We study a scenario in which multiple uncoordinated devices aim to achieve reliable transmissions within a given time frame. The devices are intermittently active and access a shared pool of channel resources in a grant-free manner by utilizing multiple transmissions (K-repetition coding). This allows them to achieve diversity and improve the reliability within a certain latency constraint. We focus on two access methods: one where devices choose K slots at random and another one where the access patterns are deterministic and follow a specific code design, namely the Steiner System. We analyze the problem under two signal models that involve different complexity for the receiver. First, collision model is considered, where only interference-free transmissions can be used and combined. Second, a model treating interference as noise is analyzed, where the receiver is capable of utilizing all K replicas, applying maximum ratio combining (MRC). For both signal models, we investigate receivers with and without successive interference cancellation (SIC). We develop approximations and bounds for the outage probabilities that very closely match simulation results. Overall, we show that deterministic access patterns have the potential to significantly outperform random selection in terms of reliability. Furthermore, deterministic access patterns offer a simplified system design.

We consider high-dimensional MIMO transmissions in frequency division duplexing (FDD) systems. For precoding, the frequency selective channel has to be measured, quantized and fed back to the base station by the users. When the number of antennas is very high this typically leads to prohibitively high quantization complexity and large feedback. In 5G New Radio (NR), a modular quantization approach has been applied for this, where first a low-dimensional subspace is identified for the whole frequency selective channel, and then subband channels are linearly mapped to this subspace and quantized. We analyze how the components in such a modular scheme contribute to the overall quantization distortion. Based on this analysis we improve the technology components in the modular approach and propose an orthonormalized wideband precoding scheme and a sequential wideband precoding approach which provide considerable gains over the conventional method. We compare the performance of the developed quantization schemes to prior art by simulations in terms of the projection distortion, overall distortion and spectral efficiency, in a scenario with a realistic spatial channel model.

We consider a two-way secret key distribution protocol in the satellite setting, where Alice, Bob and Eve each decode bits from noisy signals received from a source in their environment. Alice and Bob perform advantage distillation to find a secret key. We apply a Two-way Protocol with Parity bit Reconciliation (TPPR) where secret keys are collected from parity bits in course of advantage distillation, not only from the final distilled bits. We analyze the mutual information acquired by Eve from exploiting the original eavesdropped information together with the information leaked during the distillation protocol, as well as TPPR secret key rate. Comparing to the ParityCheck Protocol (PCP) known in the literature, TPPR provides complementary performance. In operation regions where PCP fare badly as compared to one-way protocols, TPPR provides gains in key rate.

Abstract The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice-based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of ‘structured’ LWE, trading off a hard to quantify loss of security for an increase in efficiency by working over a well-chosen ring. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a ring. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE) to replicate the addition of the ring structure taking LWE to Ring LWE by adding cyclic structure to Module LWE. We show that the security reductions expected for an LWE problem hold, namely a reduction from certain structured lattice problems to the hardness of the decision variant of the CLWE problem (under the condition of constant rank d). As a contribution of theoretic interest, we view CLWE as the first variant of Ring LWE which supports non-commutative multiplication operations. This ring structure compares favorably with Module LWE, and naturally allows a larger message space for error correction coding.

In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of so-called minimum delay asymmetric space-time codes for the$n \times m$MIMO channel. Such codes correspond to lattices in$M_{n}(\mathbb {C})$with dimension smaller than$2n^{2}$. Currently, very little is known about their DMT, except in the case$m=1$, corresponding to the multiple input single output (MISO) channel. Further, apart from the MISO case, no DMT optimal asymmetric codes are known. We first discuss previous criteria used to analyze the DMT of space-time codes and comment on why these methods fail when applied to asymmetric codes. We then consider two special classes of asymmetric codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also show that lattice codes based on$\mathbb {Q}$-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all$\mathbb {Q}$-central division algebra codes that are based on standard embeddings. While the$\mathbb {Q}$-central division algebra based codes achieve the largest possible DMT of a code restricted to either real or quaternion space, they still fall short of the optimal DMT apart from the MISO case.

We address the problem of codebook design for sparse user detection in fast fading channels, where the fading realization changes from channel use to next. In this scenario, codebook design criteria based on quasi-static fading, and/or channel state information at the receiver, become ineffective. In this paper we suggest new code design principles for signature coding in fast fading channels and provide examples of codes that are built using these methods.

We consider high-dimensional multiuser MIMO transmissions in Frequency Division Duplexing systems. For precoding, the frequency selective channel has to be measured, quantized and fed back to the base station by the users. In 5G New Radio (NR), a modular quantization approach has been applied for this, where first a low-dimensional subspace is identified for the whole frequency selective channel, and then subband channels are linearly mapped to this subspace and quantized. We analyze how the components in such a modular scheme contribute to the overall quantization distortion. Based on this analysis we improve the technology components in the modular approach. We compare the improved quantization scheme to the 5G NR standardized version by simulation in a scenario with a realistic spatial channel model. The improvements lead to a more than 25% improvement in spectral efficiency.

We consider sparse user detection in fading channels. With Rayleigh flat fading, deep fades occur with relatively high probability and it becomes challenging to provide highly reliable user detection, irrespective of the chosen multiuser detection algorithm. It has been proven that with a large number of receive antennas, this problem can be overcome and both the reliability and number of detectable users can be increased. In this paper, we show that similar improvements can be achieved by moderately increasing the number of transmit antennas at the user terminals. With multiple transmit antennas, code design becomes a problem. We provide a design criterion and show that the detection probability can be considerably improved by using the resulting well-balanced MIMO signature codes, especially in the high-reliability regime.

Two-way quantum key distribution (QKD) protocols can provide positive secret key rates for considerably higher quantum bit error rates (QBER) than one-way protocols. However, when QBER is low, only modest key rate gains have been achieved. This is one of the major obstacles for using two-way protocols. In this paper we introduce a new two-way QKD protocol which is a step towards overcoming this shortcoming. Under the assumption that the eavesdropper can only perform individual symmetric quantum attacks, our protocol performs quantum key distribution with a secret key rate that is higher than the information theoretical bound limiting the performance of any one-way protocol. This holds true also for very low QBER values.

We study medium access control layer random access under the assumption that the receiver can perform successive interference cancellation, without feedback. During recent years, a number of protocols with impressive error performance have been suggested for this channel model. However, the random nature of these protocols causes an error floor which limits their usability when targeting ultra-reliable communications. In very recent works by Paolini et al. and Boyd et. al., it was shown that if each user employs predetermined combinatorial access patterns, this error floor disappears. In this paper, we develop code design criteria for deterministic random access protocols in the ultra-reliability region, and build codes based on these principles. The suggested design methods are supported by simulations.

In this paper we consider space-time codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also point out that lattice codes based on Q-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central division algebra codes that are based on standard embeddings.

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper's channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance/minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multipleantenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates Rb- Ce- κ, where Cband Ceare Bob and Eve's channel capacities, respectively, and κ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates Rb- C̅e- κ are achievable, where C̅bis a lower bound for Bob's capacity and C̅eis an upper bound for Eve's capacity for all the channels in the set.

We explore the underlying algebraic structure of Mutually Unbiased Bases (MUBs), and their application to code design. Columns in MUBs have inner products with absolute values less or equal to 1/√N. MUBs provide a systematic way of generating optimal codebooks for various coding and precoding applications. A maximal set of MUBs (MaxMUBs) in N = 2mdimensions, with m ϵ Z, can produce codebooks of QPSK lines with good distance properties and alphabets which limit processing complexity. We expand the construction by identifying that in N = 2mdimensions there exists N(m-1)/2families of MUB, each with N matrices. Inner products of columns of these matrices are less or equal to 1/√2. As an example, we construct Grassmannian line codes from the columns of these matrices. Then decoding or encoding these codebooks can be performed without multiplications, and with a number of additions that scales linearly with the number of codewords, irrespectively of the dimension.

We consider Combinatorial Code Designs (CCD) for ensuring ultra-reliability in the random access channel. By constructing user-specific repetition patterns to be utilised over a synchronised uplink frame consisting of a number of access slots, we guarantee successive reception up to a given number of simultaneously active users. Employing advanced receivers capable of Successive Interference Cancellation (SIC) further improves reliability. As an example, we consider a system with access frames of 24 bundled slots, repetition factor 3, and reliability target 99.999%. When compared to slotted repetition ALOHA, SIC provides a 30% gain in the tolerated user activity; CCD a 30% gain; whereas CCD combined with SIC provides a gain of more than 700%. These gains come at the cost of a strict limit on the supported user population. In the given example, the system can support a total of 2024 users.

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.

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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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. 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. It is shown that a multiblock construction based on division algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. 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.

This paper shows that a family of number field lattice codes simultaneously achieves a constant gap to capacity in Rayleigh fast fading and Gaussian channels. The key property in the proof is the existence of infinite towers of Hilbert class fields with bounded root discriminant. The gap to capacity of the proposed lattice codes is determined by the root discriminant. The comparison between the Gaussian and fading case reveals that in Rayleigh fading channels the normalized minimum product distance plays an analogous role to the Hermite invariant in Gaussian channels.

This paper investigates the design of codes for multiple-input multiple-output (MIMO) multiple access channel (MAC). If a joint maximum-likelihood decoding is to be performed at the receiver, then every MIMO-MAC code can be regarded as a single-user code, where the minimum determinant criterion proposed by Tarokh et al. is useful for designing such codes and for upper bounding the maximum pairwise error probability (PEP), whenever the codes are of finite rate and operate in finite signal-to-noise ratio range. Unlike the case of single-user codes where the minimum determinant can be lower bounded by a fixed constant as code-rate grows, it was proved by Lahtonen et al. that the minimum determinant of MIMO-MAC codes decays as a function of the rates. This decay phenomenon is further investigated in this paper, and upper bounds for the decays of minimum determinant corresponding to each error event are provided. Lower bounds for the optimal decay are established and are based on an explicit construction of codes using algebraic number theory and Diophantine approximation. For some error profiles, the constructed codes are shown to meet the aforementioned upper bounds, hence they are optimal finite-rate codes in terms of PEPs associated with such error events. An asymptotic diversity-multiplexing gain tradeoff (DMT) analysis of the proposed codes is also given. It is shown that these codes are DMT optimal when the values of multiplexing gains are small.

This paper considers space-time coding over several independently Rayleigh faded blocks. In particular, we will concentrate on giving upper bounds for the coding gain of lattice space-time codes as the number of blocks grow. This problem was previously considered in the single antenna case by Bayer-Fluckiger et al. in 2006. Crucial to their work was Odlyzko's bound on the discriminant of an algebraic number field, as this provides an upper bound for the normalized coding gain of number field codes. In the MIMO context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding (noncommutative) algebra. In this paper, we will develop analogues of the Odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space-time codes. These bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases.

There is a rich theory of relations between lattices and linear codes over finite fields. However, this theory has been developed mostly with lattice codes for the Gaussian channel in mind. In particular, different versions of what is called Construction A have connected the Hamming distance of the linear code to the Euclidean structure of the lattice. This paper concentrates on developing a similar theory, but for fading channel coding instead. First, two versions of Construction A from number fields are given. These are then extended to division algebra lattices. Instead of the Euclidean distance, the Hamming distance of the finite codes is connected to the product distance of the resulting lattices, that is the minimum product distance and the minimum determinant respectively.

This paper considers shifted inverse determinant sums arising from the union bound of the pairwise error probability for space-time codes in multiple-antenna fading channels. Previous work by Vehkalahti et al. focused on the approximation of these sums for low multiplexing gains, providing a complete classification of the inverse determinant sums as a function of constellation size for the most well-known algebraic space-time codes. This work aims at building a general framework for the study of the shifted sums for all multiplexing gains. New bounds obtained using dyadic summing techniques suggest that the behavior of the shifted sums does characterize many properties of a lattice code such as the diversity-multiplexing gain trade-off, both under maximum-likelihood decoding and infinite lattice naive decoding. Moreover, these bounds allow to characterize the signal-to-noise ratio thresholds corresponding to different diversity gains.

This work considers normalized inverse determinant sums as a tool for analyzing the performance of division algebra based space-time codes for multiple antenna wireless systems. A general union bound based code design criterion is obtained as a main result. In our previous work, the behavior of inverse determinant sums was analyzed using point counting techniques for Lie groups; it was shown that the asymptotic growth exponents of these sums correctly describe the diversity-multiplexing gain trade-off of the space-time code for some multiplexing gain ranges. This paper focuses on the constant terms of the inverse determinant sums, which capture the coding gain behavior. Pursuing the Lie group approach, a tighter asymptotic bound is derived, allowing to compute the constant terms for several classes of space-time codes appearing in the literature. The resulting design criterion suggests that the performance of division algebra based codes depends on several fundamental algebraic invariants of the underlying 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.

This paper provides a general construction method for multiple-input multiple-output multiple access channel codes (MIMO MAC codes) that have so called generalized full rank property. The achieved constructions give a positive answer to the question whether it is generally possible to reach the so called pigeon hole bound, that is an upper bound for the decay of determinants of MIMO-MAC channel codes.

Earlier it was proven by Vehkalahti and Lu how the unit group and diversity-multiplexing gain trade-off (DMT) of division algebra-based space-time codes are linked to each other through inverse determinant sums. This work explores this relation further, showing that indeed the density of unit group completely determines the growth of the inverse determinant sum. In particular, in the case of Q(i)-central division algebras, the lower bound obtained from the DMT and the upper bound derived from the growth rate of units coincide.

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$\,\times\,$2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of$(4\times 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,. Several explicit constructions are presented and shown to have excellent performance through computer simulations.

Recently, extensive effort has been taken to build fast-decodable (FD) space-time (ST) codes, especially called for due to the limited power resources available for mobile receivers. Although there have been numerous successful attempts that have resulted in reduced decoding complexity for a certain number of transmit antennas, a general method for constructing FD codes for a wider range of antenna combinations and rates is still missing. Here, this problem is solved by introducing a totally general framework for constructing full-diversity FD codes with non-vanishing determinants (NVD).

In this paper we are concentrating on the diversity-multiplexing gain trade-off (DMT) of some space-time lattice codes. First we give a DMT bound for lattice codes having restricted dimension. We then recover the well known results of the DMT of algebraic number field codes and the Alamouti code by using the union bound and see that these codes do achieve the previously mentioned bound. During our analysis interesting connections to the Dedekind's zeta-function and to Dirichlet's unit theorem are revealed. Finally we prove that both the number field codes and Alamouti code are in some sense optimal codes in the multiple access channel (MAC).

Recently, several papers have been concentrating on reducing the decoding complexity of high-rate space-time codes. While the research has led to some impressive reductions in decoding complexity, the geometric methods so far used appear to have faced some fundamental limits. In this paper, we study what happens if we let go of the assumption of full diversity and study the possibility of reducing the complexity, while holding on to a high code rate, by reducing the length of codes. We will develop some tools that can be used to measure the changes we will encounter when reducing the code length. We will also study the achievable diversity-multiplexing gain trade-off (DMT) of codes with less than minimum delay (LMD) and discuss some code constructions.

Since the invention of space-time coding numerous algebraic methods have been applied in code design. In particular algebraic number theory and central simple algebras have been on the forefront of the research. In this paper we are turning the table and asking whether information theory can be used as a tool in algebra. We first show how diversity-multiplexing gain tradeoff (DMT) bounds of Zheng and Tse will give us information of the spread of determinants in matrix lattices and then apply these results to analyze unit groups of orders of division algebras. The results considering unit groups are not new or the best possible but we do find that this interesting relation between algebra and information theory is quite surprising and worth pointing out.

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$n_{t}$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$K$,$n_{t}$and$r$. It is proved that this newly proposed construction achieves the optimal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tseat high-$\hbox{SNR}$regime.

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.

Multiple-input double-output (MIDO) codes are important in future wireless communications, where the portable end-user device is physically small and will typically contain maximum two receive antennas. Especially tempting is the 4×2 channel, where the four transmitters can either be all at one station, or separated between two different stations. 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, the worst-case complexity being as high as N8, where N is the size of the complex signaling alphabet. Hence, constructions with reduced complexity are called for. One option, of course, is to use the rate-one codes such as the quasi-orthogonal codes. However, if full multiplexing, i.e., transmission of two symbols per channel use is to be maintained, this option has to be put aside. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on ad hoc methods and have resulted in a specific code instead of a more general class of codes. In this paper, it will be shown that cyclic division algebra (CDA) based codes satisfying certain criteria will always result in at least 25% worst-case complexity reduction, while maintaining full diversity and even the non-vanishing determinant (NVD) property. The reduction follows from the fact that the codes will consist of four Alamouti blocks allowing simplified decoding. At the moment, such reduction is the best known for rate-two MIDO codes,. The code proposed in was the first one to provably fulfill the related algebraic properties, and shall be repeated here as an example. Further, a new low-complexity design resulting from the proposed criteria is presented, and shown to have excellent performance through simulations.

This paper considers certain aspects of some well-known multiple-input single-output (MISO) codes. In the first section it is proved how in some special cases the n + 1 MISO channel can be seen as consisting of several parallel MISO channels having less transmit antennas. It is also pointed out that unitary conjugation does not change the diversity-multiplexing tradeoff (DMT) of a code. These simple results are then applied to analyze the DMT of several well-known MISO codes. In particular its is proved that all the considered codes are DMT optimal. As a by-product of this study it is seen that the full-diversity quasi-orthogonal codes by Su and Xia are unitarily equivalent to division algebraic constructions. This relation is then used to place the constructions by Su and Xia into a wider context. In the latter part of the paper the 2 + 1 slow fading MISO channel is considered and it is proven that one of the previously proposed MISO multi-block codes (MB-codes) has a linear worst-case sphere decoding complexity.

The paper considers the question of the normalized minimum determinant (or asymptotic coding gain) of real matrix lattices. The coding theoretic motivation for such study arises, for example, from the questions considering multiple-input multiple-output (MIMO) ultra-wideband (UWB) transmission. At the beginning, totally general coding gain bounds for real MIMO lattice codes is given by translating the problem into geometric language. Then code lattices that are produced from division algebras are considered. By applying methods from the theory of central simple algebras, coding gain bounds for code lattices coming from orders of division algebras are given. Finally, it is proven that these bounds can be reached by using maximal orders. In the case of 2 × 2 matrix lattices, this existence result proves that the general geometric bound derived earlier can be reached.

An algebraic tool from the theory of central simple algebras is proposed to obtain families of complex matrices satisfying the conditional non-vanishing determinant (CNVD) property. Such property is of great use in e.g. the design of multiuser space-time (ST) codes, in which context it is not always crucial for the transmission matrix to be invertible. On the other hand, whenever it is invertible, it is important that it has a non-vanishing determinant. Also any submatrix of any subset of users multiplied with its transpose conjugate should preferably have a non-vanishing determinant, provided it is non-zero. In recent submissions by Lu et al. it has been shown that, with suitable multiplexing, such property yields a construction of space-time codes that achieve the optimal diversity-multiplexing tradeoff (DMT) of the multiple-input multiple-output (MIMO) multiple access channel (MAC) and outperform the previously known ST codes.

In this paper we are discussing multiple-input single-output (MISO) transmission in slowly fading channel with two transmit and one receiving antenna. Particularly we are concentrating on properties of division algebra based Alamouti-like space-time block codes. The main contributions of this paper are the following. First we extend the previously done constructions by considering general orders and give a simple proof that the proposed family of codes isapproximatelyuniversalin the slow fading channel. Further we give a method to measure the pairwise error probability (PEP)-oriented normalized coding gain of the proposed codes and explore the frontiers of the achievable coding gain.

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.

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.

In [1] the authors suggested that in order to derive energy efficient space-time MIMO codes from orders of the division algebras one should use maximal orders instead of natural ones. They also described the division algebras that have the best maximal orders in terms of minimum determinant vs. average power. However, they were able to construct division algebras that were optimal only in few separate cases. In this paper we are addressing this problem and giving an explicit construction for optimal division algebras of arbitrary degree in the case when the center is Q(i). We note that all the results of this paper can be found from [2]. The goal of this paper is to give a simple representation of construction methods of [2].

We show 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 non-vanishing minimal determinant. Using results from class field theory we derive a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas). We also give examples of division algebras achieving our bound. For example, we construct a matrix lattice with QAM coefficients that has (inside 'large' subsets of the signal space) 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We also give another matrix lattice with coefficients from the hexagonal lattice with an even higher density