  A phase detection sequence is a length-$n$ cyclic sequence, such that the
location of any length-$k$ contiguous subsequence can be determined from a
noisy observation of that subsequence. In this paper, we derive bounds on the
minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence
constructions. We further consider multiple phase detection sequences, where
the location of any length-$k$ contiguous subsequence of each sequence can be
determined simultaneously from a noisy mixture of those subsequences. We study
the optimal trade-offs between the lengths of the sequences, and describe some
sequence constructions. We compare these phase detection problems to their
natural channel coding counterparts, and show a strict separation between the
fundamental limits in the multiple sequence case. Both adversarial and
probabilistic noise models are addressed.
