3,105 research outputs found
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
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
Steiner t-designs for large t
One of the most central and long-standing open questions in combinatorial
design theory concerns the existence of Steiner t-designs for large values of
t. Although in his classical 1987 paper, L. Teirlinck has shown that
non-trivial t-designs exist for all values of t, no non-trivial Steiner
t-design with t > 5 has been constructed until now. Understandingly, the case t
= 6 has received considerable attention. There has been recent progress
concerning the existence of highly symmetric Steiner 6-designs: It is shown in
[M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial
flag-transitive Steiner 6-design can exist. In this paper, we announce that
essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008,
ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in
Computer Scienc
On the existence of block-transitive combinatorial designs
Block-transitive Steiner -designs form a central part of the study of
highly symmetric combinatorial configurations at the interface of several
disciplines, including group theory, geometry, combinatorics, coding and
information theory, and cryptography. The main result of the paper settles an
important open question: There exist no non-trivial examples with (or
larger). The proof is based on the classification of the finite 3-homogeneous
permutation groups, itself relying on the finite simple group classification.Comment: 9 pages; to appear in "Discrete Mathematics and Theoretical Computer
Science (DMTCS)
Twin bent functions and Clifford algebras
This paper examines a pair of bent functions on and their
relationship to a necessary condition for the existence of an automorphism of
an edge-coloured graph whose colours are defined by the properties of a
canonical basis for the real representation of the Clifford algebra
Some other necessary conditions are also briefly examined.Comment: 11 pages. Preprint edited so that theorem numbers, etc. match those
in the published book chapter. Final post-submission paragraph added to
Section 6. in "Algebraic Design Theory and Hadamard Matrices: ADTHM,
Lethbridge, Alberta, Canada, July 2014", Charles J. Colbourn (editor), pp.
189-199, 201
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