Nearly perfect sequences with arbitrary out-of-phase autocorrelation

Abstract

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type γ and a cyclic (n, p, n, n−γ p + γ, 0, n−γ p )-DPDS for an arbitrary integer γ. Next, we present the necessary conditions for the existence of a p-ary NPS of type γ. We apply this result for excluding the existence of some p-ary NPS of period n and type γ for n ≤ 100 and |γ| ≤ 2. We also prove the similar results for an almost p-ary NPS of type γ. Finally, we show the non-existence of some almost p-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.

Almost p-ary sequences

“…Therefore we get that |T 0,j | = s j (s j − 1) for 0 ≤ j ≤ p − 1. Hence using (11), (12) and (13) we conclude that…”

“…In the following we present a known result between an almost p-ary sequence of type γ with one zero-symbol and a DPDS for an integer γ. Theorem 1. [12] a is an almost p-ary NPS of period n + 1 and type γ with one zero-symbol if and only if R a defined as in (2) is an (n + 1, p, n, n−γ−1 p + γ, 0, n−γ−1 p )-DPDS in G relative to H and P . In particular, p divides n − γ − 1.…”

“…Niu et al [9] studied a binary sequence of the periodical with a 2-level autocorrelation value, this solves three open problems given by Jungnickel and Pott [4]. Moreover, second author [12] proved an equality between a p-ary NPS of type γ and a DPDS for an arbitrary integer γ. In addition, he extended this result for an almost p-ary NPS with one zero-symbol, and proved its nonexistence cases by self-conjugacy condition.…”

Partial direct product difference sets and almost quaternary sequences

“…For nonbinary case, Ma and Ng [28] obtained a relation between an m-ary NPS of type |γ| ≤ 1 and a direct product difference set (DPDS) for a prime number m. Later, Chee et al [6] extended the methods due to Ma and Ng [28] to almost m-ary NPS of types γ = 0 and γ = −1 with one zerosymbol. The second author [49] proved an equality between an m-ary NPS of type γ and a DPDS for an arbitrary integer γ. In [5], certain p-ary sequences are linked to some partial geometric difference sets.…”

Partial direct product difference sets and sequences with ideal autocorrelation

“…For nonbinary case, Ma and Ng [10] obtained a relation between an m-ary NPS of type |γ| ≤ 1 and a direct product difference set (DPDS) for a prime number m. Later, Chee et al [2] extended the methods due to Ma and Ng [10] to almost m-ary NPS of types γ = 0 and γ = −1 with one zero-symbol. The second author [16] proved an equality between an m-ary NPS of type γ and a DPDS for an arbitrary integer γ. In [12], the authors 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.…”