Axel Kohnert

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

In this paper we construct constant dimension codes with prescribed minimum distance. There is an increased interest in subspace codes in general since a paper [13] by Kotter and Kschischang where they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to construct constant dimension codes. Using this approach we found many new constant dimension codes with a larger number of codewords than previously known codes. We finally give a table of the best constant dimension codes we found.

SYMMETRICA, an object oriented computer-algebra system for the symmetric group

Abstract This is a review of an object oriented computer algebra system which is devoted to epresentation theory, invariant theory and combinatorics of the symmetric group. Moreover, it can be used for classical multivariate polynomials via the different actions of the symmetric group on the algebra of polynomials. The review contains a brief introduction to the basic methods used. Schubert polynomials are introduced, examples are given, and some applications are described. In particular, they provide a new algorithm for the evaluation of Littlewood-Richardson coefficients via symbolic computations using integer sequences instead of partitions, tableaux or lattice permutations.

Error-correcting linear codes : classification by isometry and applications

Linear Codes.- Bounds and Modifications.- Finite Fields.- Cyclic Codes.- Mathematics and Audio Compact Discs.- Enumeration of Isometry Classes.- Solving Systems of Diophantine Linear Equations.- Linear Codes with a Prescribed Minimum Distance.- The General Case.

The Discovery of Simple 7-Designs with Automorphism Group PTL (2, 32)

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.

Optimal linear codes from matrix groups

New linear codes (sometimes optimal) over the finite field with q elements are constructed. In order to do this, an equivalence between the existence of a linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations is used. To reduce the size of the system of equations, the search for solutions is restricted to solutions with special symmetry given by matrix groups. This allows to find more than 400 new codes for the case q=2,3,4,5,7,9.

Constructing two-weight codes with prescribed groups of automorphisms

We construct new linear two-weight codes over the finite field with q elements. To do so we solve the equivalent problem of finding point sets in the projective geometry with certain intersection properties. These point sets are in bijection to solutions of a Diophantine linear system of equations. To reduce the size of the system of equations we restrict the search for solutions to solutions with special symmetries. Two-weight codes can be used to define strongly regular graphs. We give tables of the two-weight codes and the corresponding strongly regular graphs. In some cases we find new distance-optimal two-weight codes and also new strongly regular graphs.

Large sets of t-designs over finite fields

A t-(n,k,@l;q)-design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field with q elements such that each t-dimensional subspace is contained in exactly @l blocks. A partition of the complete set of k-dimensional subspaces of V into disjoint t-(n,k,@l;q) designs is called a large set of t-designs over finite fields. In this paper we give the first nontrivial construction of such a large set with t>=2.

Large sets of subspace designs

In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{\ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a

MOLGEN 5.0, a Molecular Structure Generator

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(l,s)-extension of linear codes

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