Title: On non-existence of perfect and nearly perfect sequences

Authors: Siu Lun Ma, Wei Shean Ng

Addresses: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore. ' Faculty of Information and Communication Technology, Universiti Tunku Abdul Rahman, 13 Jalan 13/6, 46200 Petaling Jaya, Selangor, Malaysia

Abstract: 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 ≥ 3, 2p^s, for s ≥ 1, and pq for prime q>p. Also, there are no ternary perfect sequences of length 3q₁q₂ for primes q₁, q₂ where 3<q₁<q₂. 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 ≤ n ≤ 50.

Keywords: perfect sequences; nearly perfect sequences; autocorrelation; difference sets; character theory.

DOI: 10.1504/IJICOT.2009.024045

International Journal of Information and Coding Theory, 2009 Vol.1 No.1, pp.15 - 38

Published online: 24 Mar 2009 *

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