35,279 research outputs found
Computing the Ball Size of Frequency Permutations under Chebyshev Distance
Let be the set of all permutations over the multiset
where
. A frequency permutation array (FPA) of minimum distance is a
subset of in which every two elements have distance at least .
FPAs have many applications related to error correcting codes. In coding
theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived
from the size of balls of certain radii. We propose two efficient algorithms
that compute the ball size of frequency permutations under Chebyshev distance.
Both methods extend previous known results. The first one runs in time and space. The second one runs in time and
space. For small constants and ,
both are efficient in time and use constant storage space.Comment: Submitted to ISIT 201
Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
Universal quantum computation on decoherence-free subspaces and subsystems
(DFSs) is examined with particular emphasis on using only physically relevant
interactions. A necessary and sufficient condition for the existence of
decoherence-free (noiseless) subsystems in the Markovian regime is derived here
for the first time. A stabilizer formalism for DFSs is then developed which
allows for the explicit understanding of these in their dual role as quantum
error correcting codes. Conditions for the existence of Hamiltonians whose
induced evolution always preserves a DFS are derived within this stabilizer
formalism. Two possible collective decoherence mechanisms arising from
permutation symmetries of the system-bath coupling are examined within this
framework. It is shown that in both cases universal quantum computation which
always preserves the DFS (*natural fault-tolerant computation*) can be
performed using only two-body interactions. This is in marked contrast to
standard error correcting codes, where all known constructions using one or
two-body interactions must leave the codespace during the on-time of the
fault-tolerant gates. A further consequence of our universality construction is
that a single exchange Hamiltonian can be used to perform universal quantum
computation on an encoded space whose asymptotic coding efficiency is unity.
The exchange Hamiltonian, which is naturally present in many quantum systems,
is thus *asymptotically universal*.Comment: 40 pages (body: 30, appendices: 3, figures: 5, references: 2). Fixed
problem with non-printing figures. New references added, minor typos
correcte
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
Correcting Charge-Constrained Errors in the Rank-Modulation Scheme
We investigate error-correcting codes for a the
rank-modulation scheme with an application to flash memory
devices. In this scheme, a set of n cells stores information in the
permutation induced by the different charge levels of the individual
cells. The resulting scheme eliminates the need for discrete
cell levels, overcomes overshoot errors when programming cells (a
serious problem that reduces the writing speed), and mitigates the
problem of asymmetric errors. In this paper, we study the properties
of error-correcting codes for charge-constrained errors in the
rank-modulation scheme. In this error model the number of errors
corresponds to the minimal number of adjacent transpositions required
to change a given stored permutation to another erroneous
one—a distance measure known as Kendall’s τ-distance.We show
bounds on the size of such codes, and use metric-embedding techniques
to give constructions which translate a wealth of knowledge
of codes in the Lee metric to codes over permutations in Kendall’s
τ-metric. Specifically, the one-error-correcting codes we construct
are at least half the ball-packing upper bound
Trellis decoding complexity of linear block codes
In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of “uniform” optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths
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