108 research outputs found
Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel
Using combinatorial arguments, we determine an upper bound on achievable
rates of stabilizer codes used over the quantum erasure channel. This allows us
to recover the no-cloning bound on the capacity of the quantum erasure channel,
R is below 1-2p, for stabilizer codes: we also derive an improved upper bound
of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for
0 < p < 1/2 and for any family of stabilizer codes whose generators have
weights bounded from above by a constant - low density stabilizer codes.
We obtain an application to percolation theory for a family of self-dual
tilings of the hyperbolic plane. We associate a family of low density
stabilizer codes with appropriate finite quotients of these tilings. We then
relate the probability of percolation to the probability of a decoding error
for these codes on the quantum erasure channel. The application of our upper
bound on achievable rates of low density stabilizer codes gives rise to an
upper bound on the critical probability for these tilings.Comment: 32 page
Constructions of Rank Modulation Codes
Rank modulation is a way of encoding information to correct errors in flash
memory devices as well as impulse noise in transmission lines. Modeling rank
modulation involves construction of packings of the space of permutations
equipped with the Kendall tau distance.
We present several general constructions of codes in permutations that cover
a broad range of code parameters. In particular, we show a number of ways in
which conventional error-correcting codes can be modified to correct errors in
the Kendall space. Codes that we construct afford simple encoding and decoding
algorithms of essentially the same complexity as required to correct errors in
the Hamming metric. For instance, from binary BCH codes we obtain codes
correcting Kendall errors in memory cells that support the order of
messages, for any constant We also construct
families of codes that correct a number of errors that grows with at
varying rates, from to . One of our constructions
gives rise to a family of rank modulation codes for which the trade-off between
the number of messages and the number of correctable Kendall errors approaches
the optimal scaling rate. Finally, we list a number of possibilities for
constructing codes of finite length, and give examples of rank modulation codes
with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor
A Construction of Quantum LDPC Codes from Cayley Graphs
We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison
and Shokrollahi. It is based on the Cayley graph of Fn together with a set of
generators regarded as the columns of the parity-check matrix of a classical
code. We give a general lower bound on the minimum distance of the Quantum code
in where d is the minimum distance of the classical code.
When the classical code is the repetition code, we are able to
compute the exact parameters of the associated Quantum code which are .Comment: The material in this paper was presented in part at ISIT 2011. This
article is published in IEEE Transactions on Information Theory. We point out
that the second step of the proof of Proposition VI.2 in the published
version (Proposition 25 in the present version and Proposition 18 in the ISIT
extended abstract) is not strictly correct. This issue is addressed in the
present versio
Large sets with small doubling modulo p are well covered by an arithmetic progression
We prove that there is a small but fixed positive integer e such that for
every prime larger than a fixed integer, every subset S of the integers modulo
p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an
arithmetic progression of length |2S|-|S|+1. This is the first result of this
nature which places no unnecessary restrictions on the size of S.Comment: 16 page
Application of coding theory to interconnection networks
AbstractWe give a few examples of applications of techniques and results borrowed from error-correcting codes to problems in graphs and interconnection networks. The degree and diameter of Cayley graphs with vertex set (Z2Z)r are investigated. The asymptotic case is dealt with in Section 2. The robustness, or fault tolerance, of the n-cube interconnection network is studied in Section 3
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