Almost p -ary sequences
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Almost p -ary sequences 1 · Oguz ¨ ˘ Yayla1 ¨ ¸ ra Ozden Bus
Received: 20 December 2018 / Accepted: 13 January 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on . It is shown that can not be less than min{s, p, n}. In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost p-ary nearly perfect sequence of type (γ1 , γ2 ) and period n + 2 with two consecutive zero-symbols and a cyclic (n + 2, p, n, n−γp2 −2 + γ2 , 0, n−γp1 −1 +γ1 , n−γp2 −2 , n−γp1 −1 ) PDPDS for arbitrary integers γ1 and γ2 . Then we prove a necessary condition on γ2 for the existence of such sequences. In particular, we show that they do not exist for γ2 ≤ −3. Keywords Almost p-ary sequence · Nearly perfect sequence · Partial direct product difference set Mathematics Subject Classification (2010) 05B10 · 94A55
1 Introduction Let ζp ∈ C be a primitive p-th root of unity for some prime number p. A sequence a = (a0 , a1 , . . . , an−1 ) of period n with ai = ζpbi for some integer bi , i = 0, 1, . . . , n−1 is called a p-ary sequence. If aij = 0 for all j = 1, 2, . . . , s where {i1 , i2 , . . . , is } ⊂ {0, 1, . . . , n−1} and ai = ζpbi for some integer bi , i ∈ {0, 1, . . . , n − 1}\{i1 , i2 , . . . , is }, then we call a an almost p-ary sequence with s zero-symbols. For instance, a = (ζ33 , ζ32 , ζ34 , ζ32 , 1) is a 3-ary sequence of period 5 and a = (0, ζ73 , 1, ζ73 , 0, 0, ζ75 , ζ76 , ζ76 , ζ75 ) is an almost 7-ary sequence with 3 zero-symbols of period 10. It is widely used that a sequence with one zero-symbol is O˘guz Yayla
[email protected] ¨ B¨us¸ra Ozden [email protected] 1
Department of Mathematics, Hacettepe University, Beytepe, 06800 Ankara, Turkey
Cryptography and Communications
called an almost p-ary sequence. But in this paper we use this notation for a p-ary sequence with s zero-symbols, for s ≥ 0. For a sequence a of period n, its autocorrelation function Ca (t) is defined as Ca (t) =
n−1
ai ai+t ,
i=0
for 0 ≤ t ≤ n − 1 where a is the complex conjugate of a. The values Ca (t) at 1 ≤ t ≤ n − 1 are called the out-of-phase autocorrelation coefficients of a. Note that the autocorrelation function of a is periodic with n. We call an almost p-ary sequence a of period n a nearly perfect sequence (NPS) of type (γ1 , γ2 ) if all out-of-phase autocorrelation coefficients of a are either γ1 or γ2 . We write NPS of type γ to denote an NPS of type (γ , γ ). Moreover, a sequence is called perfect sequence (PS) if it is an NPS of type (0, 0). We also note that there is another notion of almost perfect sequ
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