Computational methods in subspace designs

Titelangaben

Braun, Michael ; Kiermaier, Michael ; Wassermann, Alfred:
Computational methods in subspace designs.
In: Greferath, Marcus ; Pavčević, Mario Osvin ; Natalia, Silberstein ; María Ángeles, Vázquez-Castro (Hrsg.): Network Coding and Subspace Designs. - Cham : Springer , 2018 . - S. 213-244 . - (Signals and Communication Technology )
ISBN 978-3-319-70292-6
DOI: https://doi.org/10.1007/978-3-319-70293-3_9

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Abstract

Subspace designs are the q-analogs of combinatorial designs. Introduced in the 1970s, these structures gained a lot of interest recently because of their application to random network coding. Compared to combinatorial designs, the number of blocks of subspace designs are huge even for the smallest instances. Thus, for a computational approach, sophisticated algorithms are indispensible. This chapter highlights computational methods for the construction of subspace designs, in particular methods based on group theory. Starting from tactical decompositions we present the method of Kramer and Mesner which allows to restrict the search for subspace designs to those with a prescribed group of automorphisms. This approach reduces the construction problem to the problem of solving a Diophantine linear system of equations. With slight modifications it can also be used to construct large sets of subspace designs. After a successful search, it is natural to ask if subspace designs are isomorphic. We give several helpful tools which allow to give answers in surprisingly many situations, sometimes in a purely theoretical way. Finally, we will give an overview of algorithms which are suitable to solve the underlying Diophantine linear system of equations. As a companion to chapter “q-Analogs of Designs: Subspace Designs” this chapter provides an extensive list of groups which were used to construct subspace designs and large sets of subspace designs.