On the maximum size of variable-length non-overlapping codes

Non-overlapping codes are a set of codewords such that any nontrivial prefix of each codeword is not a nontrivial suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let C(n, q) be the maximum size of a fixed-length non-overlapping code of length n over an alphabet of size q. The upper bound on C(n, q) has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes whose codewords have length at most n remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a q-ary variable-length non-overlapping code is upper bounded by C(n, q). Furthermore, we prove that the minimum average codeword length of a q-ary variable-length non-overlapping code with cardinality \(\tilde{C}\), is asymptotically no shorter than \(n-2\) as q approaches \(\infty \), where n is the smallest integer such that \(C(n-1, q) < \tilde{C} \le C(n,q)\).