38,269 research outputs found
Solving Multiclass Learning Problems via Error-Correcting Output Codes
Multiclass learning problems involve finding a definition for an unknown
function f(x) whose range is a discrete set containing k > 2 values (i.e., k
``classes''). The definition is acquired by studying collections of training
examples of the form [x_i, f (x_i)]. Existing approaches to multiclass learning
problems include direct application of multiclass algorithms such as the
decision-tree algorithms C4.5 and CART, application of binary concept learning
algorithms to learn individual binary functions for each of the k classes, and
application of binary concept learning algorithms with distributed output
representations. This paper compares these three approaches to a new technique
in which error-correcting codes are employed as a distributed output
representation. We show that these output representations improve the
generalization performance of both C4.5 and backpropagation on a wide range of
multiclass learning tasks. We also demonstrate that this approach is robust
with respect to changes in the size of the training sample, the assignment of
distributed representations to particular classes, and the application of
overfitting avoidance techniques such as decision-tree pruning. Finally, we
show that---like the other methods---the error-correcting code technique can
provide reliable class probability estimates. Taken together, these results
demonstrate that error-correcting output codes provide a general-purpose method
for improving the performance of inductive learning programs on multiclass
problems.Comment: See http://www.jair.org/ for any accompanying file
Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers
We transfer the concept of linear feed-back shift registers to quantum
circuits. It is shown how to use these quantum linear shift registers for
encoding and decoding cyclic quantum error-correcting codes.Comment: 18 pages, 15 figures, submitted to Proc. R. Soc.
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
Scheme for constructing graphs associated with stabilizer quantum codes
We propose a systematic scheme for the construction of graphs associated with
binary stabilizer codes. The scheme is characterized by three main steps:
first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum
code; second, the canonical form of the CWS code is uncovered; third, the input
vertices are attached to the graphs. To check the effectiveness of the scheme,
we discuss several graphical constructions of various useful stabilizer codes
characterized by single and multi-qubit encoding operators. In particular, the
error-correcting capabilities of such quantum codes are verified in
graph-theoretic terms as originally advocated by Schlingemann and Werner.
Finally, possible generalizations of our scheme for the graphical construction
of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum
codes are briefly addressed.Comment: 42 pages, 12 figure
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size nĂn (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes
The doubly shortened perfect codes of length 13 are classified utilizing the
classification of perfect codes 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, to appear]; there are 117821 such
(13,512,3) codes. By applying a switching operation to those codes, two more
(13,512,3) codes are obtained, which are then not doubly shortened perfect
codes.Comment: v2: a correction concerning shortened codes of length 1
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