Almost p-ary sequences

Abstract

In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ 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 ℓ. It is shown that ℓ 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−γ 2 −2 p + γ2, 0, n−γ 1 −1 p + γ1, n−γ 2 −2 p , n−γ 1 −1 p ) 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 don't exist for γ2 ≤ −3.

Partial direct product difference sets and almost quaternary sequences

“…Then we present a method to construct the difference sets - Proposition 1). Similar to [34], we show that an almost m-ary NPS of type (γ 1 , γ 2 ) with non-consecutive two zerosymbols is equivalent to an -PDPDS (see Proposition 2) and their non-existence for γ 2 ≤ 3 (see Proposition 4). Next, we study the notion of multipliers and the orbit combination for -PDPDS, and we prove a necessary condition on their multiplier set (see Proposition 5).…”

“…Then, some non-existence results for NPS with two consecutive zero-symbols were proven by using their PDPDS equivalence. Here, we extend this one-to-one equivalence to the non-consecutive case and show that an NPS with two zero-symbols is related to an -PDPDS, see the consecutive zero result in [34,Theorem 3]. As its proof is very similar to the consecutive case, we do not give its proof here.…”

“…The authors in [34] studied almost m-ary NPS a with two consecutive zerosymbols and obtained that R a is a PDPDS. Then, some non-existence results for NPS with two consecutive zero-symbols were proven by using their PDPDS equivalence.…”

“…Sequences are used in satellite telecommunication, cryptographic function design, wireless networks, signal processing, radar systems, and modern cell phones (see [7,14,20,26,33,37,39]). For instance, they are used in Code Division Multiple Access (CDMA), where the sequences with small correlation are important because a signal should not be affected by other signals in order to provide high-quality communication.…”

“…In [5], certain p-ary sequences are linked to some partial geometric difference sets. Recently, the authors [34] studied the m-ary NPS of type (γ 1 , γ 2 ) with two consecutive zero-symbols, in which they proved that there is a relationship between a partial direct product difference sets (PDPDS) and an m-ary NPS of type (γ 1 , γ 2 ) with two consecutive zero-symbols. Moreover, they showed some non-existence results for these kinds of sequences.…”