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In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT) of a binary function $f:G\rightarrow {\cal R}$, where $G = (\mathbb{F}_2^n,+)$ is the group of $n$ bit vectors with bitwise modulo two addition $+$, and ${\cal R}$ is a finite commutative ring of characteristic $2$. Using the specific group structure of $G$ and a sequence of nested subgroups of $G$, we define the fast TMFT and its inverse. Since the image ${\cal R}$ of the binary functions is a ring, we can define the convolution between two functions $f:G\rightarrow {\cal R}$. We then provide the TMFT properties, including the convolution theorem, which can be used to efficiently compute convolutions. Finally, we derive the complexity of the fast TMFT and the inverse fast TMFT., Comment: to appear in IEEE Trans. on Information Theory

We construct error correcting codes for jointly transmitting a finite set of independent messages to an 'informed receiver' which has prior knowledge of the values of some subset of the messages as side information. The transmitter is oblivious to the message subset already known to the receiver and performs encoding in such a way that any possible side information can be used efficiently at the decoder. We construct and identify several families of algebraic error correcting codes for this problem using cyclic and maximum distance separable (MDS) codes. The proposed codes are of short block length, many of them provide optimum or near-optimum error correction capabilities and guarantee larger minimum distances than known codes of similar parameters for informed receivers. The constructed codes are also useful as error correcting codes for index coding when the transmitter does not know the side information available at the receivers., Comment: Accepted for publication in 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona. Keywords: Cyclic codes, index coding, informed receivers, maximum distance separable codes, side information

Lattices possess elegant mathematical properties which have been previously used in the literature to show that structured codes can be efficient in a variety of communication scenarios, including coding for the additive white Gaussian noise (AWGN) channel, dirty-paper channel, Wyner-Ziv coding, coding for relay networks and so forth. We consider the family of single-transmitter multiple-receiver Gaussian channels where the source transmits a set of common messages to all the receivers (multicast scenario), and each receiver has 'coded side information', i.e., prior information in the form of linear combinations of the messages. This channel model is motivated by applications to multi-terminal networks where the nodes may have access to coded versions of the messages from previous signal hops or through orthogonal channels. The capacity of this channel is known and follows from the work of Tuncel (2006), which is based on random coding arguments. In this paper, following the approach of Erez and Zamir, we design lattice codes for this family of channels when the source messages are symbols from a finite field 'Fp' of prime size. Our coding scheme utilizes Construction A lattices designed over the same prime field 'Fp', and uses algebraic binning at the decoders to expurgate the channel code and obtain good lattice subcodes, for every possible set of linear combinations available as side information. The achievable rate of our coding scheme is a function of the size 'p' of underlying prime field, and approaches the capacity as 'p' tends to infinity., Comment: Accepted for publication in the IEEE Transactions on Information Theory

We consider the problem of oblivious transfer (OT) over OFDM and MIMO wireless communication systems where only the receiver knows the channel state information. The sender and receiver also have unlimited access to a noise-free real channel. Using a physical layer approach, based on the properties of the noisy fading channel, we propose a scheme that enables the transmitter to send obliviously one-of-two files, i.e., without knowing which one has been actually requested by the receiver, while also ensuring that the receiver does not get any information about the other file., Comment: Submitted to Transactions on Communications

We consider the special case of index coding over the Gaussian broadcast channel where each receiver has prior knowledge of a subset of messages at the transmitter and demands all the messages from the source. We propose a concatenated coding scheme for this problem, using an index code for the Gaussian channel as an inner code/modulation to exploit side information at the receivers, and an outer code to attain coding gain against the channel noise. We derive the capacity region of this scheme by viewing the resulting channel as a multiple-access channel with many receivers, and relate it to the 'side information gain' -- which is a measure of the advantage of a code in utilizing receiver side information -- of the inner index code/modulation. We demonstrate the utility of the proposed architecture by simulating the performance of an index code/modulation concatenated with an off-the-shelf convolutional code through bit-interleaved coded-modulation., Comment: To appear in Proc. IEEE Int. Symp. Inf. Theory (ISIT) 2015, Hong Kong, Jun. 2015. 5 pages, 4 figures

We propose index codes, based on multidimensional QAM constellations, for the Gaussian broadcast channel, where every receiver demands all the messages from the source. The efficiency with which an index code exploits receiver side information in this broadcast channel is characterised by a code design metric called "side information gain". The known index codes for this broadcast channel enjoy large side information gains, but do not encode all the source messages at the same rate, and do not admit message sizes that are powers of two. The index codes proposed in this letter, which are based on linear codes over integer rings, overcome both these drawbacks and yet provide large values of side information gain. With the aid of a computer search, we obtain QAM index codes for encoding up to 5 messages with message sizes 2^m, m <= 6. We also present the simulated performance of a new 16-QAM index code, concatenated with an off-the-shelf LDPC code, which is observed to operate within 4.3 dB of the broadcast channel capacity., Comment: Accepted for publication in IEEE Communications Letters. 4 pages, 2 figures, 1 table

Line-of-sight (LoS) multiple-input multiple-output (MIMO) gives full spatial-multiplexing gain when the antenna array geometry and orientation are designed based on the inter-terminal distance. These known design methodologies, that hold for antenna arrays with fixed orientation, do not provide full MIMO gains for arbitrary array orientations. In this paper, we study LoS MIMO channels with random array orientations when the number of transmit antennas used for signalling is 2. We study the impact of common array geometries on error probability, and identify the code design parameter that describes the high signal-to-noise ratio (SNR) error performance of an arbitrary coding scheme. For planar receive arrays, the error rate is shown to decay only as fast as that of a rank 1 channel, and no better than 1/SNR^3 for a class of coding schemes that includes spatial multiplexing. We then show that for the tetrahedral receive array, which uses the smallest number of antennas among non-planar arrays, the error rate decays faster than that of rank 1 channels and is exponential in SNR for every coding scheme. Finally, we design a LoS MIMO system that guarantees a good error performance for all transmit/receive array orientations and over a range of inter-terminal distances., Comment: Accepted for publication in IEEE Transactions on Vehicular Technology. Keywords: Antenna array, array geometry, coding scheme, line-of-sight (LoS), multiple-input multiple-output (MIMO), probability of error

We present a comparative study of the performance of various polar code constructions in an additive white Gaussian noise (AWGN) channel. A polar code construction is any algorithm that selects $K$ best among $N$ possible polar bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit error rate (BER). Optimal polar code construction is hard and therefore many suboptimal polar code constructions have been proposed at different computational complexities. Polar codes are also non-universal meaning the code changes significantly with the design-SNR. However, it is not known which construction algorithm at what design-SNR constructs the best polar codes. We first present a comprehensive survey of all the well-known polar code constructions along with their full implementations. We then propose a heuristic algorithm to find the best design-SNR for constructing best possible polar codes from a given construction algorithm. The proposed algorithm involves a search among several possible design-SNRs. We finally use our algorithm to perform a comparison of different construction algorithms using extensive simulations. We find that all polar code construction algorithms generate equally good polar codes in an AWGN channel, if the design-SNR is optimized., Comment: 9 pages, submitted, under revision of an IEEE journal

We consider a point-to-point flat-fading MIMO channel with channel state information known both at transmitter and receiver. At the transmitter side, a lattice coding scheme is employed at each antenna to map information symbols to independent lattice codewords drawn from the same codebook. Each lattice codeword is then multiplied by a unitary precoding matrix ${\bf P}$ and sent through the channel. At the receiver side, an integer-forcing (IF) linear receiver is employed. We denote this scheme as unitary precoded integer-forcing (UPIF). We show that UPIF can achieve full-diversity under a constraint based on the shortest vector of a lattice generated by the precoding matrix ${\bf P}$. This constraint and a simpler version of that provide design criteria for two types of full-diversity UPIF. Type I uses a unitary precoder that adapts at each channel realization. Type II uses a unitary precoder, which remains fixed for all channel realizations. We then verify our results by computer simulations in $2\times2$, and $4\times 4$ MIMO using different QAM constellations. We finally show that the proposed Type II UPIF outperform the MIMO precoding X-codes at high data rates., Comment: 12 pages, 8 figures, to appear in IEEE-TWC

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems., Comment: 24 pages. Extends our earlier arxiv submission arxiv.1303.3475

The Unshared Secret Key Cryptography (USK), recently proposed by the authors, guarantees Shannon's ideal secrecy and perfect secrecy for MIMO wiretap channels, without requiring secret key exchange. However, the requirement of infinite constellation inputs limits its applicability to practical systems. In this paper, we propose a practical USK scheme using finite constellation inputs. The new scheme is based on a cooperative jamming technique, and is valid for the case where the eavesdropper has more antennas than the transmitter. We show that Shannon's ideal secrecy can be achieved with an arbitrarily small outage probability., Comment: 7 pages, 2 figures. arXiv admin note: text overlap with arXiv:1410.5021

While the impact of finite-rate feedback on the capacity of fading channels has been extensively studied in the literature, not much attention has been paid to this problem under secrecy constraint. In this work, we study the ergodic secret capacity of a multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channel with quantized channel state information (CSI) at the transmitter and perfect CSI at the legitimate receiver, under the assumption that only the statistics of eavesdropper CSI is known at the transmitter. We refine the analysis of the random vector quantization (RVQ) based artificial noise (AN) scheme in [1], where a heuristic upper bound on the secrecy rate loss, when compared to the perfect CSI case, was given. We propose a lower bound on the ergodic secrecy capacity. We show that the lower bound and the secrecy capacity with perfect CSI coincide asymptotically as the number of feedback bits and the AN power go to infinity. For practical applications, we propose a very efficient quantization codebook construction method for the two transmit antennas case., Comment: 9 pages, 4 figures

The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific case of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. Instances of this communication problem arise in wireless relay networks, sensor networks, and retransmissions in broadcast channels. We construct 'lattice index codes' for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a coarse lattice. We introduce a design metric called 'side information gain' that measures the advantage of a code in utilizing the side information at the receivers, and hence its goodness as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes with large side information gains using lattices over the following principal ideal domains: rational integers, Gaussian integers, Eisenstein integers, and the Hurwitz quaternions. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain. Finally, using an example, we illustrate how the proposed lattice index codes can benefit Gaussian broadcast channels with more general message demands., Comment: Accepted for publication in the IEEE Transactions on Information Theory. Two-column format, 21 pages

Current security techniques can be implemented with either secret key exchange or physical layer wiretap codes. In this work, we investigate an alternative solution for MIMO wiretap channels. Inspired by the artificial noise (AN) technique, we propose the unshared secret key (USK) cryptosystem, where the AN is redesigned as a one-time pad secret key aligned within the null space between transmitter and legitimate receiver. The proposed USK cryptosystem is a new physical layer cryptographic scheme, obtained by combining traditional network layer cryptography and physical layer security. Unlike previously studied artificial noise techniques, rather than ensuring non-zero secrecy capacity, the USK is valid for an infinite lattice input alphabet and guarantees Shannon's ideal secrecy and perfect secrecy, without the need of secret key exchange. We then show how ideal secrecy can be obtained for finite lattice constellations with an arbitrarily small outage., Comment: 13 pages, 4 figures

In this work we investigate codes in $\mathbb{Z}_{2^m}^n$ that can correct errors that occur in just one coordinate of the codeword, with a magnitude of up to a given parameter $t$. We will show upper bounds on these cross codes, derive constructions for linear codes and respective decoding algorithm. The constructions (and decoding algorithms) are given for length $n = 2$ and $n = 3$, but for general $m$ and $t$., Comment: To be published in the proceedings of ISITA 2014, IEICE copyright

The compute-and-forward (CoF) is a relaying protocol, which uses algebraic structured codes to harness the interference and remove the noise in wireless networks. We propose the use of phase precoders at the transmitters of a network, where relays apply CoF strategy. We define the {\em phase precoded computation rate} and show that it is greater than the original computation rate of CoF protocol. We further give a new low-complexity method for finding network equations. We finally show that the proposed precoding scheme increases the degrees-of-freedom (DoF) of CoF protocol. This overcomes the limitations on the DoF of the CoF protocol, recently presented by Niesen and Whiting. Using tools from Diophantine approximation and algebraic geometry, we prove the existence of a phase precoder that approaches the maximum DoF when the number of transmitters tends to infinity., Comment: Submitted with 28 Pages and 2 Figures

The artificial noise (AN) scheme, proposed by Goel and Negi, is being considered as one of the key enabling technology for secure communications over multiple-input multiple-output (MIMO) wiretap channels. However, the decrease in secrecy rate due to the increase in the number of Eve's antennas is not well understood. In this paper, we develop an analytical framework to characterize the secrecy rate of the AN scheme as a function of Eve's signal-to-noise ratio (SNR), Bob's SNR, the number of antennas in each terminal, and the power allocation scheme. We first derive a closed-form expression for the average secrecy rate. We then derive a closed-form expression for the asymptotic instantaneous secrecy rate with large number of antennas at all terminals. Finally, we derive simple lower and upper bounds on the average and instantaneous secrecy rate that provide a tool for the system design., Comment: 4 figures, submitted to IEEE Trans on Wireless Commun

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to 'jointly' find the optimum phase precoding matrix and the corresponding network equation coefficients. This is a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To solve this problem, we introduce phase precoded CoF with partial feedback. It is a quantized precoding system where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook will then be fedback to the transmitters. A "deep hole phase precoder" is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance., Comment: 5 Pages, 4 figures, submitted to ISIT 2014

In multiple-input multiple-output (MIMO) fading channels, the design criterion for full-diversity space-time block codes (STBCs) is primarily determined by the decoding method at the receiver. Although constructions of STBCs have predominantly matched the maximum-likelihood (ML) decoder, design criteria and constructions of full-diversity STBCs have also been reported for low-complexity linear receivers. A new receiver architecture called Integer-Forcing (IF) linear receiver has been proposed to MIMO channels by Zhan et al. which showed promising results for the high-rate V-BLAST encoding scheme. In this paper, we address the design of full-diversity STBCs for IF linear receivers. In particular, we are interested in characterizing the structure of STBCs that provide full-diversity with the IF receiver. Along that direction, we derive an upper bound on the probability of decoding error, and show that STBCs that satisfy the restricted non-vanishing singular value (RNVS) property provide full-diversity for the IF receiver. Furthermore, we prove that all known STBCs with the non-vanishing determinant property provide full-diversity with IF receivers, as they guarantee the RNVS property. By using the formulation of RNVS property, we also prove the existence of a full-diversity STBC outside the class of perfect STBCs, thereby adding significant insights compared to the existing works on STBCs with IF decoding. Finally, we present extensive simulation results to demonstrate that linear designs with RNVS property provide full-diversity for IF receiver., Comment: Contains extension of results in arXiv:1701.03566v1. In arXiv:1701.03566v1, we had formulated the non-vanishing singular value (NVS) property as a design criterion on codes. This submission formulates a new criterion referred to as the Restricted NVS property, which helps to identify larger class of codes that provide full-diversity with integer-forcing linear receivers

Current security techniques can be implemented either by requiring a secret key exchange or depending on assumptions about the communication channels. In this paper, we show that, by using a physical layer technique known as artificial noise, it is feasible to protect secret data without any form of secret key exchange and any restriction on the communication channels. Specifically, we analyze how the artificial noise can achieve practical secrecy. By treating the artificial noise as an unshared one-time pad secret key, we show that the proposed scheme also achieves Shannon's perfect secrecy. Moreover, we show that achieving perfect secrecy is much easier than ensuring non-zero secrecy capacity, especially when the eavesdropper has more antennas than the transmitter. Focusing on the practical applications, we show that practical secrecy and strong secrecy can be guaranteed even if the eavesdropper attempts to remove the artificial noise. We finally show the connections between traditional cryptography and physical layer security., Comment: This paper has been withdrawn by the authors

In this paper, we consider the use of artificial noise for secure communications. We propose the notion of practical secrecy as a new design criterion based on the behavior of the eavesdropper's error probability $P_E$, as the signal-to-noise ratio goes to infinity. We then show that the practical secrecy can be guaranteed by the randomly distributed artificial noise with specified power. We show that it is possible to achieve practical secrecy even when the eavesdropper can afford more antennas than the transmitter., Comment: To appear in IEEE Communications Letters, 2013, 10 pages, 2 figures

Full-rate space-time block codes (STBCs) achieve high spectral-efficiency by transmitting linear combinations of information symbols through every transmit antenna. However, the coefficients used for the linear combinations, if not chosen carefully, results in ({\em i}) large number of processor bits for the encoder and ({\em ii}) high peak-to-average power ratio (PAPR) values. In this work, we propose a new class of full-rate STBCs called Integer STBCs (ICs) for multiple-input multiple-output (MIMO) fading channels. A unique property of ICs is the presence of integer coefficients in the code structure which enables reduced numbers of processor bits for the encoder and lower PAPR values. We show that the reduction in the number of processor bits is significant for small MIMO channels, while the reduction in the PAPR is significant for large MIMO channels. We also highlight the advantages of the proposed codes in comparison with the well known full-rate algebraic STBCs., Comment: 10 pages and 3 figures

In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper's correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algebraic number theory before, they are now brought into novel practice in the context of fading channel communications. Hence, the interest here is in small-dimensional lattices and finite constellations rather than in the asymptotic behavior.

Integer-forcing (IF) linear receiver has been recently introduced for multiple-input multiple-output (MIMO) fading channels. The receiver has to compute an integer linear combination of the symbols as a part of the decoding process. In particular, the integer coefficients have to be chosen based on the channel realizations, and the choice of such coefficients is known to determine the receiver performance. The original known solution of finding these integers was based on exhaustive search. A practical algorithm based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice reduction algorithms was also proposed recently. In this paper, we propose a low-complexity method based on complex LLL algorithm to obtain the integer coefficients for the IF receiver. For the 2 X 2 MIMO channel, we study the effectiveness of the proposed method in terms of the ergodic rate. We also compare the bit error rate (BER) of our approach with that of other linear receivers, and show that the suggested algorithm outperforms the minimum mean square estimator (MMSE) and zero-forcing (ZF) linear receivers, but trades-off error performance for complexity in comparison with the IF receiver based on exhaustive search or on HKZ and Minkowski lattice reduction algorithms., Comment: To appear in the proceedings of the IEEE Australian Communication Theory Workshop-2013, Adelaide, Australia

A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multiple-output (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the $2 \times 2$ and $4\times 4$ MIMO channels, we compare the coded-block error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors., Comment: 9 figures and 11 pages. Modified the title, abstract and some parts of the paper. Major change from v1: Added new results on applicability of the CLLL reduction

In this paper, we address the design of high spectral-efficiency Barnes-Wall (BW) lattice codes which are amenable to low-complexity decoding in additive white Gaussian noise (AWGN) channels. We propose a new method of constructing complex BW lattice codes from linear codes over polynomial rings, and show that the proposed construction provides an explicit method of bit-labeling complex BW lattice codes. To decode the code, we adapt the low-complexity sequential BW lattice decoder (SBWD) recently proposed by Micciancio and Nicolosi. First, we study the error performance of SBWD in decoding the infinite lattice, wherein we analyze the noise statistics in the algorithm, and propose a new upper bound on its error performance. We show that the SBWD is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise-trimming technique. This is the first work that showcases the error performance of SBWD in decoding BW lattice codes of large block lengths., Comment: 30 pages with 10 figures and 1 table

Unipolar communications systems can transmit information using only real and positive signals. This includes a variety of physical channels ranging from optical (fiber or free-space), to RF wireless using amplitude modulation with non-coherent reception, to baseband single wire communications. Unipolar OFDM techniques enable to efficiently compensate frequency selective distortion in the unipolar communication systems. One of the leading examples of unipolar OFDM is asymmetric clipped optical OFDM (ACO-OFDM) originally proposed for optical communications. Flip-OFDM is an alternative approach that was proposed in a patent, but its performance and full potentials have never been investigated in the literature. In this paper, we first compare Flip-OFDM and ACO-OFDM, and show that both techniques have the same performance but different complexities (Flip-OFDM offers 50% saving). We then propose a new detection scheme, which enables to reduce the noise at the Flip-OFDM receiver by almost 3dB. The analytical performance of the noise filtering schemes is supported by the simulation results., Comment: 19 pages, 8 pages (re-uploaded with corrected Fig 2a)

We consider the root lattice $A_n$ and derive explicit formulae for the moments of its Voronoi cell. We then show that these formulae enable accurate prediction of the error probability of lattice codes constructed from $A_n$., Comment: submitted to IEEE Transactions on Information Theory

We consider two uniploar OFDM techniques for optical wireless communications: asymmetric clipped optical OFDM (ACO-OFDM) and Flip-OFDM. Both techniques can be used to compensate multipath distortion effects in optical wireless channels. However, ACO-OFDM has been widely studied in the literature, while the performance of Flip-OFDM has never been investigated. In this paper, we conduct the performance analysis of Flip-OFDM and propose additional modification to the original scheme in order to compare the performance of both techniques. Finally, it is shown by simulation that both techniques have the same performance but different hardware complexities. In particular, for slow fading channels, Flip-OFDM offers 50% saving in hardware complexity over ACO-OFDM at the receiver., Comment: published in IEEE Information Theory Workshop, Paraty Brazil, Sept 2011

In this paper, we address the design of codes which achieve modulation diversity in block fading single-input single-output (SISO) channels with signal quantization at receiver and low-complexity decoding. With an unquantized receiver, coding based on algebraic rotations is known to achieve modulation coding diversity. On the other hand, with a quantized receiver, algebraic rotations may not guarantee diversity. Through analysis, we propose specific rotations which result in the codewords having equidistant component-wise projections. We show that the proposed coding scheme achieves maximum modulation diversity with a low-complexity minimum distance decoder and perfect channel knowledge. Relaxing the perfect channel knowledge assumption we propose a novel training/estimation and receiver control technique to estimate the channel. We show that our coding/training/estimation scheme and minimum distance decoding achieve an error probability performance similar to that achieved with perfect channel knowledge.