Distance-Regular Graphs

Finite Generalized Quadrangles

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

/pdf/the-geometry-of-two-weight-codes-nnndeshbae.pdf

The Geometry of Two-Weight Codes

https://jupiter.math.nctu.edu.tw/~weng/references/distance-regular graphs/cospec_2003.pdf?origin=publication_detail

Which graphs are determined by their spectrum

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276---279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 m?r in binary Reed---Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752---759, 1970) and Kasami et al. (Inf Control 30(4):380---395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65---74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303---324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.

/pdf/minimal-codewords-in-reed-muller-codes-30labcefyv.pdf

Minimal codewords in Reed---Muller codes

An axiom system is presented to characterize the Grassmannians of buildings of spherical type having hyperlines of rank 3 only; some open problems remain in the infinite case.

Pseudopolar spaces of polar rank three

Generalized quadrangles may be considered from two points of view, the group theoretical one and the combinatorial one. The present survey attempts to deal mainly with what might be called the combinatorial properties of the finite generalized quadrangles : first, given an abstract generalized quadrangle, what restriction on the parameters, etc., are there; second, what conditions of a combinatorial nature on an abstract generalized quadrangle force it to be isomorphic to one of the known examples.

Combinatorics of Finite Generalized Quadrangles : A Survey

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

Let S be a thick generalized quadrangle and let G be a group of automorphisms of S. If G acts transitively on the set of non-degenerate ordered pentagons, then S is one of the classical generalized quadrangles W (q), Q(4, q), Q(5, q) or H(3, q2). The possibilities for G in each case are determined. We do not use the classification of the finite simple groups (from which this result also follows).

/pdf/the-classification-of-finite-generalized-quadrangles-54azm4u0gc.pdf

Joseph A. Thas

Papers

Minimal codewords in Reed---Muller codes

Pseudopolar spaces of polar rank three

Combinatorics of Finite Generalized Quadrangles : A Survey

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

The Classification of Finite Generalized Quadrangles Admitting a Group Acting Transitively on Ordered Pentagons