Roope Vehkalahti

On the Densest MIMO Lattices From Cyclic Division Algebras

It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory, a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving the bound are given. For example, a matrix lattice with quadrature amplitude modulation (QAM) coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant is constructed. Also, a general algorithm due to Ivanyos and Ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed. Also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed.

Fast-Decodable Asymmetric Space-Time Codes From Division Algebras

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4×2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4×4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the nonvanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes [5], [6]. Several explicit constructions are presented and shown to have excellent performance through computer simulations.simulations.

On the algebraic structure of the Silver code: A 2 × 2 perfect space-time block code

Recently, a family of full-rate, full-diversity space-time block codes (STBCs) for 2 times 2 multiple-input multiple-output (MIMO) channels was proposed in the works of Tirkkonen et al., using a combination of Clifford algebra and Alamouti structures, namely twisted space-time transmit diversity code. This family was recently rediscovered by Paredes et al., and they pointed out that such STBCs enable reduced-complexity maximum-likelihood (ML) decoding. Independently, the same STBCs were found in the work of Samuel and Fitz (2007) and named multi-strata space-time codes. In this paper we show how this code can be constructed algebraically from a particular cyclic division algebra (CDA). This formulation enables to prove that the code has the non-vanishing determinant (NVD) property and hence achieves the diversity-multiplexing tradeoff (DMT) optimality. The fact that the normalized minimum determinant is 1/radic(7) places this code in the second position with respect to the golden code, which exhibits a minimum determinant of 1/radic(5), and motivates the name silver code.

Fast-decodable MIDO codes from crossed product algebras

The goal of this paper is to design fast-decodable space-time codes for four transmit and two receive antennas. The previous attempts to build such codes have resulted in codes that are not full rank and hence cannot provide full diversity or high coding gains. Extensive work carried out on division algebras indicates that in order to get, not only non-zero but perhaps even non-vanishing determinants (NVD) one should look at division algebras and their orders. To further aid the decoding, we will build our codes so that they consist of four generalized Alamouti blocks which allows decoding with reduced complexity. As far as we know, the resulting codes are the first having both reduced decoding complexity, and at the same time allowing one to give a proof of the NVD property.

New Space–Time Code Constructions for Two-User Multiple Access Channels

This paper addresses the problem of constructing multiuser multiple-input multiple-output (MU-MIMO) codes for two users. The users are assumed to be equipped with nt transmit antennas, and there are nr antennas available at the receiving end. A general scheme is proposed and shown to achieve the optimal diversity-multiplexing gain tradeoff (DMT). Moreover, an explicit construction for the special case of nt = 2 and nr = 2 is given, based on the optimization of the code shape and density. All the proposed constructions are based on cyclic division algebras and their orders and take advantage of the multi-block structure. Computer simulations show that both the proposed schemes yield codes with excellent performance improving upon the best previously known codes. Finally, it is shown that the previously proposed design criteria for DMT optimal MU-MIMO codes are sufficient but in general too strict and impossible to fulfill. Relaxed alternative design criteria are then proposed and shown to be still sufficient for achieving the multiple-access channel diversity-multiplexing tradeoff.

Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain tradeoff is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well-known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain tradeoff and point counting in Lie groups.

DMT Optimal Codes Constructions for Multiple-Access MIMO Channel

Explicit code constructions for multiple-input multiple-output (MIMO) multiple-access channels (MAC) with K users are presented in this paper. The first construction is dedicated to the case of symmetric MIMO-MAC where all the users have the same number of transmit antennas nt and transmit at the same level of per-user multiplexing gain r. Furthermore, we assume that the users transmit in an independent fashion and do not cooperate. The construction is systematic for any values of K, nt and r. It is proved that this newly proposed construction achieves the optimal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tse at high-SNR regime. In the second part of the paper we take a further step to investigate the MAC-DMT of a general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and can transmit at different levels of multiplexing gain. The exact optimal MAC-DMT of such channel is explicitly characterized in this paper. Interestingly, in the general MAC-DMT, some users might not be able to achieve their single-user DMT performance as in the symmetric case, even when the multiplexing gains of the other users are close to 0. Detailed explanations of such unexpected result are provided in this paper. Finally, by generalizing the code construction for the symmetric MIMO-MAC, explicit code constructions are provided for the general MIMO-MAC and are proved to be optimal in terms of the general MAC-DMT.

An algebraic MIDO-MISO code construction

Multiple-input double-output (MIDO) codes are important in future wireless communications, where the portable end-user device is physically small and uses only two receive antennas. In this paper, we address the design of 4×2 MIDO codes. Starting from a 4×4 space-time block code matrix built from a cyclic division algebra, two ways of puncturing the code are presented, resulting in either a well-shaped MIDO code, or a MIDO code with some orthonormal columns, yielding fast maximum-likelihood (ML) decodability. The well-shaped MIDO code outperforms the fast decodable one through simulations, an indication that the shaping property stays an important code design criterion. We then provide a slightly modified version of the well-shaped MIDO code which both preserves the shaping and increases the orthogonality of its columns in an attempt to speed up the decoding of the code. Finally, we show that a multiple-input single output (MISO) code is actually embedded in the MIDO code, allowing the transmitter to choose between sending a MIDO or MISO code, without having to change the encoder. All the proposed codes have the non-vanishing determinant (NVD) property.

Remarks on the criteria of constructing MIMO-MAC DMT optimal codes

In this paper we investigate the criteria proposed by Coronel et al. for constructing MIMO MAC-DMT optimal codes over several classes of fading channels. We first show by a counterexample that their DMT result might not be correct when the channel has selective fading. For the case of symmetric MIMO-MAC flat fading channels, we study their criteria for constructing MAC-DMT optimal codes when the number of receive antennas is sufficiently large. We show their criterion is equivalent to requiring the codes of any subset of users to satisfy a joint non-vanishing determinant criterion when operating in the antenna pooling regime. Finally an upper bound on the product of minimum eigenvalues of the difference matrices is provided, and is used to show MIMO-MAC lattice codes satisfying their criterion exist only when the target multiplexing gain is small.

Almost Universal Codes Achieving Ergodic MIMO Capacity Within a Constant Gap

This paper addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. A design criterion based on the normalized minimum determinant is proposed for division algebra multi-block space-time codes over fading channels; this plays a similar role to the Hermite invariant for Gaussian channels. Under maximum likelihood decoding, it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels. Moreover, if the number of receive antennas is greater than or equal to the number of transmit antennas, the same constant gap is achieved under naive lattice decoding as well. In the case of independent identically distributed Rayleigh fading, the error probability vanishes exponentially fast. In contrast to the standard approach in the literature, which employs random lattice ensembles, the existence results in this paper are derived from the number theory. First, the gap to capacity is shown to depend on the discriminant of the chosen division algebra; then, class field theory is applied to build families of algebras with small discriminants. The key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants.