  Sequences play an important role in many engineering applications and
systems. Searching sequences with desired properties has long been an
interesting but also challenging research topic. This article proposes a novel
method, called HpGAN, to search desired sequences algorithmically using
generative adversarial networks (GAN). HpGAN is based on the idea of zero-sum
game to train a generative model, which can generate sequences with
characteristics similar to the training sequences. In HpGAN, we design the
Hopfield network as an encoder to avoid the limitations of GAN in generating
discrete data. Compared with traditional sequence construction by algebraic
tools, HpGAN is particularly suitable for intractable problems with complex
objectives which prevent mathematical analysis. We demonstrate the search
capabilities of HpGAN in two applications: 1) HpGAN successfully found many
different mutually orthogonal complementary code sets (MOCCS) and optimal
odd-length Z-complementary pairs (OB-ZCPs) which are not part of the training
set. In the literature, both MOCSSs and OB-ZCPs have found wide applications in
wireless communications. 2) HpGAN found new sequences which achieve four-times
increase of signal-to-interference ratio--benchmarked against the well-known
Legendre sequence--of a mismatched filter (MMF) estimator in pulse compression
radar systems. These sequences outperform those found by AlphaSeq.
