Characterizations and construction methods for linear functional-repair storage codes

We present a precise characterization of linear functional-repair storage codes in terms of 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.