On non-existence of perfect and nearly perfect sequences (Q622767)

Summary: We study the complex \(p\)-ary perfect and nearly perfect sequences where \(p\) is an odd prime and show that the existence of such sequences is equivalent to the existence of certain kinds of difference sets. Using results from difference sets, there are no \(p\)-ary perfect sequences of length \(p^s\) for \(s\geq 3 \), \(2p^s \) for \(s\geq 1 \), and \(pq\) for prime \(q>p\). Also, there are no ternary perfect sequences of length \(3q_1q_2 \) for primes \(q_1,q_2 \) where \(3<q_1<q_2 \). Using the standard 'self-conjugate' conditions, more non-existence results on perfect and nearly perfect sequences are obtained. To summarise the works, tables of existence and non-existence of perfect and nearly perfect sequences of length \(n\) are listed for \(2\leq n\leq 50\).