14,287 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
Singularities of Type-Q ABS Equations
The type-Q equations lie on the top level of the hierarchy introduced by
Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts
of KdV-type integrable partial differential equations. We ask what
singularities are possible in the solutions of these equations, and examine the
relationship between the singularities and the principal integrability feature
of multidimensional consistency. These questions are considered in the global
setting and therefore extend previous considerations of singularities which
have been local. What emerges are some simple geometric criteria that determine
the allowed singularities, and the interesting discovery that generically the
presence of singularities leads to a breakdown in the global consistency of
such systems despite their local consistency property. This failure to be
globally consistent is quantified by introducing a natural notion of monodromy
for isolated singularities.Comment: contribution to the SIDE-9 special issue of SIGM
Classification of GHZ-type, W-type and GHZ-W-type multiqubit entanglements
We propose the concept of SLOCC-equivalent basis (SEB) in the multiqubit
space. In particular, two special SEBs, the GHZ-type and the W-type basis are
introduced. They can make up a more general family of multiqubit states, the
GHZ-W-type states, which is a useful kind of entanglement for quantum
teleporatation and error correction. We completely characterize the property of
this type of states, and mainly classify the GHZ-type states and the W-type
states in a regular way, which is related to the enumerative combinatorics.
Many concrete examples are given to exhibit how our method is used for the
classification of these entangled states.Comment: 16 pages, Revte
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