18,848 research outputs found
On Optimal Binary One-Error-Correcting Codes of Lengths and
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that
triply-shortened and doubly-shortened binary Hamming codes (which have length
and , respectively) are optimal. Properties of such codes are
here studied, determining among other things parameters of certain subcodes. A
utilization of these properties makes a computer-aided classification of the
optimal binary one-error-correcting codes of lengths 12 and 13 possible; there
are 237610 and 117823 such codes, respectively (with 27375 and 17513
inequivalent extensions). This completes the classification of optimal binary
one-error-correcting codes for all lengths up to 15. Some properties of the
classified codes are further investigated. Finally, it is proved that for any
, there are optimal binary one-error-correcting codes of length
and that cannot be lengthened to perfect codes of length
.Comment: Accepted for publication in IEEE Transactions on Information Theory.
Data available at http://www.iki.fi/opottone/code
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
The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 was recently carried out in
[P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary
one-error-correcting codes of length 15: Part I--Classification," IEEE Trans.
Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying
work, the classified codes are studied in great detail, and their main
properties are tabulated. The results include the fact that 33 of the 80
Steiner triple systems of order 15 occur in such codes. Further understanding
is gained on full-rank codes via switching, as it turns out that all but two
full-rank codes can be obtained through a series of such transformations from
the Hamming code. Other topics studied include (non)systematic codes, embedded
one-error-correcting codes, and defining sets of codes. A classification of
certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of
coordinates fixed by symmetries of codes), added and extended many other
result
Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories
Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation
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
Sub-Classifier Construction for Error Correcting Output Code Using Minimum Weight Perfect Matching
Multi-class classification is mandatory for real world problems and one of
promising techniques for multi-class classification is Error Correcting Output
Code. We propose a method for constructing the Error Correcting Output Code to
obtain the suitable combination of positive and negative classes encoded to
represent binary classifiers. The minimum weight perfect matching algorithm is
applied to find the optimal pairs of subset of classes by using the
generalization performance as a weighting criterion. Based on our method, each
subset of classes with positive and negative labels is appropriately combined
for learning the binary classifiers. Experimental results show that our
technique gives significantly higher performance compared to traditional
methods including the dense random code and the sparse random code both in
terms of accuracy and classification times. Moreover, our method requires
significantly smaller number of binary classifiers while maintaining accuracy
compared to the One-Versus-One.Comment: 7 pages, 3 figure
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