# Voronoi-Based Coding

Sammanfattning: The performance of a digital communication system can generally be improved by increasing the number of variables being jointly coded. In this sense, it is desirable to have, e.g., higher-dimensional quantizers, longer channel codes, and more users in a multiple-access system. However, increasing the number of variables results in higher complexity of encoding and decoding, which are two limiting factors in the choice of coding methods. In this dissertation, which comprises seven published or submitted articles, algorithms are discussed for three related applications in telecommunications: vector quantizer encoding, multiuser detection, and soft-decision channel decoding. A unified approach is obtained through a common formulation in terms of discrete optimization, or, taking on a geometric viewpoint, searching in multidimensional point sets. A convenient instrument to characterize the structure of these sets is the Voronoi diagram.

The first considered application is vector quantization, with some focus on lattices. Lattices are regular structures of infinitely many points that after truncation can be employed as quantizers. An algorithm is developed to optimize the quantization performance of lattices. Employing the algorithm, two lattices are found that improve on previous results. It is also discovered that the so-called D+ tessellation, which is a union of two lattices, is superior to any known single lattice in dimensions 7 and 9. Truncated lattices are also analyzed, revealing that their distortion in relation to that of optimal quantization increases with the number of points. An algorithm for quantizer design is introduced that maintains close to minimal distortion by suitably stretching the truncated lattice. The work on vector quantization also includes theory and algorithms for index assignment, which is a way to incorporate error-robustness into the quantizer design through a careful codevector ordering. Another contribution is a comparison of the complexity of some encoding methods.

Multiuser detection in CDMA systems is formulated as the geometric problem of searching a point set having a certain structure. Properties of Voronoi diagrams of such structures are given, thus making it possible to apply a known Voronoi-based search algorithm for detection.

Soft-decision decoding of block channel codes is the third application being studied, again by means of the Voronoi diagram. A fast algorithm is presented to investigate the Voronoi diagram of binary linear block codes. Several well-known codes are characterized. Asymptotic theory shows that for most classes of long codes, the complexity of the Voronoi diagram as a function of the bit rate displays a distinct threshold at the rate of one half. Voronoi-based soft decision decoding is essentially practicable only for rates above this value.

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