Mimicking the idea of the generalized Hamming weight of linear codes, we introduce a new lattice invariant, the generalized theta series. Applications range from identifying stable lattices to the lattice isomorphism problem. Moreover, we provide counterexamples for the secrecy gain conjecture on isodual lattices, which claims that the ratio of the theta series of an isodual (and more generally, formally unimodular) lattice by the theta series of the integer lattice ${\mathbb{Z}^n}$ is minimized at a (unique) symmetry point.

We present novel techniques to verify the linearity of$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes, i.e., the binary codes obtained as the image of the generalized Gray map of$\mathbb{Z}_{2^{L}}-\mathbf{additive}$codes. The central idea is the definition of two auxiliary binary codes, which we denote by associated and decomposition codes. Since$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes can be linear or nonlinear, as a consequence of our contributions, we are able to construct families of linear$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes from nested Reed-Muller and cyclic codes. This work expands on previous results from the literature, where the linearity of$\mathbb{Z}_{2^{L}}. \mathbf{linear}$codes was established with respect to the kernel of the underlying$\mathbb{Z}_{2^{L}}-\mathbf{additive}$code and/or operations on$\mathbb{Z}_{2^{L}}$.

Recently, a design criterion depending on a lattice’s volume and theta series, called the secrecy gain, was proposed to quantify the secrecy-goodness of the applied lattice code for the Gaussian wiretap channel. To address the secrecy gain of Construction A4 lattices from formally self-dual$ \mathbb {Z}_{4}$-linear codes, i.e., codes for which the symmetrized weight enumerator (swe) coincides with the swe of its dual, we present new constructions of$ \mathbb {Z}_{4}$-linear codes which are formally self-dual with respect to the swe. For even lengths, formally self-dual$ \mathbb {Z}_{4}$-linear codes are constructed from nested binary codes and double circulant matrices. For odd lengths, a novel construction called odd extension from double circulant codes is proposed. Moreover, the concepts of Type I/II formally self-dual codes/unimodular lattices are introduced. Next, we derive the theta series of the formally unimodular lattices obtained by Construction A4 from formally self-dual$ \mathbb {Z}_{4}$-linear codes and describe a universal approach to determine their secrecy gains. The secrecy gain of Construction A4 formally unimodular lattices obtained from formally self-dual$ \mathbb {Z}_{4}$-linear codes is investigated, both for even and odd dimensions. Numerical evidence shows that for some parameters, Construction A4 lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Results concerning the flatness factor, another security criterion widely considered in the Gaussian wiretap channel, are also discussed.

In contrast to binary codes, odd-length self-dual codes exist over the integers modulo 4. Lately, the use of lattices constructed from codes over ℤ4to guarantee secure communication in a Gaussian wiretap channel was proposed and shown to exceed the performance of lattices from binary codes. This performance is measured regarding the secrecy gain, a criterion that depends on a lattice’s volume and theta series. Formally unimodular lattices, i.e., lattices with the same theta series as their dual, have presented promising results with respect to the secrecy gain. While previous contributions in the literature were mainly focused on even-dimensional lattices, this paper addresses the secrecy gain of odd-dimensional formally unimodular lattices obtained from codes over ℤ4, together with a novel construction of such codes.

This paper introduces the family of lattice-like packings, which generalizes lattices, consisting of packings possessing periodicity and geometric uniformity. The subfamily of formally unimodular (lattice-like) packings is further investigated. It can be seen as a generalization of the unimodular and isodual lattices, and the Construction A formally unimodular packings obtained from formally self-dual codes are presented. Recently, lattice coding for the Gaussian wiretap channel has been considered. A measure called the secrecy function was proposed to characterize the eavesdropper’s probability of correctly decoding. The aim is to determine the global maximum value of the secrecy function, called (strong) secrecy gain. We further apply lattice-like packings to coset coding for the Gaussian wiretap channel and show that the family of formally unimodular packings shares the same secrecy function behavior as unimodular and isodual lattices. We propose a universal approach to determine the secrecy gain of a Construction A formally unimodular packing obtained from a formally self-dual code. From the weight distribution of a code, we provide a necessary condition for a formally self-dual code such that its Construction A formally unimodular packing is secrecy-optimal. Finally, we demonstrate that formally unimodular packings/lattices can achieve higher secrecy gain than the best-known unimodular lattices.

Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction A4from codes over ${{\mathbb{Z}}_4}$ is derived, and we provide a universal approach to determine their secrecy gains. Initial results indicate that Construction A4lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Furthermore, a new code construction of formally self-dual ${{\mathbb{Z}}_4}$-linear codes is presented.

The nearest lattice point problem in$\mathbb {R}^{n}$is formulated in a distributed network with$n$nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication. Algorithms with a single as well as an unbounded number of rounds of communication are considered for the case$n=2$. For the algorithm with a single round, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice structure and that zero error requires an infinite number of communicated bits. In contrast, with an infinite number of allowed communication rounds, the nearest lattice point can be determined without error with a finite average number of communicated bits and a finite average number of rounds of communication. In two dimensions, the hexagonal lattice, which is most efficient for communication and compression, is found to be the most expensive in terms of communication cost.

Motivated by applications in reliable and secure communication, we address the problem of tiling (or partitioning) a finite constellation in$\mathbb{Z}_{2^{L}}^{n}$by subsets, in the case that the constellation does not possess an abelian group structure. The property that we do require is that the constellation is generated by a linear code through an injective mapping. The intrinsic relation between the code and the constellation provides a sufficient condition for a tiling to exist. We also present a necessary condition. Inspired by a result in group theory, we discuss results on tiling for the particular case when the finer constellation is an abelian group as well.

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector${\mathbf{X}}\in \mathbb {R}^{n}$in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of$\boldsymbol {X}$are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point, motivated by the fact that for some cases, a distributed algorithm for finding the Babai point is sufficient for finding the nearest lattice point itself. Two different probability models for$\boldsymbol {X}$are considered—uniform and Gaussian. For the uniform model, in dimensions two and three, the error probability is seen to grow with the packing density, and we demonstrate that the densest lattice in dimension two presents the worst error probability. For higher dimensions, we develop probabilistic concentration bounds as well as bounds based on geometric arguments for the error probability. The probabilistic bounds lead to the conclusion that for lattices which generate suitably thin coverings of$\mathbb {R}^{n}$(which includes lattices that meet Rogers’ bound on the covering radius), the error probability goes to unity as$n$grows. Probabilistic and geometric bounds are also used to estimate the error probability under the uniform model for various lattices including the$A_{n}$family and the Leech lattice,$\Lambda _{24}$. On the other hand, for the Gaussian model, the error probability goes to zero as the lattice dimension tends to infinity, provided the noise variance is sufficiently small.

Construction C* was recently introduced as a generalization of the multilevel Construction C (or Forney's code-formula), such that the coded levels may be dependent. Both constructions do not produce a lattice in general, hence the central idea of this paper is to present a 3-level lattice Construction C* scheme that admits an efficient nearest-neighborhood decoding. In order to achieve this objective, we choose coupled codes for levels 1 and 3, and set the second level code C2as an independent linear binary self-dual code, which is known to have a rich mathematical structure among families of linear codes. Our main result states a necessary and sufficient condition for this construction to generate a lattice. We then present examples of efficient lattices and also non-lattice constellations with good packing properties.

We prove that the ensemble of random Generalized Low-Density (GLD) lattices can attain the Poltyrev limit for an alphabet size increasing polylogarithmically with the lattice dimension. Our main theorem imposes no constraints on the normalized minimum distance of the code associated to the lattice ensemble, any asymptotically good code is suitable. This is a great improvement with respect to the first theorem on Poltyrev goodness of GLD lattices (2015). Our new bound is based on a new method referred to as the buckets approach where we employ the asymptotics of the restricted compositions of the Hamming weight. The new bound has applications in many coding areas beyond the specific lattice ensemble considered in this paper.

Lattice and special nonlattice multilevel constellations constructed from binary codes, such as Constructions A, C, and D, have relevant applications in Mathematics (sphere packing) and in Communication (multi-stage decoding and efficient vector quantization). In this work, we explore some properties of Construction C, in particular its geometric uniformity. We then propose a new multilevel construction, inspired by bit interleaved coded modulation (BICM), that we call Construction$C^\star $. We investigate the geometric uniformity, latticeness, and minimum distance properties of Construction$C^\star $, and discuss its superior packing efficiency when compared to Construction C.

We consider the closest lattice point problem in a distributed network setting and study the communication cost and the error probability for computing an approximate nearest lattice point, using the nearest-plane algorithm, due to Babai. Two distinct communication models, centralized and interactive, are considered. The importance of proper basis selection is addressed. Assuming a reduced basis for a two-dimensional lattice, we determine the approximation error of the nearest plane algorithm. The communication cost for determining the Babai point, or equivalently, for constructing the rectangular nearest-plane partition, is calculated in the interactive setting. For the centralized model, an algorithm is presented for reducing the communication cost of the nearest plane algorithm in an arbitrary number of dimensions.

We consider the problem of distributed computation of the nearest lattice point for a two-dimensional lattice. An interactive model of communication is considered. We address the problem of reconfiguring a specific rectangular partition, a nearest plane, or Babai, partition, into the Voronoi partition. Expressions are derived for the error probability as a function of the total number of communicated bits. With an infinite number of allowed communication rounds, the average cost of achieving zero error probability is shown to be finite. For the interactive model, with a single round of communication, expressions are obtained for the error probability as a function of the bits exchanged. We observe that the error exponent depends on the lattice.

Construction C (also known as Forney's multi-level code formula) forms a Euclidean code for the additive white Gaussian noise (AWGN) channel from L binary code components. If the component codes are linear, then the minimum distance is the same for all the points, although the kissing number may vary. In fact, while in the single level (L = 1) case it reduces to lattice Construction A, a multi-level Construction C is in general not a lattice. We show that the two-level (L = 2) case is special: a two-level Construction C satisfies Forney's definition for a geometrically uniform constellation. Specifically, every point sees the same configuration of neighbors, up to a reflection of the coordinates in which the lower level code is equal to 1. In contrast, for three levels and up (L ≥ 3), we construct examples where the distance spectrum varies between the points, hence the constellation is not geometrically uniform.