (ST) codes, unified construction.

The problem of constructing space--time (ST) block codes over a fixed, desired signal constellation is considered. In this situation, there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code. The transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal-to-noise ratio (SNR). In the setting of a quasi-static channel model, let [n.sub.t] denote the number of transmit antennas and T the block interval. For any [n.sub.t] [less than or equal to] T, a unified construction of ([n.sub.t] x T) ST codes is provided here, for a class of signal constellations that includes the familiar pulse-amplitude (PAM), quadrature-amplitude (QAM), and [2.sup.K]-ary phase-shift-keying (PSK) modulations as special cases. The construction is optimal as measured by the rate--diversity tradeoff and can achieve any given integer point on the rate-diversity tradeoff curve. An estimate of the coding gain realized is given. Other results presented here include i) an extension of the optimal unified construction to the multiple fading block case, ii) a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes, iii) the providing of a linear dispersion form for the underlying binary block codes, iv) a Gray-mapped version of the unified construction, and v) a generalization of construction of the [??]-ary case corresponding to constellations of size [[??].sup.-K]. Items ii) and iii) are aimed at simplifying the decoding of this class of ST codes. Index Terms--Diversity gain advantage, multiple antennas, multiple-input multiple-output (MIMO), rate-diversity tradeoff, space--time (ST) codes, unified construction.

This correspondence presents a new asymptotically exact lower bound on pairwise error probability of a space-time code as well as an example code that outperforms the comparable orthogonal-design-based space-time (ODST) code. Also contained in the correspondence are an exact expression for pairwise error probability (PEP), signal design guidelines, and some observations relating to the reception of ODST codes. Index Terms--Orthogonal design, pairwise error probability (PEP), space-time codes.

In this correspondence, we show that for any (Q x M) space-time code S having a fixed, finite signal constellation, there is a tradeoff between the transmission rate R and the transmit diversity gain v achieved by the code. The tradeoff is characterized by R [less than or equal to] Q - v + 1, where Q is the number of transmit antennas. When either binary phase-shift keying (BPSK) or quaternary phase-shift keying (QPSK) is used as the signal constellation, a systematic construction is presented to achieve the maximum possible rate for every possible value of transmit diversity gain. Index Terms--Rate-diversity tradeoff, space-time codes.