1,007 research outputs found
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
An introduction of the theory of nonlinear error-correcting codes
Nonlinear error-correcting codes are the topic of this thesis. As a class of codes, it has been investigated far less than the class of linear error-correcting codes. While the latter have many practical advantages, it the former that contain the optimal error-correcting codes. In this project the theory (with illustrative examples) of currently known nonlinear codes is presented. Many definitions and theorems (often with their proofs) are presented thus providing the reader with the opportunity to experience the necessary level of mathematical rigor for good understanding of the subject. Also, the examples will give the reader the additional benefit of seeing how the theory can be put to use. An introduction to a technique for finding new codes via computer search is presented
Simple Quantum Error Correcting Codes
Methods of finding good quantum error correcting codes are discussed, and
many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where
C_1 and C_2 are classical codes, is used to obtain codes for up to 16
information qubits with correction of small numbers of errors. The results are
tabulated. More efficient codes are obtained by allowing C_1 to have reduced
distance, and introducing sign changes among the code words in a systematic
manner. This systematic approach leads to single-error correcting codes for 3,
4 and 5 information qubits with block lengths of 8, 10 and 11 qubits
respectively.Comment: Submitted to Phys. Rev. A. in May 1996. 21 pages, no figures. Further
information at http://eve.physics.ox.ac.uk/ASGhome.htm
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