3,995 research outputs found
Multiset Combinatorial Batch Codes
Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai,
mimic a distributed storage of a set of data items on servers, in such
a way that any batch of data items can be retrieved by reading at most some
symbols from each server. Combinatorial batch codes, are replication-based
batch codes in which each server stores a subset of the data items.
In this paper, we propose a generalization of combinatorial batch codes,
called multiset combinatorial batch codes (MCBC), in which data items are
stored in servers, such that any multiset request of items, where any
item is requested at most times, can be retrieved by reading at most
items from each server. The setup of this new family of codes is motivated by
recent work on codes which enable high availability and parallel reads in
distributed storage systems. The main problem under this paradigm is to
minimize the number of items stored in the servers, given the values of
, which is denoted by . We first give a necessary and
sufficient condition for the existence of MCBCs. Then, we present several
bounds on and constructions of MCBCs. In particular, we
determine the value of for any , where
is the maximum size of a binary constant weight code of length
, distance four and weight . We also determine the exact value of
when or
Some new constructions of optimal linear codes and alphabet-optimal -locally repairable codes
In distributed storage systems, locally repairable codes (LRCs) are designed
to reduce disk I/O and repair costs by enabling recovery of each code symbol
from a small number of other symbols. To handle multiple node failures,
-LRCs are introduced to enable local recovery in the event of up to
failed nodes. Constructing optimal -LRCs has been a
significant research topic over the past decade. In \cite{Luo2022}, Luo
\emph{et al.} proposed a construction of linear codes by using unions of some
projective subspaces within a projective space. Several new classes of Griesmer
codes and distance-optimal codes were constructed, and some of them were proved
to be alphabet-optimal -LRCs.
In this paper, we first modify the method of constructing linear codes in
\cite{Luo2022} by considering a more general situation of intersecting
projective subspaces. This modification enables us to construct good codes with
more flexible parameters. Additionally, we present the conditions for the
constructed linear codes to qualify as Griesmer codes or achieve distance
optimality. Next, we explore the locality of linear codes constructed by
eliminating elements from a complete projective space. The novelty of our work
lies in establishing the locality as , , or -locality,
in contrast to the previous literature that only considered -locality.
Moreover, by combining analysis of code parameters and the C-M like bound for
-LRCs, we construct some alphabet-optimal -LRCs which
may be either Griesmer codes or not Griesmer codes. Finally, we investigate the
availability and alphabet-optimality of -LRCs constructed from our
modified framework.Comment: 25 page
Framework for classifying logical operators in stabilizer codes
Entanglement, as studied in quantum information science, and non-local
quantum correlations, as studied in condensed matter physics, are fundamentally
akin to each other. However, their relationship is often hard to quantify due
to the lack of a general approach to study both on the same footing. In
particular, while entanglement and non-local correlations are properties of
states, both arise from symmetries of global operators that commute with the
system Hamiltonian. Here, we introduce a framework for completely classifying
the local and non-local properties of all such global operators, given the
Hamiltonian and a bi-partitioning of the system. This framework is limited to
descriptions based on stabilizer quantum codes, but may be generalized. We
illustrate the use of this framework to study entanglement and non-local
correlations by analyzing global symmetries in topological order, distribution
of entanglement and entanglement entropy.Comment: 20 pages, 9 figure
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