Distinct 8-QAM+ Perfect Arrays

This paper investigates the construction of 8-QAM+ perfect arrays with at least an odd integer in their sizes, and gives the sufficient condition producing distinct 8-QAM+ perfect arrays. For different choices of odd integers in a given size of at least an odd integer, the resultant distinct 8-QAM+perfect arrays have their own sizes. For each choice, the number of the resultant distinct 8-QAM+perfect arrays is exactly equivalent to this chosen odd integer. This paper can provide more candidates for the applications of 8-QAM+ perfect arrays in communications, radar, and so forth. Introduction Perfect arrays have the property of impulse-like autocorrelation, which results in the fact that perfect arrays are widely applied to communications, radar, time-frequency-coding, map matching, synchronization, built-in tests of VLSI-circuits, and so on [1]. There are a large number of perfect binary, ternary, and ployphase array [1]. However, 8-QAM+ perfect arrays are rare, whose main reason lies at the late introduction of 8-QAM+ constellation [2]. In 2010, Boztas and Parampalli introduced the definition of 8-QAM+ signal set [2]. From that time on, the sequences over 8-QAM+ constellation are rapidly developed. Perfect and odd perfect 8-QAM+ sequences are given in 2012 and 2014 , respectively [3] [4]. 8-QAM+ complementary sequences are presented in 2012 and 2014, respectively [5] [6]. ZCZ 8-QAM+ sequences appeared in 2012 [7]. Recently, 8-QAM+ perfect arrays are proposed in 2014 and 2015 [8] [9]. The aim of this paper is to investigate 8-QAM+ perfect arrays, to increase their family size, and to reflect the state-of-the-art of this area. A sufficient condition producing distinct 8-QAM+ perfect arrays will be introduced. The number of the resultant 8-QAM+ perfect arrays will be improved. The rest of this paper will organized as follows. In Section 2, we will review some necessary concepts and the known results on the constructions of 8-QAM+ perfect arrays. In Section 3, a sufficient condition producing distinct 8-QAM+ perfect arrays will be given. The examples will appear in Section 4. Finally, we conclude this paper in Section 5. Necessary Concepts and Known Constructions of 8-QAM+ Perfect Arrays In a mathematical term, an array A of n dimension with the size 1 2 n N N N × × ×  is 1 2 , , , | 0 ,1 n i i i k k A a i N k n   = ≤ < ≤ ≤    , (1) whose periodic autocorrelation is defined by