283 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
Constructing flag-transitive, point-imprimitive designs
We give a construction of a family of designs with a specified point-partition and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric 2-(1408,336,80) design with automorphism group 2^12:((3⋅M22):2) and a construction of one of the families of the symplectic designs (the designs S^−(n) ) exhibiting a flag-transitive, point-imprimitive automorphism group.PostprintPeer reviewe
Flag-transitive L_h.L*-geometries
The classification of finite flag-transitive linear spaces, obtained by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl [20] at the end of the eighties, gave new impulse to the program of classifying various classes of locally finite flag-transitive geometries belonging to diagrams obtained from a Coxeter diagram by putting a label L or L ∗ on some (possibly, all) of the singlebond strokes for projective planes
Double transitivity of Galois Groups in Schubert Calculus of Grassmannians
We investigate double transitivity of Galois groups in the classical Schubert
calculus on Grassmannians. We show that all Schubert problems on Grassmannians
of 2- and 3-planes have doubly transitive Galois groups, as do all Schubert
problems involving only special Schubert conditions. We use these results to
give a new proof that Schubert problems on Grassmannians of 2-planes have
Galois groups that contain the alternating group. We also investigate the
Galois group of every Schubert problem on Gr(4,8), finding that each Galois
group either contains the alternating group or is an imprimitive permutation
group and therefore fails to be doubly transitive. These imprimitive examples
show that our results are the best possible general results on double
transitivity of Schubert problems.Comment: 25 page
Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6
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