Power sum polynomials and the ghosts behind them

The power sum polynomial associated to a multi-subset of the projective plane PG (2 , q) is the sum of the (q- 1) -th powers of the R\ue9dei factors of the points in the multi-subset. The classification of multi-subsets having the same power sum polynomial passes through the determination of those multi-subsets associated to the zero polynomial, called ghosts. In this paper we provide new classes of ghosts and compute the dimension of the ghost subspace by exploiting the linear code generated by the lines of PG (2 , q) and its dual. Moreover, we explicitly enumerate and classify ghosts for planes of order 2,\ua03,\ua04

Codes of Desarguesian projective planes of even order, projective triads and (q+t,t)-arcs of type (0,2,t)

AbstractWe study the binary dual codes associated with Desarguesian projective planes PG(2,q), with q=2h, and their links with (q+t,t)-arcs of type (0,2,t), by considering the elements of Fq as binary h-tuples. Using a correspondence between (q+t,t)-arcs of type (0,2,t) and projective triads in PG(2,q), q even, we present an alternative proof of the classification result on projective triads. We construct a new infinite family of (q+t,t)-arcs of type (0,2,t) with t=q4, using a particular form of the primitive polynomial of the field Fq

On four codes with automorphism group P Sigma L(3,4) and pseudo-embeddings of the large Witt designs

A pseudo-embedding of a point-line geometry is a representation of the geometry into a projective space over the field F-2 such that every line corresponds to a frame of a subspace. Such a representation is called homogeneous if every automorphism of the geometry lifts to an automorphism of the projective space. In this paper, we determine all homogeneous pseudo-embeddings of the three Witt designs that arise by extending the projective plane PG(2, 4). Along our way, we come across some codes with automorphism group P Sigma L(3, 4) and sets of points of PG(2, 4) that have a particular intersection pattern with Baer subplanes or hyperovals

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

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4