1,982 research outputs found
Quartic Curves and Their Bitangents
A smooth quartic curve in the complex projective plane has 36 inequivalent
representations as a symmetric determinant of linear forms and 63
representations as a sum of three squares. These correspond to Cayley octads
and Steiner complexes respectively. We present exact algorithms for computing
these objects from the 28 bitangents. This expresses Vinnikov quartics as
spectrahedra and positive quartics as Gram matrices. We explore the geometry of
Gram spectrahedra and we find equations for the variety of Cayley octads.
Interwoven is an exposition of much of the 19th century theory of plane
quartics.Comment: 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8,
other minor change
Configurations of lines and models of Lie algebras
The automorphism groups of the 27 lines on the smooth cubic surface or the 28
bitangents to the general quartic plane curve are well-known to be closely
related to the Weyl groups of and . We show how classical
subconfigurations of lines, such as double-sixes, triple systems or Steiner
sets, are easily constructed from certain models of the exceptional Lie
algebras. For and we are lead to
beautiful models graded over the octonions, which display these algebras as
plane projective geometries of subalgebras. We also interpret the group of the
bitangents as a group of transformations of the triangles in the Fano plane,
and show how this allows to realize the isomorphism in terms of harmonic cubes.Comment: 31 page
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
Switching codes and designs
AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters
Resolving sets for Johnson and Kneser graphs
A set of vertices in a graph is a {\em resolving set} for if, for
any two vertices , there exists such that the distances . In this paper, we consider the Johnson graphs and Kneser
graphs , and obtain various constructions of resolving sets for these
graphs. As well as general constructions, we show that various interesting
combinatorial objects can be used to obtain resolving sets in these graphs,
including (for Johnson graphs) projective planes and symmetric designs, as well
as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems
and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl
SOCIETY TRANSFORMATION AND SOCIAL DEVELOPMENT THROUGH UNIVERSITY-COMMUNITY TRANSFORMATION CENTRE (UCTC) VIA UNIVERSITI TUN HUSSEIN ONN MALAYSIA EXPERIENCE
Society transformation implies to various means in which globalising forces give impacts upon local communities and national societies with highly diverse historical backgrounds, social and economic patterns, political institutions and cultures. Hence, social development means a commitment that development processes need to benefit people, particularly but not only the poor, but also a recognition that people, and the way they interact in groups and society, and the norms that facilitates such interaction, shape development processes. Malaysia has a large base of middle income household segment. However, there are 94% working population which contributes to 32% on the Gross Domestic Product. In comparison to the countries at Western Europe, they attain a level of 65% working population that contributes to 51% GDP. This signifies that a majority of workforce contributes to the significant half of the wealth of the nations. Hence, it is important for Malaysia to attain these development levels through the leverage of information technology, the Internet and social media for the well-being of the community. University Community Transformation Centre (UCTC) is a mean to deliver more university intellectual and physical values to wider communities for their higher quality of living, activity outcome and income throughput. Academicians can also transfer and share their knowledge and expertise to community members in solving their constraints and problems. This paper highlights on UTHM community programmes that complement society daily problems and challenges, hence uplifting the well-being of community and business alike
New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
This paper presents several new construction techniques for low-density
parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on
specific classes of combinatorial designs, the improved code design focuses on
high-rate structured codes with constant column weights 3 and higher. The
proposed codes are efficiently encodable and exhibit good structural
properties. Experimental results on decoding performance with the sum-product
algorithm show that the novel codes offer substantial practical application
potential, for instance, in high-speed applications in magnetic recording and
optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications
Frequency permutation arrays
Motivated by recent interest in permutation arrays, we introduce and
investigate the more general concept of frequency permutation arrays (FPAs). An
FPA of length n=m lambda and distance d is a set T of multipermutations on a
multiset of m symbols, each repeated with frequency lambda, such that the
Hamming distance between any distinct x,y in T is at least d. Such arrays have
potential applications in powerline communication. In this paper, we establish
basic properties of FPAs, and provide direct constructions for FPAs using a
range of combinatorial objects, including polynomials over finite fields,
combinatorial designs, and codes. We also provide recursive constructions, and
give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design
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