Strongly Regular Graphs

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.

Two-weight and three-weight linear codes constructed from Weil sums

<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

Three-Dimensional Image Reconstruction for Virtual Talent Training Scene

In real training, the training conditions are often undesirable, and the use of equipment is severely limited. These problems can be solved by virtual practical training, which breaks the limit of space, lowers the training cost, while ensuring the training quality. However, the existing methods work poorly in image reconstruction, because they fail to consider the fact that the environmental perception of actual scene is strongly regular by nature. Therefore, this paper investigates the three-dimensional (3D) image reconstruction for virtual talent training scene. Specifically, a fusion network model was deigned, and the deep-seated correlation between target detection and semantic segmentation was discussed for images shot in two-dimensional (2D) scenes, in order to enhance the extraction effect of image features. Next, the vertical and horizontal parallaxes of the scene were solved, and the depth-based virtual talent training scene was reconstructed three dimensionally, based on the continuity of scene depth. Finally, the proposed algorithm was proved effective through experiments.

Distance-transitive strongly regular graphs

We present a complete description of strongly regular graphs admitting a distance-transitive group of automorphisms. Parts of the list have already appeared in the literature; however, this is the first time that the complete list appears in one place. The description is complemented, where possible, with the discussion of the corresponding distance-transitive groups and some further properties of the graphs. We also point out an open problem.

Strongly regular points of mappings

In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.