Non-overlapping codes

We say that a $q$-ary length $n$ code is \emph{non-overlapping} if the set of non-trivial prefixes of codewords and the set of non-trivial suffices of codewords are disjoint. These codes were first studied by Levenshtein in 1964, motivated by applications in synchronisation. More recently these codes were independently invented (under the name \emph{cross-bifix-free} codes) by Baji\'c and Stojanovi\'c. We provide a simple construction for a class of non-overlapping codes which has optimal cardinality whenever $n$ divides $q$. Moreover, for all parameters $n$ and $q$ we show that a code from this class is close to optimal, in the sense that it has cardinality within a constant factor of an upper bound due to Levenshtein from 1970. Previous constructions have cardinality within a constant factor of the upper bound only when $q$ is fixed. Chee, Kiah, Purkayastha and Wang showed that a $q$-ary length $n$ non-overlapping code contains at most $q^n/(2n-1)$ codewords; this bound is weaker than the Levenshtein bound. Their proof appealed to the application in synchronisation: we provide a direct combinatorial argument to establish the bound of Chee \emph{et al}. We also consider codes of short length, finding the leading term of the maximal cardinality of a non-overlapping code when $n$ is fixed and $q\rightarrow \infty$. The largest cardinality of non-overlapping codes of lengths $3$ or less is determined exactly.