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 $2^n$ 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 $n\to\infty$. We prove the existence of functions with the trivial inertia group in $GA_n$ for all $n\ge 7$. An upper bound for the asymmetry of a $k$-uniform hypergraph is obtained.