报告题目： Codes, groups and designs

摘要：

The adjacency code of a rank three group contains PBIBDs by construction. When we are lucky we obtain a BIBD  also known as 2-design. This is the case for instance of the one point stabilizer of PSL(2,41) in its action on the binary QR(41). When we are even luckyer we obtain a 3-design in the extended code. This is the case of the XQR(41) in length 42. By considering known families of rank 3 groups we obtain in that way 111 2-design and nine 3-designs. The supporting codes are constructed by modular representation theory (joint works with Bonnecaze and with Rodrigues). Another group action method applies to isodual ternary codes the automorphism group of which admits two orbits on triples (joint work with Shi and Liu.) Interesting examples are provided by Generalized Quadratic Residue codes, a family of abelian codes discussed by van Lint and MacWilliams. Since none of the discussed designs can be explained by the Assmus-Mattson theorem or transitivity arguments, all of this work is based on computer calculations in Magma. Recently a theoretical explanation of the designs in the weight 10 codewords of the XQR(41) was provided by Munemasa based on harmonic weight enumerators.