16 research outputs found
Repairable Replication-based Storage Systems Using Resolvable Designs
We consider the design of regenerating codes for distributed storage systems
at the minimum bandwidth regeneration (MBR) point. The codes allow for a repair
process that is exact and uncoded, but table-based. These codes were introduced
in prior work and consist of an outer MDS code followed by an inner fractional
repetition (FR) code where copies of the coded symbols are placed on the
storage nodes. The main challenge in this domain is the design of the inner FR
code.
In our work, we consider generalizations of FR codes, by establishing their
connection with a family of combinatorial structures known as resolvable
designs. Our constructions based on affine geometries, Hadamard designs and
mutually orthogonal Latin squares allow the design of systems where a new node
can be exactly regenerated by downloading packets from a subset
of the surviving nodes (prior work only considered the case of ).
Our techniques allow the design of systems over a large range of parameters.
Specifically, the repetition degree of a symbol, which dictates the resilience
of the system can be varied over a large range in a simple manner. Moreover,
the actual table needed for the repair can also be implemented in a rather
straightforward way. Furthermore, we answer an open question posed in prior
work by demonstrating the existence of codes with parameters that are not
covered by Steiner systems
Replication based storage systems with local repair
We consider the design of regenerating codes for distributed storage systems
that enjoy the property of local, exact and uncoded repair, i.e., (a) upon
failure, a node can be regenerated by simply downloading packets from the
surviving nodes and (b) the number of surviving nodes contacted is strictly
smaller than the number of nodes that need to be contacted for reconstructing
the stored file.
Our codes consist of an outer MDS code and an inner fractional repetition
code that specifies the placement of the encoded symbols on the storage nodes.
For our class of codes, we identify the tradeoff between the local repair
property and the minimum distance. We present codes based on graphs of high
girth, affine resolvable designs and projective planes that meet the minimum
distance bound for specific choices of file sizes
On highly regular digraphs
We explore directed strongly regular graphs (DSRGs) and their connections to association schemes and finite incidence structures. More specically, we study flags and antiflags of finite
incidence structures to provide explicit constructions of DSRGs. By using this connection between the finite incidence structures and digraphs, we verify the existence and non-existence of -designs with certain parameters by the existence and non-existence of corresponding digraphs, and vice versa. We also classify DSRGs of given parameters according to isomorphism classes. Particularly, we examine the actions of automorphism groups to provide explicit
examples of isomorphism classes and connection to association schemes. We provide infinite families of vertex-transitive DSRGs in connection to non-commutative association schemes.
These graphs are obtained from tactical configurations and coset graphs
Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures
Recently, locally repairable codes has gained significant interest for their
potential applications in distributed storage systems. However, most
constructions in existence are over fields with size that grows with the number
of servers, which makes the systems computationally expensive and difficult to
maintain. Here, we study linear locally repairable codes over the binary field,
tolerating multiple local erasures. We derive bounds on the minimum distance on
such codes, and give examples of LRCs achieving these bounds. Our main
technical tools come from matroid theory, and as a byproduct of our proofs, we
show that the lattice of cyclic flats of a simple binary matroid is atomic.Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018.
This extended arxiv version includes corrected versions of Theorem 1.4 and
Proposition 6 that appeared in the IZS 2018 proceeding
Invariants of Quadratic Forms and applications in Design Theory
The study of regular incidence structures such as projective planes and
symmetric block designs is a well established topic in discrete mathematics.
Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the
Hasse-Minkowski local-global theory for quadratic forms to derive non-existence
results for certain design parameters. Several combinatorialists have provided
alternative proofs of this result, replacing conceptual arguments with
algorithmic ones. In this paper, we show that the methods required are purely
linear-algebraic in nature and are no more difficult conceptually than the
theory of the Jordan Canonical Form. Computationally, they are rather easier.
We conclude with some classical and recent applications to design theory,
including a novel application to the decomposition of incidence matrices of
symmetric designs.Comment: 23 page