We consider compound multi-input multi-output (MIMO) wiretap channels where minimal channel state information at the transmitter (CSIT) is assumed. Code construction is given for the special case of isotropic mutual information, which serves as a conservative strategy for general cases. Using the flatness factor for MIMO channels, we propose lattice codes universally achieving the secrecy capacity of compound MIMO wiretap channels up to a constant gap (measured in nats) that is equal to the number of transmit antennas. The proposed approach improves upon existing works on secrecy coding for MIMO wiretap channels from an error probability perspective, and establishes information theoretic security (in fact semantic security). We also give an algebraic construction to reduce the code design complexity, as well as the decoding complexity of the legitimate receiver. Thanks to the algebraic structures of number fields and division algebras, our code construction for compound MIMO wiretap channels can be reduced to that for Gaussian wiretap channels, up to some additional gap to secrecy capacity.

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We generalize bounds on the radius of perfect codes in a generic lattice, previously known for the cubic lattice. The new bounds are based on covering density, and covering radius of the ambient lattices, and, along with algebraic methods, allow to characterise all perfect codes in small dimension for a given ambient lattice. We provide case studies for some well known ambient lattices, such as the hexagonal lattice, and the checkerboard lattices. In contrast to the cubic lattice, these case studies show that, by changing the ambient lattice, one can find rich sets of perfect codes.

In this paper, we show that any sequence of infinite lattice constellations which is good for the unconstrained Gaussian channel can be shaped into a capacity-achieving sequence of codes for the power-constrained Gaussian channel under lattice decoding and non-uniform signaling. Unlike previous results in the literature, our scheme holds with no extra condition on the lattices (e.g., quantization-goodness or vanishing flatness factor), thus establishing a direct implication between AWGN-goodness, in the sense of Poltyrev and capacity-achieving codes. Our analysis uses properties of the discrete Gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates. In particular, we obtain a simple characterization of the finite-blocklength behavior of the scheme, showing that it approaches the optimal dispersion coefficient for high signal-to-noise ratio. We further show that for low signal-to-noise ratio, the discrete Gaussian over centered lattice constellations cannot achieve capacity, and thus a shift (or “dither”) is essentially necessary.

The compute-and-forward (C&F) protocol in quasi-static channels normally employs lattice codes based on the rational integers ℤ, the Gaussian integers ℤ[i], or the Eisenstein integers ℤ[ω], while its extension to more general channels often assumes channel state information at transmitters (CSIT). In this paper, we propose a novel scheme for C&F in block-fading channels without CSIT, which is referred to as ring C&F because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Owing to the multiplicative closure of the algebraic lattices employed, a relay is able to decode an algebraic-integer linear combination of lattice codewords. We analyze its achievable computation rates and show it outperforms conventional C&F based on the ℤ-lattices. By investigating the effect of the Diophantine approximation by algebraic conjugates, we prove that the degrees of freedom (DoFs) of the optimized computation rate are n/L, where n is the number of blocks and L is the number of users.

We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e., with packing density comparable to the Minkowski-Hlawka lower bound). Specializing to number field lattices, we obtain a number of interesting corollaries - for instance, the best known packing density of ideal lattices, and an elementary coding-theoretic construction of asymptotically dense Hurwitz lattices. All results are algorithmically effective, in the sense that, for any dimension, a finite family containing dense lattices is exhibited. For suitable constructions based on Craig's lattices, this family is smaller, in terms of alphabet-size, than previous ensembles in the literature.

We propose a coding scheme that achieves the capacity of the compound MIMO channel with algebraic lattices. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned channel realizations. To shape the constellation, a discrete Gaussian distribution over the lattice points is applied. These techniques, along with algebraic properties of the proposed lattices, are then used to construct a sub-optimal de-coupled coding scheme that achieves a constant gap to compound capacity by decoding in a lattice that does not dependent on the channel realization. The gap is characterized in terms of algebraic invariants of the code and is shown to be significantly smaller than previous schemes in the literature. We also exhibit alternative algebraic constructions that achieve the capacity of ergodic (SISO) fading channels.

The Golomb-Welch conjecture (1968) on the non-existence of perfect Lee codes in Znwith radius e ≥ 2 and dimensions n ≥ 3, widely believed to be true, has been up to now only proved for large radius in any dimension, for small dimensions, and for some small radii and specific n. The main result of this paper is that for radius e = 2, there are no perfect Lee linear codes in Znfor infinitely many values of n.

We consider explicit constructions of multi-level lattice codes that universally approach the capacity of the compound block-fading channel. Specifically, building on algebraic partitions of lattices, we show how to construct codes with negligible probability of error for any channel realization and normalized log-density approaching the Poltyrev limit. Capacity analyses and numerical results on the achievable rates for each partition level are provided. The proposed codes have several enjoyable properties such as constructiveness and good decoding complexity, as compared to random one-level codes. Numerical results for finite-dimensional multi-level lattices based on polar codes are exhibited.

Previous approaches to compute-and-forward (C&F) are mostly based on quantizing channel coefficients to integers. In this work, we investigate the C&F strategy over block fading channels using Construction A over rings, so as to allow better quantization for the channels. Advantages in decoding error probabilities and computation rates are demonstrated, and the construction is shown to outperform the C&F strategy over the integers Z.

We propose a lattice coding scheme that achieves the capacity of the compound block-fading channel. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned channel realizations. To shape the constellation, a discrete Gaussian distribution over the lattice points is applied. A by-product of our results is a refined analysis of the probability of error of the lattice Gaussian distribution in the AWGN channel.

In this work we show that algebraic lattices constructed from error-correcting codes achieve the ergodic capacity of the fading channel. The main ingredients for our construction are a generalized version of the Minkowski-Hlawka theorem and shaping techniques based on the lattice Gaussian distribution. The structure of the ring of integers in a number field plays an important role in the proposed construction. In the case of independent and identically distributed fadings, the lattices considered exhibit full diversity and an exponential decay of the probability of error with respect to the blocklength.

We consider the probability of data loss, or equivalently, the reliability function for an erasure coded distributed data storage system under worst case conditions. Data loss in an erasure coded system depends on probability distributions for the disk repair duration and the disk failure duration. In previous works, the data loss probability of such systems has been studied under the assumption of exponentially distributed disk failure and disk repair durations, using well-known analytic methods from the theory of Markov processes. These methods lead to an estimate of the integral of the reliability function. Here, we address the problem of directly calculating the data loss probability for general repair and failure duration distributions. A closed limiting form is developed for the probability of data loss, and it is shown that the probability of the event that a repair duration exceeds a failure duration is sufficient for characterizing the data loss probability. For the case of constant repair duration, we develop an expression for the conditional data loss probability given the number of failures experienced by a each node in a given time window. We do so by developing a geometric approach that relies on the computation of volumes of a family of polytopes that are related to the code. An exact calculation is provided, and an upper bound on the data loss probability is obtained by posing the problem as a set avoidance problem. Theoretical calculations are compared with simulation results.

Bounds are developed on the probability that the Cartesian product of a given number of finite random sets does not intersect (avoids) a given fixed set. These bounds are then used to estimate the probability of data loss in a distributed storage system that uses erasure codes to protect against data loss when disks fail. These are the first bounds on the probability of data loss that we are aware of. We compare our upper bound on the probability of data loss to approximations that are used in the literature, and show that our bounds are tighter and the gap is significant in some cases. Our bounds also suggest that in some cases, a more efficient (higher rate) code will suffice to meet a data loss probability target than that predicted by approximations widely used in the industry.

We consider the probability of data loss in an erasure coded distributed storage system. Data loss in an erasure coded system depends on the repair duration and the failure probability of individual disks. This dependence on the repair duration complicates the data loss probability analysis. In previous work, the data loss probability of such systems has been studied under the assumption of exponentially distributed disk life and disk repair durations, using well-known analytic methods from the theory of Markov processes. Here, we assume that the repair duration is a constant and derive an upper bound on the probability of data loss by calculating the volumes of specific polytopes that are determined by the code. Closed form bounds are exhibited for some example codes.

We address the problem of constructing explicit mappings from a k-dimensional continuous alphabet source to an n-dimensional Gaussian channel. The source is assumed to be uniformly distributed on the unit cube [0, 1)k. The scheme considered is based on a family of piecewise linear mappings and its performance is shown to be related to specific projected lattices of Zn. We study sufficient conditions for the mean squared error of such mappings to scale optimally with the signal-to-noise ratio of the channel and present an explicit construction for the case k = n-1. However, in some other cases our scheme requires the source to be uniformly distributed over a fundamental region of a specific lattice, that may be not congruent to [0, 1)k. A dissection technique is presented in order to overcome the source support mismatch and the MSE degradation of such a transformation is analyzed. An example construction of a 2 : n expansion mapping using the dissection technique is presented and is shown to exhibit optimal scaling of the MSE with the channel SNR.

In this paper, we consider the problem of transmitting a continuous alphabet discrete-time source over an additive white Gaussian noise channel in the bandwidth expansion case. We propose a constructive scheme based on a set of curves on the surface of a 2N-dimensional sphere. Our approach shows that the design of good codes for this communication problem relies on geometrical properties of spherical codes and projections of N-dimensional rectangular lattices. Theoretical comparisons with some previous works in terms of the mean squared error as a function of the channel SNR, as well as simulations, are provided.

In this paper we consider the problem of transmitting a continuous alphabet discrete-time source over an AWGN channel. The design of good curves for this purpose relies on geometrical properties of spherical codes and projections of N-dimensional lattices. We propose a constructive scheme based on a set of curves on the surface of a 2N-dimensional sphere and present comparisons with some previous works.

Q-ary lattices can be obtained from q-ary codes using the so-called Construction A. We investigate these lattices in the Lee metric and show how their decoding process can be related to the associated codes. For prime q we derive a Lee sphere decoding algorithm for q-ary lattices, present a brief discussion on its complexity and some comparisons with the classic sphere decoding.

Some hard problems from lattices, like LWE (Learning with Errors), are particularly suitable for application in Cryptography due to the possibility of using worst-case to average-case reductions as evidence of strong security properties. In this work, we show two LWE-based constructions of zero-knowledge identification schemes and discuss their performance and security. We also highlight the design choices that make our solution of both theoretical and practical interest.