386 research outputs found
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
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
The multicovering radius problem for some types of discrete structures
The covering radius problem is a question in coding theory concerned with
finding the minimum radius such that, given a code that is a subset of an
underlying metric space, balls of radius over its code words cover the
entire metric space. Klapper introduced a code parameter, called the
multicovering radius, which is a generalization of the covering radius. In this
paper, we introduce an analogue of the multicovering radius for permutation
codes (cf. Keevash and Ku, 2006) and for codes of perfect matchings (cf. Aw and
Ku, 2012). We apply probabilistic tools to give some lower bounds on the
multicovering radii of these codes. In the process of obtaining these results,
we also correct an error in the proof of the lower bound of the covering radius
that appeared in Keevash and Ku (2006). We conclude with a discussion of the
multicovering radius problem in an even more general context, which offers room
for further research.Comment: To appear in Designs, Codes and Cryptography (2012
Reconstruction of permutations distorted by single transposition errors
The reconstruction problem for permutations on elements from their
erroneous patterns which are distorted by transpositions is presented in this
paper. It is shown that for any an unknown permutation is uniquely
reconstructible from 4 distinct permutations at transposition distance at most
one from the unknown permutation. The {\it transposition distance} between two
permutations is defined as the least number of transpositions needed to
transform one into the other. The proposed approach is based on the
investigation of structural properties of a corresponding Cayley graph. In the
case of at most two transposition errors it is shown that
erroneous patterns are required in order to reconstruct an unknown permutation.
Similar results are obtained for two particular cases when permutations are
distorted by given transpositions. These results confirm some bounds for
regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200
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