Abstract

An m-sequence is the one of the largest period among those produced by a linear feedback shift register. It possesses several desirable features of pseudorandomness such as balance, uniform pattern distribution and ideal autocorrelation for applications to communications. However, it also possesses undesirable features such as low linear complexity. Here we prove a nontrivial upper bound on its arithmetic autocorrelation, another figure of merit introduced by Mandelbaum for error-correcting codes and later investigated by Goresky and Klapper for FCSRs. The upper bound is close to half of the period and hence rather large, which gives an undesirable feature.

摘要

m序列是线性反馈移位寄存器产生的周期中最大的周期之一。它具有伪随机性的几个理想特征，例如平衡、均匀模式分布和通信应用的理想自相关性。然而，它也具有一些不良特征，例如线性复杂度低。在这里，我们证明了算术自相关的一个重要上限，这是 Mandelbaum 为纠错码引入的另一个品质因数，后来由 Goresky 和 ​​Klapper 针对 FCSR 进行了研究。上限接近周期的一半，因此相当大，这给出了不期望的特征。