Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima - the norms of the shortest independent vectors of a lattice - and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.
翻译:在数学研究中,拉特点是一个广受欢迎的研究领域,但在诸如加密学或多投入/多输出传输(MIMO)等更实际的领域也是如此。在数学理论中,多数情况下考虑的是真实数字的拉特点。然而,在通信中,通常会有兴趣的是复杂价值的处理。此外,通过使用双极传输以及两个时间档或频率的组合,四维(量值)方法变得越来越重要。因此,在普遍化领域,众所周知的拉特点算法和相关概念在这项工作中是普遍的。为此,对复杂的算术,包括Gaussian和Eisenstein整数的组合进行简要审查,并介绍四级估值数字,包括Lipschitz和Hurwitz整数的组合。在此基础上,两种重要算法的通用变式越来越重要:首先,多货币-时间级LLLL算法,因此,通过进行特别的Latticreal-alalalalal-alvalue应用的拉特值的缩基点,其普通值值值值值值的LLLLaldal-al-lvalatealalal-altialal-altialalalalalalationalalalalalalmas mas mas lautismlation real demas mas laismlate la lax是其内部总总算法系为独立的缩算法系的缩算法系的缩缩算法系的缩算法系的缩缩缩算法系的缩算法系。