645 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
Wet paper codes and the dual distance in steganography
In 1998 Crandall introduced a method based on coding theory to secretly embed
a message in a digital support such as an image. Later Fridrich et al. improved
this method to minimize the distortion introduced by the embedding; a process
called wet paper. However, as previously emphasized in the literature, this
method can fail during the embedding step. Here we find sufficient and
necessary conditions to guarantee a successful embedding by studying the dual
distance of a linear code. Since these results are essentially of combinatorial
nature, they can be generalized to systematic codes, a large family containing
all linear codes. We also compute the exact number of solutions and point out
the relationship between wet paper codes and orthogonal arrays
Locally Testable Codes and Cayley Graphs
We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than
and has no short linear dependencies, but yields a shortest-path metric that
embeds into with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into .
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, monomials and
trinomials over finite fields with even characteristic are employed to
construct a number of families of binary cyclic codes. Lower bounds on the
minimum weight of some families of the cyclic codes are developed. The minimum
weights of other families of the codes constructed in this paper are
determined. The dimensions of the codes are flexible. Some of the codes
presented in this paper are optimal or almost optimal in the sense that they
meet some bounds on linear codes. Open problems regarding binary cyclic codes
from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687,
arXiv:1206.437
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