589 research outputs found
Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes
We consider linear cyclic codes with the locality property, or locally
recoverable codes (LRC codes). A family of LRC codes that generalize the
classical construction of Reed-Solomon codes was constructed in a recent paper
by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper
we focus on optimal cyclic codes that arise from this construction. We give a
characterization of these codes in terms of their zeros, and observe that there
are many equivalent ways of constructing optimal cyclic LRC codes over a given
field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes)
and establish several results about their locality and minimum distance. The
locality parameter of a cyclic code is related to the dual distance of this
code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this
submission was posted earlier as arXiv:1502.01414 and was published in
Proceedings of the 2015 IEEE International Symposium on Information Theory,
Hong Kong, China, June 14-19, 2015, pp. 1262--126
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields
We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the dimension for
a given distance and length. Four new constructions of optimal linear codes
over small fields with locality properties are developed. The first two
approaches give binary cyclic codes with locality two. While the first
construction has availability one, the second binary code is characterized by
multiple available repair sets based on a binary Simplex code. The third
approach extends the first one to q-ary cyclic codes including (binary)
extension fields, where the locality property is determined by the properties
of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear
codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem,
Israe
Cyclic LRC Codes and their Subfield Subcodes
We consider linear cyclic codes with the locality property, or locally
recoverable codes (LRC codes). A family of LRC codes that generalizes the
classical construction of Reed-Solomon codes was constructed in a recent paper
by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014;
arXiv:1311.3284). In this paper we focus on the optimal cyclic codes that arise
from the general construction. We give a characterization of these codes in
terms of their zeros, and observe that there are many equivalent ways of
constructing optimal cyclic LRC codes over a given field. We also study
subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish
several results about their locality and minimum distance.Comment: Submitted for publicatio
Quantum Convolutional BCH Codes
Quantum convolutional codes can be used to protect a sequence of qubits of
arbitrary length against decoherence. We introduce two new families of quantum
convolutional codes. Our construction is based on an algebraic method which
allows to construct classical convolutional codes from block codes, in
particular BCH codes. These codes have the property that they contain their
Euclidean, respectively Hermitian, dual codes. Hence, they can be used to
define quantum convolutional codes by the stabilizer code construction. We
compute BCH-like bounds on the free distances which can be controlled as in the
case of block codes, and establish that the codes have non-catastrophic
encoders.Comment: 4 pages, minor changes, accepted for publication at the 10th Canadian
Workshop on Information Theory (CWIT'07
Gradient Coding from Cyclic MDS Codes and Expander Graphs
Gradient coding is a technique for straggler mitigation in distributed
learning. In this paper we design novel gradient codes using tools from
classical coding theory, namely, cyclic MDS codes, which compare favorably with
existing solutions, both in the applicable range of parameters and in the
complexity of the involved algorithms. Second, we introduce an approximate
variant of the gradient coding problem, in which we settle for approximate
gradient computation instead of the exact one. This approach enables graceful
degradation, i.e., the error of the approximate gradient is a
decreasing function of the number of stragglers. Our main result is that
normalized adjacency matrices of expander graphs yield excellent approximate
gradient codes, which enable significantly less computation compared to exact
gradient coding, and guarantee faster convergence than trivial solutions under
standard assumptions. We experimentally test our approach on Amazon EC2, and
show that the generalization error of approximate gradient coding is very close
to the full gradient while requiring significantly less computation from the
workers
- …