The concept of well-rounded lattices has recently found important applications in the setting of a fading single-input single-output (SISO) wiretap channel. It has been shown that, under this setup, the property of being well-rounded is critical for minimizing the eavesdropper's probability of correct decoding in lower SNR regimes. The superior performance of coset codes constructed from well-rounded lattices has been illustrated in several simulations. In the present article, this work is extended to fading multiple-input multiple-output (MIMO) wiretap channels, and similar design criteria as in the SISO case are derived. Further, explicit coset codes for Rayleigh fading MIMO wiretap channels are designed. In particular, it is shown through extensive simulations that sublattices of the well-known Alamouti code and Golden code which meet our design criteria perform better than scalar multiples of the code lattice for the same parameters.
The previous ECDP-based design criteria are commonly based on relatively coarse approximations, resulting in the so-called in the (SISO) case @cite_22 , also studied in @cite_19 , or the @cite_11 in the MIMO setting. See @cite_16 for related work in the reliability setting. In @cite_6 so-called @math -th coding gains are defined which were used in @cite_17 to derive a simple geometric criterion for the design of coset codes.
@cite_22 @cite_17 @cite_6 @cite_19 @cite_16 @cite_11
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Coding for the Gaussian Wiretap Channel can be done with nested lattices. The fine lattice must be designed in the same way as lattices used on the regular Gaussian Channel while the coarse one must be a lattice whose theta series is minimized. We present a criterion of design of both the fine and coarse lattice when used on a Rayleigh fading wiretap channel In a distributed communications scenario, a relay applying the compute-and-forward strategy for the real-valued channel model aims to decode an integer linear combination of transmitted messages, a task of very complex nature for which general efficient algorithms for dimension n > 1 have not yet been developed. Nonetheless, the maximum-likelihood decoding metric related to solving for the desired lattice point exhibits interesting properties which lead to partial design criteria for lattice codes in compute-and-forward. This article generalizes maximum-likelihood decoding at the relay to allow for arbitrary real lattice codes at the transmitters, and studies the behavior of the resulting decoding metric using an approximation of the theta series of a lattice, which is itself derived in this article. For the first time, the resulting random sums of lattices over whose points the relay needs to perform a sum are analyzed and furthermore, previous related work is extended to the case of K > 2 transmitters. The specific cases K = 2 and K = 3 are studied empirically. Recently, several papers have been concentrating on reducing the decoding complexity of high-rate space-time codes. While the research has led to some impressive reductions in decoding complexity, the geometric methods so far used appear to have faced some fundamental limits. In this paper, we study what happens if we let go of the assumption of full diversity and study the possibility of reducing the complexity, while holding on to a high code rate, by reducing the length of codes. We will develop some tools that can be used to measure the changes we will encounter when reducing the code length. We will also study the achievable diversity-multiplexing gain trade-off (DMT) of codes with less than minimum delay (LMD) and discuss some code constructions. 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. 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. We consider MIMO (Multiple Input Multiple Output) wiretap channels, where a legitimate transmitter Alice is communicating with a legitimate receiver Bob in the presence of an eavesdropper Eve, and communication is done via MIMO channels. We suppose that Alice's strategy is to use an infinite lattice codebook, which then allows her to perform coset encoding. We analyze Eve's probability of correctly decoding the message Alice meant to Bob, and from minimizing this probability, we derive a code design criterion for MIMO lattice wiretap codes. The case of block fading channels is treated similarly, and fast fading channels are derived as a particular case. The Alamouti code is carefully studied as an illustration of the analysis provided.