Constructing k-ary orientable sequences with asymptotically optimal length

An orientable sequence of order n over an alphabet\(\{0,1,\ldots , k{-}1\}\) is a cyclic sequence such that each length-n substring appears at most once in either direction. When \(k= 2\), efficient algorithms are known to construct binary orientable sequences, with asymptotically optimal length, by applying the classic cycle-joining technique. The key to the construction is the definition of a parent rule to construct a cycle-joining tree of asymmetric bracelets. Unfortunately, the parent rule does not generalize to larger alphabets. Furthermore, unlike the binary case, a cycle-joining tree does not immediately lead to a simple successor-rule when \(k \ge 3\) unless the tree has certain properties. In this paper, we derive a parent rule to derive a cycle-joining tree of k-ary asymmetric bracelets. This leads to a successor rule that constructs asymptotically optimal k-ary orientable sequences in O(n) time per symbol using O(n) space. In the special case when \(n=2\), we provide a simple construction of k-ary orientable sequences of maximal length.