On ternary complementary pairs
نویسندگان
چکیده
Let A = {ao, ... , ae-d, B = {bO, ••• , be-d be two finite sequences of length £. Their nonperiodic autocorrelation function N A,B (s) is defined as: £-1-8 £-1-8 NA,B(S) = L aiai+s + L bibi+s' S = 0, ... , £ 1, i=O i=O where x* is the complex conjugate of x. If NA,B(S) = ° for S = 1, ... , £-1 then A, B is called a complementary pair. If, furthermore, ai, bi E {-I, I}, i = 0, ... , £-1, or, ai, bi E {-I, 0, I}, i = 0, ... , £-1, then A, B is called a binary complementary pair (BCP), or, a ternary complementary pair (TCP), respectively. A BCP is also called a Golay sequence. A TCP is a generalisation of a BCP. Since Golay sequences are only known to exist for lengths n = 2a lOb26c , a, b, c ~ 0, recent papers have focused on TCP's. The purpose of this paper is to give an overview of existing constructions and techniques and present a variety of new constructions, new restrictions on the deficiencies and new computational results for TCP's. In particular: • We give many new constructions which concatenate shorter groups of sequences to obtain longer sequences. Many of these constructions can be applied recursively and lead to infinite families of TCP's . • We give many new restrictions on TCP's of lengths £ and deficiencies 8 = 2x, where x == £ mod 4. * This research was carried out while the first author was at the University of Wollongong. Australasian Journal of Combinatorics 23(2001), pp.153-170 • We settle all the cases for existence/non-existence of TCP's of lengths R ::; 20 and weights w ::; 40 . • We give TCP's with minimum deficiencies for all lengths R:::; 22.
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ورودعنوان ژورنال:
- Australasian J. Combinatorics
دوره 23 شماره
صفحات -
تاریخ انتشار 2001