A novel design of MIMO downlink communications that is assisted by modern intelligent reflecting surfaces (IRS) is proposed in this paper. The new design employs successive zero-forcing dirty paper coding at base station and allows multiple receive antennas to be used at users, making it much more realistic than existing works, which are applicable only to cases of single receive antenna. Furthermore, the new design requires IRS elements with only finite phase resolution and is much more practical for implementation. Simulation results show that the proposed design can significantly improve the performance of IRS-aided downlink communications. In particular, for the classical scenario of single receive antenna at users, our design, using only a QPSK phases for IRS elements, increases by more than 200-540% the energy efficiency of several existing works that assume IRS elements with infinite resolution for phases.

The tradeoff between sum-rate and fairness for MIMO downlink communications employing dirty paper coding (DPC) is investigated in this paper. More specifically, we first apply the luml1-norm fairness measure to formulate the tradeoff as a certain optimization problem, which is unfortunately nonconvex and cannot be efficiently solved. To overcome the difficulty, we make use of the uplink-downlink duality to transform the problem back and forth between uplink and downlink communications. An efficient, iterative waterfilling based algorithm is then provided to yield achievable points on the sum rate-fairness tradeoff. Simulation results show surprisingly that the proposed DPC-based approach offers a significant gain in the achievable sum-rates for a wide range of fairness values, when compared to successive zero-forcing DPC-based coding schemes.

Based on a recently proposed L_1-norm quantitive fairness measure, the optimal tradeoff between sum rate and fairness for MISO downlink communication employing zero-forcing dirty paper coding is provided in this paper. The corresponding achievable schemes are also given for every point lying on the tradeoff . In particular, a novel power allocation scheme that takes into account both sum rate and fairness is proposed. Compared to the popular approach of proportional fairness, simulation results show that the new scheme can significantly and simultaneously improve upon the achieved sum rate and fairness.

The weight generating functions associated with convolutional codes (CCs) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing-Luerssen and Schneider in the case of minimal encoders, and generalized by Forney. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity WAMs of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum CCs.

A novel transmission scheme for cooperative relay networks is presented in this paper. The proposed scheme is based on the non-orthogonal selection decode-and-forward protocol with an additional assumption of having a low rate feedback channel from the destination to relays. Benefited from the feedback information, an optimal distributed code that has an extremely short delay equal to four is constructed, and the same code is applicable to networks with the arbitrary number of relays to yield optimal cooperative diversity. The proposed code is sphere decodable with a decoding complexity again independent of the number of relays in high SNR regime. In particular, when operating at multiplexing gain ≥(1/2), the lattice decoder at the destination has a zero complexity exponent, meaning a constant decoding complexity and independent of transmission rate. Analyses for the decoding complexity of other existing diversity-optimal distributed codes are also provided. It is shown that these codes have a linear growth in delay and an exponential growth in decoding complexity as the number of relays increases.

A novel transmission scheme based on the non-orthogonal selection decode-and-forward protocol is presented in this paper for cooperative relay networks. The proposed scheme assumes a low rate feedback channel from the destination to the relays. Benefited from the feedback information, an optimal distributed code that has an extremely short delay equal to four is constructed, and the same code can be applied to networks with arbitrary number of relays to yield optimal cooperative diversity gains. The proposed code is sphere-decodable with decoding complexity again independent of the number of relays. In particular, when operating at multiplexing gain ≥ 1/2, the lattice decoder at the destination has a zero complexity exponent, meaning a constant decoding complexity and independent of transmission rate. Analyses for the decoding complexity of other existing diversity-optimal distributed codes are also provided. It is shown that these codes have a linear growth in delay and an exponential growth in decoding complexity as the number of relays increases.

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.

In this paper, two selection schemes are proposed for coded transmission over multiple-input multiple-output (MIMO) multiple-access channels (MAC) to yield a much higher diversity-multiplexing gain tradeoff (DMT) performance. These schemes require a channel feedback, but at an extremely low rate. The first scheme is based on user selection and can be easily implemented in the existing MIMO-MAC systems. Upper bounds on the minimal computational complexity required by sphere decoders to decode DMT-optimal codes for this scheme as well as for MIMO MAC without feedback are given. It is shown that this scheme can offer both a much larger DMT and an exponential reduction on decoding complexity, compared with the latter. The second scheme selects jointly the users and their transmit antennas. It requires an additional design of rate assignments for performance optimization. A very general framework on the design of optimal rate assignments is thus provided. It is shown that this scheme can yield DMT performances far superior to the optimal MIMO-MAC DMT without channel feedback. The simulation results confirm that in some cases, this scheme can provide an astonishing SNR gain of 14.64 dB at outage probability 10-6compared with the optimal MIMO-MAC coding schemes without feedback.

In a MIMO MAC where the base station has fewer receive antennas than the transmit antennas of all users, sphere (lattice) decoding for the existing MIMO-MAC codes requires an exhaustive search of exponentially large size before processing the root of a sphere-decoding tree. In this paper, two coding schemes are proposed and are shown to yield a constant sphere-decoding complexity, independent of the numbers of users and transmit antennas. The schemes require a channel feedback, but only at an extremely low rate. The first scheme is based on user selection, and the second scheme selects jointly users and transmit antennas, using a fast antenna selection algorithm recently proposed by Jiang and Varanasi. It also involves a design of rate-assignments that maximizes the overall DMT performance. It is shown that both schemes yield DMT performances far superior to the optimal MIMO-MAC DMT without channel feedback in certain multiplexing gain regime. Simulation results confirm that in some cases the second proposed scheme can provide an astonishing SNR gain of 14:5 dB at outage probability 10-6compared to the optimal coding schemes without feedback.

In this paper, we prove a MacWilliams identity for the weight adjacency matrices based on the constraint codes of a convolutional code (CC) and its dual. Our result improves upon a recent result by Gluesing-Luerssen and Schneider, where the requirement of a minimal encoder is assumed. We can also establish the MacWilliams identity for the input-parity weight adjacency matrices of a systematic CC and its dual. Most importantly, we show that a type of Hamming weight enumeration functions of all codewords of a CC can be derived from the weight adjacency matrix, which thus provides a connection between these two very different notions of weight enumeration functions in the convolutional code literature. Finally, the relations between various enumeration functions of a CC and its dual are summarized in a diagram. This explains why no MacWilliams identity exists for the free-distance enumerators.

In this paper, we consider the uniquely decodable one-to-one code (UDOOC) that is obtained by inserting a comma indicator, termed the unique word (UW), between consecutive one-to-one codewords for separation. As such, we analyze a class of UDOOCs and present practical algorithms for encoding and decoding such codes. Specifically, for various cases of UWs, we investigate the number of length-n codewords of UDOOCs and their asymptotic growth rates in n. The proposed encoding and decoding algorithms of UDOOCs can be implemented in parallel at low computational complexity without storing the codebook. Simulation results show that for proper choices of UWs, UDOOCs can achieve better compression efficiency than Lempel-Ziv codes even when the source is not statistically independent.

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.

In this paper, we answer several open questions related to diversity-multiplexing tradeoffs (DMTs) for point-to-point and multiple-access (MAC) MIMO channels. By analyzing the DMT performance of a simple code, we show that the optimal MAC-DMT holds even when the channel remains fixed for less thanKnt+nr-1 channel uses, whereKis the number of users,ntis the number of transmit antennas of each user, andnris the number of receive antennas at receiver. We also prove that the simple code is MAC-DMT optimal. A general code design criterion for constructing MAC-DMT optimal codes that is much more relaxed than the previously known design criterion is provided. Finally, by changing some design parameters, the simple code is modified for use in point-to-point MIMO channels. We show the modified code achieves the same DMT performance as the Gaussian random code.

In this paper, we investigate the diversity multiplexing tradeoff (DMT) performance of MIMO linear receivers with general colored Gaussian input. By varying the rank of the covariance matrix of the channel input and allowing temporal coding across multiple channel uses, we find the DMT performance of MIMO linear receivers can be significantly improved and be much better than the currently known.

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).

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.

In this paper we analyze the generalized degrees of freedom (GDOF) and the DMT performances of both the symmetric and asymmetric MIMO Gaussian interference fading channel with a fixed-power-split HK scheme for two transmit-receive pairs. Exact characterizations of both the GDOF and DMT performance measures are given. It is shown in the SIMO case that when the number of receive antennas is at least two, full region of GDOF can be achieved, as if the non-intending transmitter does not exist. The same conclusion is applied to the DMT measure as well. In particular, if the channel is symmetric, both receivers are able to achieve the single-user performance regime.

In diversity embedded coding, information streams are divided into two sub-streams with different priorities. If the optimal DMT performance of each coded stream can be achieved, then such code is said to be successive refinable. For the cases of SISO, SIMO, and MISO Rayleigh slow fading channels, Diggavi and Tse had shown that superposition coding with successive cancellation receiver achieves successive refinability in these channels. However, such optimality might not be extended to MIMO channel due to the strictly sub-optimality of successive cancellation receiver. In this paper, we first provide an explicit construction of MIMO diversity embedded codes that is sphere decodable. We then show that the proposed code is approximately universal, if joint ML decoding is used, and hence extend the notion of successive refinability to general MIMO channels.

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.

In multiple-input-multiple-output (MIMO) communications, the notion of asymmetric channel refers to the situation when the number of transmit antennas is strictly larger than the number of receive antennas. Such channels can often be found in MIMO downlink transmissions. While existing cyclic-division-algebra (CDA)-based codes can still be employed to achieve the optimal diversity-multiplexing tradeoff (DMT) at high signal-to-noise ratio (SNR) regime, such codes cannot be directly decoded using, for example, the pure sphere decoding method. Although other means of decoding methods such as minimum mean square error generalized decision feedback equalizer (MMSE-GDFE) with lattice search and regularized lattice decoding are available, an alternative approach is to constrain the number of active transmit antennas in each channel use to be no larger than the number of receive antennas. The resulting system is coined constrained asymmetric MIMO system. Two general types of asymmetrical channels are considered in this paper, namely, 1) when there are two receive antennas and the number of transmit antennas is arbitrary, and 2) when the number of transmit antennas is one larger than the number of receive antennas. Explicit optimal transmission schemes as well as the corresponding code constructions for such constrained asymmetric MIMO channels are presented, and are shown to achieve the same DMT performance as their unconstrained counterparts.

An asynchronous cooperative network where different time delays exist among nodes is considered in this paper. Assuming the signals are OFDM modulated, it is first shown that the diversity-multiplexing tradeoff (DMT) achieved by the non-orthogonal selection decode-and-forward (NSDF) protocol for this network is the same as that for the synchronous one. In contrast to the complicated approximately universal "matrix" codes, where each relay uses a different codebook, a systematic construction of an extremely simple "vector" code is proposed. Given the transmitted codeword vector, this vector will be used by all nodes in the network for signal transmission; hence, the proposed coding scheme greatly reduces the complexity of relay deployment and decoding. Furthermore, it is proven that the proposed scheme is optimal in terms of the DMT of the NSDF protocol for this asynchronous network, provided all time delays are distinct. Finally, it is shown that the proposed code design can be extended to the orthogonal selection decode-and-forward protocol and remains to be DMT optimal.

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.

An explicit, systematic code construction for the symmetric MIMO (multi-input multi-output) multiple access (MAC) channel with any number of users and any numbers of transmit and receive antennas is presented in this paper. The users are assumed to transmit at the same level of multiplexing gain. This newly constructed code is proved to achieve the optimal MIMO-MAC diversity-multiplexing tradeoff.

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. Divsalariquests 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.

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 with4Tx+2Rxantennas. 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.

In a MIMO downlink channel it is often that the number of transmit antennas is strictly larger than the number of receive, and such channel is termed asymmetric MIMO channel. To employ simple decoding techniques in this channel, such as zero-forcing or sphere decoding, the number of active transmit antennas must be constrained to be no larger than the number of receive, and the resulting system is coined "constrained asymmetric MIMO system." For the case of two receive antennas and for any number of transmit antennas, an optimal transmission scheme is presented in this paper and is shown to achieve the same performance as the unconstrained ones in terms of the diversity-multiplexing tradeoff. The construction of optimal constrained codes is also provided.

A method for constructing asymmetric space-time block codes (ASTBC) is proposed. This Smart Puncturing Method (SPM) generalizes the so-called subfield construction method (SCM) introduced in earlier work and applies to any antenna combination with #Rx≪#Tx as opposed to SCM, where the requirement is #Tx= m#Rx for some integer m. It has been shown that e.g. for 4Tx+2Rx antennas, the SCM performs equally or even better than the trivial puncturing method (TPM), but admits at the same time lower peak-to-mean power ratio. This is due to the fact that there are no zero slots in the code matrix but the information symbols are evenly spread into the matrix slots. The generalized method proposed in this paper is also based on cyclic division algebras (CDAs) and allows us to do the same for any number of receiving antennas #Rx≪#Tx.

A systematic construction of fully-diverse, high-rate OFDM-based space-frequency coding schemes for the cooperative relay network communication is presented in this paper. Assuming an asynchronous communication between the intermediate relays and the destination node, it is shown that the proposed scheme can achieve full cooperative diversity for any possible time delays between the relays and the destination node if the number of subcarriers used in the OFDM modulation and the oversampling ratio used at the destination node are both powers of a prime. Such conditions are naturally met in all existing OFDM-based communication standards. Comparing to the space-time trellis codes proposed by Xia et al. , the present scheme not only achieves full diversity gain at a much higher rate but also are more flexible in both code and signal designs. A design example targeting at the latest OFDM-based standards is also given.

An asynchronous cooperative network where different time delays among nodes exist is considered in this paper. With OFDM modulation, it is first shown that the diversity-multiplexing tradeoff (DMT) achieved by the non-orthogonal selection decode-and-forward (NSDF) protocol for this network is the same as that for the synchronous one. In sharp contrast to the currently available complicated approximately universal ‘matrix’ codes, a systematic construction of an extremely simple ‘vector’ code is proposed. Given the transmitted codeword vector, the same codeword will be sent by all nodes during transmission, hence it will greatly reduce the complexities of relay deployment and decoding. Furthermore, it is proved that the proposed scheme achieves the optimal DMT of the NSDF protocol for this asynchronous cooperative network.

In this paper, a systematic construction of asymmetric space-time codes that is a promising solution to the asymmetric coding problem is presented. By the asymmetric coding problem we mean that the number of receive antennas is strictly less than the number of transmit antennas in a MIMO communication system, and the task is to design a coding scheme that has an efficient decoding using e.g. a sphere decoder. Specifically, given any desired antenna selection pattern, the proposed construction will yield codes that are transmitted using exactly the specified pattern, and that can be easily decoded using sphere decoding or MMSE techniques. Our construction can be applied to any kinds of antenna selection patterns, including the one that equally-likely uses all possible n-subsets of the transmit antennas for some n. Moreover, the resulting codes are proved to achieve the diversity-multiplexing tradeoff associated with the designated selection pattern.

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).

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.

A construction of low complexity multi-block space- time codes that is optimal with respect to the diversity- multiplexing tradeoff (DMT) when the coding is applied over any number of independent fading blocks is presented in this paper. Specifically, for a MIMO system over a quasi-static Rayleigh block fading channel with quasi-static interval T, the present construction requires a number field that is cyclic Galois over Q(i) of degree only T, while the previously known construction requires a number field of degree mT, where m is the number of coded fading blocks. Thus, the present construction provides a great reduction on both the encoding and decoding complexities of multi-block space-time codes comparing to earlier constructions, and it achieves complexity at the same level as that of the conventional DMT-optimal single block space-time codes.

Recently an algebraic construction of (nttimes Q) space-frequency (SF) codes over finite field Fqwas proposed for use in MIMO-OFDM systems, where nt is the number of transmit antenna and Q = qnt- 1 is the number of subcarriers employed in the code. One inconvenience arising from that construction is that the number of subcarriers Q can sometimes be insufficient for constructing codes of large minimum column distance. To completely eliminate this disadvantage, an alternative construction of SF codes with Q = qm- 1 is provided in this paper, whenever m is a multiple of nt. Lower bounds on the minimum rank and column distances of the proposed construction are also given. Simulation results show that the newly constructed codes provide a significant improvement in SNR compared to other SF codes available in the literature.

A SIMO-OFDM system with nrreceive antennas and Q subcarriers is considered in this paper. Under a frequency-selective Rayleigh fading channel with L multipaths, Lout(r) = Lnr(1-r). Finally we prove that a newly constructed multi-block space-frequency code can be used to reach the optimal diversity-multiplexing gain tradeoff of this channel with optimal diversity gain d-(r) sime Lnr(1-r).

Network coding over a delay-free acyclic communication network with single source is considered in this paper. The network is modeled as a directed acyclic graph G where each edge in G is assumed to have unit link capacity. Previous works on network coding require a sufficiently large field such that the network has either a linear multicast, a linear broadcast, or a linear dispersion solution. For certain graphs, it is also known that linear network codes over a field of smaller size might not exist. In this paper, we propose a linear network code with memory and show that for any directed acyclic network with single source, there always exists a binary linear dispersion network code. Such code can be explicitly constructed and requires only the binary field for realization. Thus, this approach would dramatically reduce the hardware complexity for code implementation. Also contained in this paper is an explicit construction of binary linear broadcast network code for any directed acyclic networks.

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^a-1,$$\omega=q,$and$\lambda=1$.

Constructions of multi-block space-time coding schemes that are optimal with respect to the diversity-multiplexing tradeoff when coding is applied over any number of independent fading blocks are presented in this paper. The constructions are based on the left-regular representation of elements of 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, and the resultant scheme is termed minimal delay multi-block space-time coding scheme. Variations of this construction corresponding to the cases of non-minimal 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 MIMO communication with vanishing error probability.

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$T geq n_t+n_r-1$where$T$is the number of time slots over which coding takes place and$n_t,n_r$are the number of transmit and receive antennas, respectively. For$T ≪ n_t+n_r-1$, 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 case when$T=n_t=n$, 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,$T ≫ n_t$case, 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$Tgeq n_t$is the same as that previously known to hold for$T geq n_t + n_r -1$.

A generalized p-radii construction for space-time codes achieving the optimal rate-diversity tradeoff is presented in this paper. The new construction is obtained by extending Hammons' dyadic dual-radii construction to the cases when the size of the constellation A is a power of a prime p, p /spl ges/ 2. The resulting space-time code is optimal in terms of achieving the rate-diversity tradeoff and has an AM-PSK constellation with signal alphabets distributed over p-concentric circles in the complex plane, i.e., there are p radii. Finally, we present the generalized super-unified construction by generalizing the super-unified construction by Hammons (2004). The generalized results are readily to be extended to cater to the constructions of both optimal space-time block and trellis codes and even to the constructions of optimal codes over multiple-fading blocks.

In the recent landmark paper of Zheng and Tse it is shown for the quasi-static, Rayleigh-fading MIMO channel with n/sub t/ transmit and n/sub r/ receive antennas, that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the diversity-multiplexing gain (D-MG) tradeoff. This paper presents the first explicit construction of space-time (ST) codes for an arbitrary number of transmit and/or receive antennas that achieve the D-MG tradeoff. It is shown here that ST codes constructed from cyclic-division-algebras (CDA) and satisfying a certain non-vanishing determinant (NVD) property, are optimal under the D-MG tradeoff for any n/sub t/,n/sub r/. Furthermore, this optimality is achieved with minimum possible value of the delay or block-length parameter T = n/sub t/. CDA-based ST codes with NVD have previously been constructed for restricted values of n/sub t/. A unified construction of D-MG optimal CDA-based ST codes with NVD is given here, for any number n/sub t/ of transmit antennas. The CDA-based constructions are also extended to provide D-MG optimal codes for all T /spl ges/ n/sub t/, again for any number nt of transmit antennas. This extension thus presents rectangular D-MG optimal space-time codes that achieve the D-MG tradeoff. Taken together, the above constructions also extend the region of T for which the D-MG tradeoff is precisely known from T /spl ges/ n/sub t/ + n/sub r/ - 1 to T /spl ges/ n/sub t/.

Constructions of space-frequency (SF) codes for MIMO-OFDM systems with n/sub t/ transmit antennas and Q subcarriers are considered in this paper. Arising from the pairwise-error-probability analysis, in addition to the rank distance criterion, the minimum column distance of (n/sub t/ /spl times/ Q) SF codes serves as another benchmark in code design. Codes with larger minimum column distance are expected to have better performance. Following this observation, two 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 /spl times/ Q) nonsingular matrix. Codes obtained from this construction are called linearly transformed MRD (LT-MRD) codes in this paper. Minimum column distance of the LT-MRD codes, when averaged over all code ensembles, is shown to meet the Gilbert-Varshamov bound. The second 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 and bounds on the minimal column distance of these codes are presented.

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 multiradii 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.

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/ /spl les/ T, a unified construction of (n/sub t/ /spl times/ 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.

Bounds on the diversity-multiplexing gain tradeoff of a recent space-time block code construction are provided. This construction makes use of binary codes which is optimum in terms of the rate-diversity tradeoff. The code can be decoded using sphere decoding techniques.

In this paper, a systematic method for constructing space-time block codes that are optimal in terms of achieving the rate-diversity tradeoff over a large variety of signal constellations whose sizes are power of a prime p is presented. The resulting signal constellation includes the p/sup K/-ary PAM, QAM, and PSK signallings as special cases. The construction is a generalization of the unified construction proposed by the authors earlier. It consists of a generalized unified mapper and a class of maximal, rank-d, p-codes over F/sub p/. The generalized unified construction can also be applied to build optimal space-time block and trellis codes.

This paper presents an optimal construction of space-time codes over multiple fading blocks and over a variety of constellations including PAM, QAM and 2/sup K/-ary PSK. The constructions can be used for any numbers of transmit antennas and for any desirable transmit diversity gain.

For any space-time code having a fixed, finite signal constellation, there is a tradeoff between the transmission rate and the transmit diversity gain achieved by the code. For any number of transmit antennas, a unified construction of space-time codes is provided, for a class of signal constellations that includes pulse-amplitude-modulation (PAM), quadrature-amplitude-modulation (QAM) and 2/sup K/-ary phase-shift-keying (PSK) as special cases. The construction is optimal as measured by the rate-diversity tradeoff.

We first show the criteria for designing binary space-time trellis codes that achieve the optimal rate-diversity tradeoff. Based on the criteria, two systematic constructions of binary space-time trellis codes for any number of transmit antennas and any desirable rate are provided. Finally, by extending our work on the unified construction of space-time codes, these newly constructed binary space-time trellis codes are generalized to a series of codes over a much larger signal constellation, for instance, PAM, QAM and PSK modulations.

We show that for any (Q/spl times/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 /spl nu/ achieved by the code. The tradeoff is characterized by R/spl les/Q-/spl nu/+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.

This article 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 article are an exact expression for pairwise error probability (PEP), signal design guidelines, and some observations relating to the reception of ODST codes.

This paper provides an overview of the results in a recent journal submission by the same authors. The first part of that paper studies the pairwise error probability of codewords (PEP) of a space-time code over a quasistatic channel, using an approach that allows both known and unknown channel cases to be considered simultaneously. A closed-form expression for the PEP is provided, and given a constraint on the sum of the squares of the singular values of the difference signal matrix, it is shown that the PEP is minimized by choosing signals with equal singular values. A useful sequence of simple upper and lower bounds that converge to the PEP is also provided. An example space-time code is introduced and shown using this sequence of bounds to outperform the corresponding orthogonal-design-based space-time (ODST) code at all values of SNR. Exact expressions, based on the PEP, are given for the asymptotic coding and diversity gain. It is shown that the diversity gain remains unchanged if the PEP is replaced by either the codeword error probability (CEP) or else the message symbol error probability (SEP). Signal-design implications of the above results are also discussed. The second part deals with ODST codes. It is shown that ODST codes represent an instance of orthogonal signaling. This observation is used to derive a closed-form expression for the pairwise error probability of message symbols (PEP-ms) of these codes, as well as an expression for coding gain based on PEP-ms, that is exact in the case of BPSK signaling.