Characterizations and construction methods for linear functional-repair storage codes

We present a precise characterization of linear functional-repair storage codes in terms of {\em admissible states/}, with each state made up from a collection of vector spaces over some fixed finite field. To illustrate the usefulness of our characterization, we provide several applications. We first describe a simple construction of functional-repair storage codes for a family of code parameters meeting the cutset bound outside the MBR and MSR points; these codes are conjectured to have optimal rate with respect to their repair locality. Then, we employ our characterization to develop a construction method to obtain functional repair codes for given parameters using symmetry groups, which can be used both to find new codes and to improve known ones. As an example of the latter use, we describe a beautiful functional-repair storage code that was found by this method, with parameters belonging to the family investigated earlier, which can be specified in terms of only eight different vector spaces.Comment: ISIT 2013, Istanbul, Turkey, 8-12 July 201

A Generalisation of Isomorphisms with Applications

In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs. We start with a study of invariance under TF-isomorphisms. In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs. We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph. By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants. This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group.Comment: 27 pages, 8 figure