Binary Codes Formed by Functions with Nontrivial Inertia Groups

Let K be a permutation group acting on binary vectors of length n and F_K be a code of length 2ⁿ consisting of all binary functions with nontrivial inertia group in K. We obtain upper and lower bounds on the covering radii of F_K, where K are certain subgroups of the affine permutation group GA_n. We also obtain estimates for distances between F_K and almost all functions in n variables as n . We prove the existence of functions with the trivial inertia group in GA_n for all n 7. An upper bound for the asymmetry of a k-uniform hypergraph is obtained.