Searching for binary orientable sequences

Source abstract

An aperiodic binary orientable sequence is a finite sequence of zeros and ones for which, for some integer $n$, all the subsequences of $n$ consecutive symbols, read either forwards or backwards, are distinct. Analogously, a periodic binary orientable sequence is an infinite periodic sequence of zeros and ones for which, for some integer $n$, all the subsequences of $n$ consecutive symbols, read either forwards or backwards, are distinct within a single period of the sequence. Such sequences have potential application in position-location applications. For applications, the longer the sequence (or the greater the period in the periodic case) the greater its utility, but yet while constructions are known for such sequences, and upper bounds on their length (or period) exist, the precise maximum length (or period) was previously unknown for anything other than small values of $n$. In this note the results of LLM-conducted searches for the longest possible aperiodic and periodic binary orientable sequences for $n\leq16$ are reported. In each case an example of a sequence of the maximum discovered length is also provided, and the maximum possible length is now determined for every $n\leq16$ for both the periodic and aperiodic cases.