De Paoli-Iseppi, R., Johansson, P., Menzies, A., Dias, K., Pupo, G., Kakavand, H., Wilmott, J., Mann, G., Hayward, N., Dinger, M., Long, G., Scolyer, R. (2016). Comparison of whole-exome sequencing of matched fresh and formalin fixed paraffin embedded melanoma tumours: implications for clinical decision making. Pathology, 48(3), 261-266. ...

Some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the Hooman-Singleton graph and the Higman-Sims graph. Some of the codes are optimal or nearly optimal for the given length and dimension. The dual codes admit majority logic decoding.

We show, by introducing an appropriate basis, that a one-parameter family of Hopf algebras introduced by Foissy [Adv. Math. 218 (2008) 136-162] interpolates beetween the Faà di Bruno algebra and the Farahat-Higman algebra. Its structure constants in this basis are deformation of the top connection coefficients, for which we obtain analogues of Macdonald’s formulas.

We prove that the only finite factor-representations of the HigmanThompson groups {Fn,r}, {Gn,r} are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that any measure-preserving ergodic action of a simple Higman-Thompson group must be essentially free. Finite factor representations of other classes of groups are also discussed.