We present a new method of constructing three dimensional periodic arrays by composing a two dimensional periodic array with a sequence of shifts consisting of a cyclic group of points on an elliptic curve over a prime field \({\mathbb {F}}_p\). For every base array B with period (c, c) and every cyclic group G of order C there are \(\phi (C)\) families, each of size \(p^2\), of 3D arrays with period (c, c, C). We illustrate our method using a Legendre array as base array. The resulting 3D arrays have period (p, p, C), peak auto-correlation value \(C(p^2-1)\), and non-peak auto-correlation and cross-correlation values of the form \(kp^2-C\) where C is the order of the group and, in the general case, \(k\le 3\). We present experimental results that show that \(k\le 2\) for a certain type of cyclic group of points in \({\mathbb {F}}_p\) when \(p<1000\). Finally, we show that the linear complexity of our constructions compare favorably with other known constructions.

我们提出了一种构造三维周期数组的新方法，通过将二维周期数组与一系列移位组成，该移位序列由素数域上的椭圆曲线上的循环点组组成 \({\mathbb {F}}_p\ )。对于每个具有周期 ( c ,  c ) 的基本数组B和每个 C 阶循环群G，都有\ (\phi (C)\) 个族，每个族的大小为\(p^2\)，具有周期 ( c ,  c ,  C )。我们使用勒让德数组作为基本数组来说明我们的方法。生成的 3D 数组具有周期 ( p ,  p ,  C )、峰值自相关值\(C(p^2-1)\)以及形式为\(kp的非峰值自相关值和互相关值^2-C\)其中C是群的阶，一般情况下为\(k\le 3\)。我们提出的实验结果表明，当\(p<1000\)时，\({\mathbb {F}}_p\)中某种类型的循环点群的\(k\le 2\)。最后，我们表明我们的结构的线性复杂度与其他已知结构相比毫不逊色。