Cyclic Division Algebras Research Articles

Space–Time Codes From Sum-Rank Codes

Just as rank-metric or Gabidulin codes may be used to construct rate–diversity tradeoff optimal space–time codes, a recently introduced generalization for the sum-rank metric—linearized Reed–Solomon codes—accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate–diversity optimal multiblock space–time codes as an application of linearized Reed–Solomon codes. We also provide sequential decoders for these codes and, more generally, space–time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations.

Quotients of orders in algebras obtained from skew polynomials with applications to coding theory

ABSTRACTWe describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.

Nonassociative cyclic extensions of fields and central simple algebras

We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.

Open-Loop End-to-End Transmission for Multihop Opportunistic Networks With Energy-Harvesting Devices

Networks formed by energy-harvesting devices impose new technological challenges on data transmission due to uncertainty of the amount of energy that can be harvested. This is even more challenging with multihop networks, in which transmission outage occurs when one single device along the relay path cannot harvest enough energy. A virtual multiple-input multiple-output (MIMO) model for multihop, multipath networks has recently been developed to facilitate reliable open-loop end-to-end transmissions, making networks robust to random transmission outages without the necessity of bandwidth-consuming and complicated end-to-end feedback and control. In this paper, a framework based on a virtual MIMO model of multihop, multipath opportunistic networks formed by energy-harvesting devices is developed. We propose rotated-algebraic path-time codes (RA-PTC), by which data are encoded using Givens rotation and cyclic division algebras. Without rate loss, a form of time diversity is exploited by repeatedly transmitting the RA-PTC-coded data. Extensive theoretical analyses are carried out using amount of fading and diversity as metrics. Both the performance enhancement due to the proposed RA-PTC and the performance loss due to energy shortage are quantified. Furthermore, a simple yet effective cyclic power control is proposed to improve transmission reliability. Numerical results demonstrate that RA-PTC and cyclic power control enable efficient and reliable end-to-end transmission in multipath, multihop opportunistic networks formed by energy-harvesting devices.

On skew polynomial codes and lattices from quotients of cyclic division algebras

We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak{p}\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak{p}$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak{p}\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.

Construction of Block Orthogonal STBCs and Reducing Their Sphere Decoding Complexity

A new class of Space Time Block Codes (STBCs) known as block orthogonal STBCs (BOSTBCs) was recently presented by Ren et al., which could be exploited by a QR decomposition decoder with M paths (QRDM decoder) to achieve significant decoding complexity reduction without performance loss. The block orthogonal property of the codes constructed, was however only shown via simulations. In this paper, we give analytical proofs for the block orthogonal structure of various existing codes in literature including codes formed as the sum of Clifford Unitary Weight Designs (CUWDs). We also show that, construction methods from Coordinate Interleaved Orthogonal Designs (CIODs), Cyclic Division Algebras (CDAs) and Crossed-Product Algebras (CPAs) can lead to BOSTBCs. In addition, we show that the block orthogonal STBCs offer a reduced decoding complexity when used in tandem with a fast sphere decoder using a depth first search approach. Simulation results involving decoding complexity show a 30% reduction in the number of floating point operations (FLOPS) of BOSTBCs as compared to STBCs without the block orthogonal structure.

Cyclic Division Algebras with Non-norm Elements

In this paper, we construct some cyclic division algebras (K/F, σ, γ). We obtain a necessary and sufficient condition of a non-norm element γ provided that F = ℚ and K is a subfield of a cyclotomic field ℚ(ζpu), where p is a prime and ζpu is a puth primitive root of unity. As an application for space time block codes, we also construct cyclic division algebras (K/F, σ, γ), where F = ℚ(i), [Formula: see text], K is a subfield of ℚ(ζ4pu) or [Formula: see text], and γ = 1+i. Moreover, we describe all cyclic division algebras (K/F,σ,γ) such that F= ℚ(i), K is a subfield of [Formula: see text] and γ=1+i, where [Formula: see text], d = 2 or 4, φ is the Euler totient function, and p1, p2 ≤ 100 are distinct odd primes.

Generalized Distributive Law for ML Decoding of Space–Time Block Codes

The problem of designing good space-time block codes (STBCs) with low maximum-likelihood (ML) decoding complexity has gathered much attention in the literature. All the known low ML decoding complexity techniques utilize the same approach of exploiting either the multigroup decodable or the fast-decodable (conditionally multigroup decodable) structure of a code. We refer to this well-known technique of decoding STBCs as conditional ML (CML) decoding . In this paper, we introduce a new framework to construct ML decoders for STBCs based on the generalized distributive law (GDL) and the factor-graph-based sum-product algorithm. We say that an STBC is fast GDL decodable if the order of GDL decoding complexity of the code, with respect to the constellation size M, is strictly less than Mλ, where λ is the number of independent symbols in the STBC. We give sufficient conditions for an STBC to admit fast GDL decoding, and show that both multigroup and conditionally multigroup decodable codes are fast GDL decodable. For any STBC, whether fast GDL decodable or not, we show that the GDL decoding complexity is strictly less than the CML decoding complexity. For instance, for any STBC obtained from cyclic division algebras which is not multigroup or conditionally multigroup decodable, the GDL decoder provides about 12 times reduction in complexity compared to the CML decoder. Similarly, for the Golden code, which is conditionally multigroup decodable, the GDL decoder is only half as complex as the CML decoder.

Structured dispersion matrices from division algebra codes for space-time shift keying

We propose a novel method of constructing Dispersion Matrices (DM) for Coherent Space-Time Shift Keying (CSTSK) relying on arbitrary PSK signal sets by exploiting codes from division algebras. We show that classic codes from Cyclic Division Algebras (CDA) may be interpreted as DMs conceived for PSK signal sets. Hence various benefits of CDA codes such as their ability to achieve full diversity are inherited by CSTSK. We demonstrate that the proposed CDA based DMs are capable of achieving a lower symbol error ratio than the existing DMs generated using the capacity as their optimization objective function for both perfect and imperfect channel estimation.

On the construction of perfect codes from HEX signal constellations

Recently, minimum and non-minimum delay perfect codes were proposed for any channel of dimension n. Their construction appears in the literature as a subset of cyclic division algebras over Q(ζ3) only for the dimension n=2sn1, where s∈{0,1}, n1 is odd and the signal constellations are isomorphic to Z[ζ3]n. In this work, we propose an innovative methodology to extend the construction of minimum and non-minimum delay perfect codes as a subset of cyclic division algebras over Q(ζ3), where the signal constellations are isomorphic to the hexagonal A2n-rotated lattice, for any channel of any dimension n such that gcd(n,3)=1.

Construction of New Delay-Tolerant Space-Time Codes

Perfect space-time codes (STC) are optimal codes in their original construction for multiple-input multiple-output (MIMO) systems. Based on cyclic division algebras (CDA), they are full-rate, full-diversity codes, have non-vanishing determinants (NVD) and hence achieve diversity-multiplexing tradeoff (DMT). In addition, these codes have led to optimal distributed space-time codes when applied in cooperative networks under the assumption of perfect synchronization between relays. However, they lose their diversity when delays are introduced and thus are not delay-tolerant. In this paper, using the cyclic division algebras of perfect codes, we construct new codes that maintain the same properties as perfect codes in the synchronous case. Moreover, these codes preserve their full-diversity in asynchronous transmission.

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

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <formula formulatype="inline"><tex Notation="TeX">$n_{t}$</tex> </formula> transmit antennas, and there are <formula formulatype="inline"> <tex Notation="TeX">$n_{r}$</tex></formula> 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 <formula formulatype="inline"><tex Notation="TeX">$n_{t}=2$</tex></formula> and <formula formulatype="inline"><tex Notation="TeX">$n_{r}=2$</tex></formula> 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. </para>

High-Rate Space–Time Coded Large-MIMO Systems: Low-Complexity Detection and Channel Estimation

In this paper, we present a low-complexity algorithm for detection in high-rate, non-orthogonal space-time block coded (STBC) large-MIMO systems that achieve high spectral efficiencies of the order of tens of bps/Hz. We also present a training-based iterative detection/channel estimation scheme for such large STBC MIMO systems. Our simulation results show that excellent bit error rate and nearness-to-capacity performance are achieved by the proposed multistage likelihood ascent search (M-LAS) detector in conjunction with the proposed iterative detection/channel estimation scheme at low complexities. The fact that we could show such good results for large STBCs like 16x16 and 32x32 STBCs from Cyclic Division Algebras (CDA) operating at spectral efficiencies in excess of 20 bps/Hz (even after accounting for the overheads meant for pilot based training for channel estimation and turbo coding) establishes the effectiveness of the proposed detector and channel estimator. We decode perfect codes of large dimensions using the proposed detector. With the feasibility of such a low-complexity detection/channel estimation scheme, large-MIMO systems with tens of antennas operating at several tens of bps/Hz spectral efficiencies can become practical, enabling interesting high data rate wireless applications.

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.

Construction Methods for Asymmetric and Multiblock Space–Time Codes

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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4Tx</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+2Rx</i> 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.

Generally Explicit Space–Time Codes With Nonvanishing Determinants for Arbitrary Numbers of Transmit Antennas

Nonvanishing determinants have emerged as an attractive criterion enabling a space-time code achieve the optimal diversity-multiplexing gains tradeoff. It seems that cyclic division algebras play the most crucial role in designing a space-time code with nonvanishing determinants. In this paper, we explicitly construct space-time codes for arbitrary numbers of transmit antennas that achieve nonvanishing determinants and the optimal diversity-multiplexing gains tradeoff over Z [i]. Unlike previous methods usually arising a field compositum for two or more fields, our scheme, which only requires one simple extension, constitutes a much more efficient and feasible advancement whether in theory or practice.

Construction of Multiblock Space–Time Codes From Division Algebras With Roots of Unity as Nonnorm Elements

The authors give a construction of multiblock space-time block codes from cyclic division algebras with a given <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> th root of unity as the nonnorm element. The construction uses local class field theory. Multiblock codes with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> blocks can achieve an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -fold increase in diversity.

Maximal Orders in the Design of Dense Space-Time Lattice Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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). </para>

On the Amplify-and-Forward Cooperative Diversity with Time-Hopping Ultra-Wideband Communications

In this paper, we extend the Amplify-and-Forward cooperative diversity scheme to the context of impulse radio ultra wideband (IR-UWB). In particular, we present the construction of three families of minimal-delay and totally-real distributed algebraic space-time (ST) codes suitable for IR-UWB. The first family encodes adjacent symbols and is based on totally-real cyclic division algebras. The second family encodes the pulses used to transmit one information symbol and permits to achieve high performance levels with lower complexity. Both families of codes achieve full rate, full diversity with non-vanishing determinants for various number of relays. These schemes can be associated with pulse position modulation (PPM), pulse amplitude modulation (PAM) and hybrid pulse position and amplitude modulation (PPM-PAM). The third family of codes is information-lossless and does not require any pulse repetitions. It is specific to M-PPM-M'-PAM with M ges 3 and for all values of M'. Simulations performed over realistic indoor UWB channels are provided to verify the theoretical results.

MMSE optimal algebraic space-time codes

Design of Space-Time Block Codes (STBCs) for Maximum Likelihood (ML) reception has been predominantly the main focus of researchers. However, the ML decoding complexity of STBCs becomes prohibitive large as the number of transmit and receive antennas increase. Hence it is natural to resort to a suboptimal reception technique like linear Minimum Mean Squared Error (MMSE) receiver. Barbarossa et al and Liu et al have independently derived necessary and sufficient conditions for a full rate linear STBC to be MMSE optimal, i.e achieve least Symbol Error Rate (SER). Motivated by this problem, certain existing high rate STBC constructions from crossed product algebras are identified to be MMSE optimal. Also, it is shown that a certain class of codes from cyclic division algebras which are special cases of crossed product algebras are MMSE optimal. Hence, these STBCs achieve least SER when MMSE reception is employed and are fully diverse when ML reception is employed.