A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of`-elementary… (More)
A lattice in euclidean space which is an orthogonal sum of nontrivial sub-lattices is called decomposable. We present an algorithm to construct a lattice's decomposition into indecomposable sublattices. Similar methods are used to prove a covering theorem for generating systems of lattices and to speed up variations of the LLL algorithm for the computation… (More)
I have a great suspicion that for example Euler today would spend much more of his time on writing software because he spent so much of his time e.g., in efforts of calculating tables of moon positions. And I believe that Gauß as well would spend much more time sitting in front of the screen.
In this short note we give incremental algorithms for the following lattice problems: finding a basis of a lattice, computing the successive minima, and determining the orthogonal decomposition. We prove an upper bound for the number of update steps for every insertion order. For the determination of the orthogonal decomposition we efficiently implement an… (More)
We have adopted the incremental construction algorithm paradigm which is a standard technique in computational geometry for lattice problems. The considered lattice problems are: lattice basis, successive minima, orthogonal decomposition. For these problems we give algorithms which are reasonable fast. Using a classical theorem of MINKOWSKI we prove a… (More)