A perfect code on the Schreier graphs of an automaton group

The vertices of the $n$-th orbital Schreier graph of the so-called Tangled Odometers group, which is an example of an automaton group acting on the ternary rooted tree, can be identified with the elements of the vector space $\mathbb{F}_3^n$, where $\mathbb{F}_3$ denotes the finite field of $3$ elements. On such a graph, we explicitly construct a perfect $1$-error-correcting code; equivalently, we determine an efficient dominating vertex subset. We prove that this code is unique and that it is linear. Our proofs strongly use the group structure which is intrinsic in the graph. The decoding process associating with every non-codeword the corresponding unique codeword can be described both in terms of the action of the group generators, and by means of the adjacency matrix of the graph, by considering the elements of $\mathbb{F}_3^n$ as ternary expansions of integers from $0$ to $3^n -1$.