Strongly Regular Graphs Constructed from pp-ary Bent Functions

In this paper, we generalize the construction of strongly regular graphs in [Y. Tan et al., Strongly regular graphs associated with ternary bent functions, J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to $p$-ary bent functions, where $p$ is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic $p$-ary bent functions, our constructions can give rise to new strongly regular graphs for small parameters.Comment: to appear in Journal of Algebraic Combinatoric

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

Doubly transitive lines II: Almost simple symmetries

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper, the second in a series, classifies those lines that exhibit almost simple symmetries. To perform this classification, we introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group

Crystalline boundedness principle

We prove that an $F$-crystal (M,\vph) over an algebraically closed field $k$ of characteristic $p>0$ is determined by (M,\vph) mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of (M,\vph). We also extend this result to triples (M,\vph,G), where $G$ is a flat, closed subgroup scheme of ${\bf GL}_M$ whose generic fibre is connected and has a Lie algebra normalized by \vph. We get two purity results. If {\got C} is an $F$-crystal over a reduced ${\bf F}_p$-scheme $S$, then each stratum of the Newton polygon stratification of $S$ defined by {\got C}, is an affine $S$-scheme (a weaker result was known before for $S$ noetherian). The locally closed subscheme of the Mumford scheme {\Ma_{d,1,N}}_k defined by the isomorphism class of a principally quasi-polarized $p$-divisible group over $k$ of height 2d, is an affine {\Ma_{d,1,N}}_k-scheme.Comment: Final version (63 pages) accepted for publication in Ann. Sci. Ec. Norm. Su

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