Decoding complexity and trellis structure of lattices

Abstract

Decoding operation is the major obstacle associated with using a lattice in communication applications. There are two general methods for lattice decoding: i) The integer programming method based on geometry of numbers, and ii) The trellis method. This thesis has contributions to both methods, and provides results which make the cmoparison between the two methods possible.

Regarding method (i), Kannan's algorithm, which is currently known as the fastest method for decoding of a general lattice, is analyzed. Based on a geometrical interpretation of this algorithm, it is shown that it is a special case of a wider category of algorithms, called recursive cube search (RCS) algorithms. In this category, we improve Kannan's algorithm, and establish tight upper and lower bounds on the decoding complexity of lattices. The lower bounds prove that the RCS decoding complexity of any sequence of lattices with possible application in communications increases at least exponentially with dimension and coding gain.

Regarding method (ii), we discuss and develop a universal approach to the construction and analysis of the trellis diagrams of lattices using their bases. Based on this approach, we derive tight upper bounds on the trellis complexity of lattices, and study the problem of finding minimal trellis diagrams for lattices. The upper bounds both improve and generalize the previously known similar results. Minimal trellis diagrams for many important lattices are also constructed. These trellises, which are novel in many cases, can be employed to efficiently decode the lattices via the Viterbi algorithm. Moreover, we establish tight lower bounds on the trellis complexity of lattices. For many of the obtained trellises, these lower bounds provide a proof for minimality.

Finally, we derive some results in lattice theory with possible application in communications. These include an upper bound on covering radius of a lattice in terms of its successive minima, and an inequality on the coding gain of densest lattice packings in successive dimensions.