Distribution of Linear Rules in Cellular Automata Rule Space

We attempt to find a good linear (that is, additive) approximation of a given rule by looking at the geometric-combinatorial properties of rule space with a Hamming distance. We show that all linear rules are equidistant from one another for a prime number of states, and that several parameters describing the geometry of rule space are essentially preserved in the subspace of linear rules . This is particularly true of the average and maximum distance between rules. The fit improves as the number of states grows toward the thermodynamic limit. In particular, linear rules provide a welldistributed initial population for a genetic search of CA rule space. We achieve asymptotically exact solutions for the nearest approximation to a given rule from an initial population of piecewise linear rules obtained by splicing, a modification of the standard genetic crossover operator.