A formulation of the Hexad algorithm using the geometry of the Fano projective plane

We have described a new formulation of the ‘Hexad Algorithm’ for decomposing a given 3D symmetric positive-definite aspect tensor on a lattice into its six unique local line generators and corresponding projected rank-one aspect tensor contributions. The new formulation adheres strictly to the symmetries possessed by the associated Fano projective plane and Galois field, GF(8). A bonus of adopting these constraints is that the computer code in the new algorithm is relatively succinct. The motivation for this work has been to provide a supporting algorithmic framework for the production of a computationally efficient way to generate anisotropic and spatially inho-mogeneous covariances, initially for NOAAs new 3D RTMA, but also for more general data assimilation applications later. 2020 NWS (National Weather Service) NCEP (National Centers for Environmental Prediction) Submitted https://doi.org/10.25923/xrzp-x016 Public Domain 2285