820 research outputs found
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
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
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
Perfect binary codes: classification and properties
An r-perfect binary code is a subset of ℤ2n such that for any word, there is a unique codeword at Hamming distance at most r. Such a code is r-error-correcting. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and adding a constant vector. The main result of this thesis is a computer-aided classification, up to equivalence, of the 1-perfect binary codes of length 15.
In an extended 1-perfect code, the neighborhood of a codeword corresponds to a Steiner quadruple system. To utilize this connection, we start with a computational classification of Steiner quadruple systems of order 16. This classification is also used to establish the nonexistence of Steiner quintuple systems S(4, 5, 17).
The classification of the codes is used for computational examination of their properties. These properties include occurrences of Steiner triple and quadruple systems, automorphisms, ranks, structure of i-components and connections to orthogonal arrays and mixed perfect codes.
It is also proved that extended 1-perfect binary codes are equivalent if and only if their minimum distance graphs are isomorphic
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
The classification of flag-transitive Steiner 3-designs
We solve the long-standing open problem of classifying all 3-(v,k,1) designs
with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom.
Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F.
Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating
back to 1965). Our result relies on the classification of the finite
2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry
Block-Transitive Designs in Affine Spaces
This paper deals with block-transitive - designs in affine
spaces for large , with a focus on the important index case. We
prove that there are no non-trivial 5- designs admitting a
block-transitive group of automorphisms that is of affine type. Moreover, we
show that the corresponding non-existence result holds for 4- designs,
except possibly when the group is one-dimensional affine. Our approach involves
a consideration of the finite 2-homogeneous affine permutation groups.Comment: 10 pages; to appear in: "Designs, Codes and Cryptography
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