813 research outputs found
Load-Balanced Fractional Repetition Codes
We introduce load-balanced fractional repetition (LBFR) codes, which are a
strengthening of fractional repetition (FR) codes. LBFR codes have the
additional property that multiple node failures can be sequentially repaired by
downloading no more than one block from any other node. This allows for better
use of the network, and can additionally reduce the number of disk reads
necessary to repair multiple nodes. We characterize LBFR codes in terms of
their adjacency graphs, and use this characterization to present explicit
constructions LBFR codes with storage capacity comparable existing FR codes.
Surprisingly, in some parameter regimes, our constructions of LBFR codes match
the parameters of the best constructions of FR codes
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
Irreducibility of configurations
In a paper from 1886, Martinetti enumerated small -configurations. One
of his tools was a construction that permits to produce a
-configuration from a -configuration. He called configurations
that were not constructible in this way irreducible configurations. According
to his definition, the irreducible configurations are Pappus' configuration and
four infinite families of configurations. In 2005, Boben defined a simpler and
more general definition of irreducibility, for which only two
-configurations, the Fano plane and Pappus' configuration, remained
irreducible. The present article gives a generalization of Boben's reduction
for both balanced and unbalanced -configurations, and proves several
general results on augmentability and reducibility. Motivation for this work is
found, for example, in the counting and enumeration of configurations
New Classes of Partial Geometries and Their Associated LDPC Codes
The use of partial geometries to construct parity-check matrices for LDPC
codes has resulted in the design of successful codes with a probability of
error close to the Shannon capacity at bit error rates down to . Such
considerations have motivated this further investigation. A new and simple
construction of a type of partial geometries with quasi-cyclic structure is
given and their properties are investigated. The trapping sets of the partial
geometry codes were considered previously using the geometric aspects of the
underlying structure to derive information on the size of allowable trapping
sets. This topic is further considered here. Finally, there is a natural
relationship between partial geometries and strongly regular graphs. The
eigenvalues of the adjacency matrices of such graphs are well known and it is
of interest to determine if any of the Tanner graphs derived from the partial
geometries are good expanders for certain parameter sets, since it can be
argued that codes with good geometric and expansion properties might perform
well under message-passing decoding.Comment: 34 pages with single column, 6 figure
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