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 non-vanishing 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 [4], [5]. Several explicit constructions are presented and shown to have excellent performance through computer simulations. Index Terms-Coding gain, cyclic division algebra, digital video broadcasting next generation handheld (DVB-NGH), fast maximum-likelihood (ML) sphere decoding, Hamiltonian quaternions, Hasse invariants, lattices, lowcomplexity space-time block codes (STBCs), multiple-input single/double/multiple-output (MISO/MIDO/MIMO), nonvanishing determinant (NVD), orders.
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 our bound are given. E.g. we construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We describe a general algorithm due to Ivanyos and Rónyai for finding maximal orders within a cyclic division algebra and discuss our enhancements to this algorithm. We also consider general methods for finding cyclic division algebras of a prescribed index achieving our lower bound. Index TermsCyclic division algebras, dense lattices, discriminants, Hasse invariants, maximal orders, multiple-input multipleoutput (MIMO) channels, multiplexing, space-time block codes (STBCs).I. OVERVIEW Multiple-antenna wireless communication promises very high data rates, in particular when we have perfect channel state information (CSI) available at the receiver. In [1] the design criteria for such systems were developed, and further on the evolution of space-time (ST) codes took two directions: trellis codes and block codes. Our work concentrates on the latter branch.We are interested in the coherent multiple input-multiple output (MIMO) case. A lattice is a discrete finitely generated free abelian subgroup L of a real or complex finite dimensional vector space V, called the ambient space. In the space-time setting a natural ambient space is the space M n (C) of complex n×n matrices. We only consider full rank lattices that have a basis x 1 , x 2 , . . . , x 2n 2 consisting of matrices that are linearly independent over the field of real numbers. We can form a 2n 2 × 2n 2 matrix M having rows consisting of the real and imaginary parts of all the basis elements. It is well known that the measure, or hypervolume, m(L) of the fundamental parallelotope of the lattice then equals the absolute value of det(M). Alternatively we may use the Gram matrix 2 . From the pairwise error probability (PEP) point of view [2], the performance of a space-time code is dependent on two parameters: diversity gain and coding gain. Diversity gain is the minimum of the rank of the difference matrix X − X ′ taken over all distinct code matrices X, X ′ ∈ C, also called the rank of the code C. When C is full-rank, the coding gain is proportional to the determinant of the matrix
This work concentrates on the study of 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 trade-off 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 trade-off and point counting in Lie groups.
This work 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 multiblock space-time codes over fading channels; this plays a similar role to the Hermite invariant for Gaussian channels. It is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for slow fading channels and ergodic fading channels. This performance is achieved both under maximum likelihood decoding and naive lattice decoding. In the case of independent identically distributed Rayleigh fading, it is also shown that 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 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.
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. The level of reduction depends on whether one is willing to compromise the ML performance. As far as we know, the resulting codes are the first having reduced decoding complexity and at the same time allow one to give a proof of the NVD property.
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.
We consider a fading wiretap channel model where the transmitter has only statistical channel state information, and the legitimate receiver and eavesdropper have perfect channel state information. We propose a sequence of non-random lattice codes which achieve strong secrecy and semantic security over ergodic fading channels. The construction is almost universal in the sense that it achieves the same constant gap to secrecy capacity over Gaussian and ergodic fading models
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