A formulation of the Decad algorithm using the symmetries of the Galois field, GF (16)

The problem we have addressed is to find a systematic way of generalizing the existing Triad and Hexad algorithms of two and three dimensions to the true Decad algorithm in four dimensions. We have succeeded by exploiting the symmetry properties of the Galois field, GF(16) and its associated finite project geometry, PG(3, 2), especially the restricted group of the automorphisms that we have called Aut+{PG(3, 2)} that seem best adapted to our problem. The outcome is a reliable algorithm which, given any valid 4D symmetric aspect tensor with its ten independent components, will provide a unique set (the decad) of generalized grid line directions (the integer 4-vector generators). In addition the algorithm supplies precisely the amount (measured by the rank-one projected aspect tensor weight) of Gaussian smoothing that is required to be applied along the space-time grid lines oriented parallel to these generators, in the sequence orchestrated by the different Galois colors, so that the product smoothing operation is the Gaussian convolution with exactly the second moment ‘aspect tensor’ attribute intended. The expected application of this method is primarily to a data analysis generalizing what is already being done in the 2D and 3D RTMA (Pondeca et al., 2011), to a new kind of ‘nowcasting’ analysis in four dimensions covering a period of time too brief to fit well with the use of a costly forecast model initialization and integration. 2020 NWS (National Weather Service) NCEP (National Centers for Environmental Prediction) Submitted https://doi.org/10.25923/4nhy-x681 Public Domain 2285