# BlackJAX: Composable Bayesian inference in JAX

Lead Authors
Alberto Cabezas<EMAIL_ADDRESS>
Adrien Corenflos<EMAIL_ADDRESS>
Junpeng Lao<EMAIL_ADDRESS>
Rémi Louf<EMAIL_ADDRESS>

Technical Contributors
Antoine Carnec, Kaustubh Chaudhari, Reuben Cohn-Gordon,
Jeremie Coullon, Wei Deng, Sam Duffield, Gerardo Durán-Martín,
Marcin Elantkowski, Dan Foreman-Mackey, Michele Gregori, Carlos Iguaran,
Ravin Kumar, Martin Lysy, Kevin Murphy, Juan Camilo Orduz,
Karm Patel, Xi Wang, Rob Zinkov

*Authors are ordered alphabetically within groups Department of Mathematics and Statistics, Lancaster University, UKDepartment of Electrical Engineering and Automation, Aalto University, FinlandGoogle, Switzerland.txt, France

###### Abstract

BlackJAX is a library implementing sampling and variational inference
algorithms commonly used in Bayesian computation. It is designed for ease of
use, speed, and modularity by taking a functional approach to the algorithm’s
implementation. Designed from basic components to specific iterative
procedures, BlackJAX allows the end user to build and experiment with new
algorithms by composition. BlackJAX is written in Python using JAX to compile
and run NumpPy-like samplers and variational methods on CPUs, GPUs and TPUs.
The library integrates well with probabilistic programming languages by
working directly with the (un-normalized) target log density function. The
library is intended as a collection of low-level, composable implementations
of basic statistical ‘atoms’ that can be combined to perform well-defined
Bayesian inference. It is designed for users who need cutting-edge methods,
researchers who want to create complex sampling methods, and people who want
to learn how these work.

Keywords: Bayesian computation, variational inference, Markov chain Monte
Carlo, sequential Monte Carlo, approximate Bayesian inference

##  1 Introduction

Sampling from a probability distribution, either manually defined or
constructed using probabilistic programming languages (PPLs), is a recurring
topic in statistics and machine learning. Automatic sampling software has
historically been limited to Gibbs-type methods (Meyer and Yu, 2000; Lunn et
al., 2000; Depaoli et al., 2016), requiring knowledge of the model structure.
Black-box samplers, typically relying on Hamiltonian Monte Carlo (HMC, Duane
et al., 1987), allowed general, model-agnostic improvement in the
applicability of the method. This was spearheaded by Stan (Carpenter et al.,
2017), which leveraged the development of automatic differentiation. The same
developments have allowed for automatically learning rich approximations via
variational inference (VI, Jordan et al., 1999) to the models of interest. Put
together, these led to the creation of an array of modern PPLs that have
pushed the boundaries on the feasibility of Bayesian computation (Abril-Pla et
al., 2023; Bingham et al., 2019; Phan et al., 2019). While black-box samplers
have paved the way, we believe that inference in today’s models increasingly
requires reintroducing structure-aware algorithms.

To achieve this, BlackJAX provides users with composable inferential building
blocks written using JAX (Bradbury et al., 2018), such as Metropolis–Hastings
(Metropolis et al., 1953; Hastings, 1970; Robert, 2016) accept/reject step,
Hamiltonian or Langevin (Besag, 1994) dynamics, stochastic gradient utilities,
resampling and tempering mechanisms for use within sequential Monte Carlo
(SMC, Del Moral et al., 2006), or mean field approximations (Jordan et al.,
1999), as well as other mechanisms. These components are unified under a
convenient, functionally-oriented API that can be combined to form new or
existing algorithms immediately applicable to sequential and parallel modern
computer architectures.

##  2 Design principles

BlackJAX supports sampling algorithms such as MCMC, SMC, and Stochastic
Gradient MCMC (SGMCMC) and approximate inference algorithms such as VI. In all
cases, BlackJAX takes a Markovian approach, whereby all the information to
obtain the next iteration of an algorithm is contained in its current state.
This naturally results in a functionally pure (Lonsdorf, 2020) structure,
where no side-effects are allowed, simplifying parallelisation. For
efficiency, auxiliary information may be included in the state too.

Once the sampling algorithm has been chosen (or designed), it is instantiated
on the target function, often given by its log density. The sampling (or VI
calibration) procedure is then carried out as a loop, updating the previous
state into the current, which can be collected to compute statistics.
Additional information (for example, the acceptance probability of the
proposed MCMC state) is also returned for debugging or diagnostic purposes.

⬇

1# Generic sampling algorithm:

2sampling_algorithm = blackjax.sampling_algorithm(logdensity_fn, ...)

3state = sampling_algorithm.init(initial_position)

4new_state, info = sampling_algorithm.step(rng_key, state)

5

6# Generic approximate inference algorithm:

7approx_algorithm = blackjax.approx_algorithm(logdensity_fn, optimizer, ...)

8state = approx_algorithm.init(initial_position)

9new_state, info = approx_algorithm.step(rng_key, state)

10position_samples = approx_algorithm.sample(rng_key, state, num_samples)

###  2.1 Lower-level API

Users might need a tailored algorithm for the model they are trying to sample
or approximate from, or they might try out different Markov transition kernels
running in parallel with various particles in an SMC algorithm, or they might
use optimization to approximate the hyperparameters of the models while
sampling from the rest. For any of these cases, BlackJAX provides access to a
lower-level API, giving access to functions implementing methods with more
parameters. This allows for implementing more complex methods than those
available at the top level of the library. The user-facing interface then
resembles (using MCMC for illustration, but it can be any algorithm type):

⬇

1# Lower-level sampling algorithm:

2init = blackjax.mcmc.sampling_algorithm.init

3state = init(initial_position, logdensity_fn)

4kernel = blackjax.mcmc.sampling_algorithm.build_kernel(...)

5new_state, info = kernel(rng_key, state, logdensity_fn, ...)

6

7# Lower-level approximate inference algorithm:

8init = blackjax.vi.approx_algorithm.init

9state = init(initial_position, optimizer, ...)

10step = blackjax.vi.approx_algorithm.step

11new_state, info = step(rng_key, state, logdensity_fn, optimizer, ...)

12sample = blackjax.vi.approx_algorithm.sample

13position_samples = sample(rng_key, state, num_samples)

The functional design of the library, where programs are constructed by
applying and composing functions, allows the end user to build and experiment
with new algorithms by applying the same mathematical logic used to design
them. This is used, for instance, to apply different base kernels in an SMC
setting or combine optimization within sampling algorithms. For a detailed
example, see the implementation of the window_adaptation scheme for adaptation
of the step-size and mass-matrix appearing in HMC (Betancourt, 2016).

###  2.2 Basic components and the compositional paradigm

All inference algorithms are composed of basic components which provide the
lowest level of algorithm abstraction and are available to the user. With
BlackJAX, researchers and practitioners can leverage all provided basic
components already implemented to construct their method. For instance,
BlackJAX contains two variants of the MH accept/reject step, starting from the
computation of the accept/reject probability: the simpler safe_energy_diff if
the proposal transition kernel is symmetric and the more general
compute_asymmetric_acceptance_ratio if the proposal transition kernel is
asymmetric. Hence, the HMC algorithm uses the former while the Metropolis
adjusted Langevin algorithm (MALA Besag, 1994) uses the latter. After the
acceptance probability of the proposal is computed, the proposal is accepted
or rejected using static_binomial_sampling. In BlackJAX, this staple of MCMC
can be immediately swapped for the non-reversible slice sampling algorithm of
Neal (2020) simply by replacing static_binomial_sampling with
nonreversible_slice_sampling.

Because all algorithms in BlackJAX are implemented as composites of these
public low-level components, the end-user can then build custom algorithms
that are not direct combinations of already “well-formed” methods such as HMC.
For instance, one can use leapfrog dynamics independently of the acceptance
step, giving the user more flexibility in terms of their end algorithm.

###  2.3 Existing sampling libraries

BlackJAX is the only Python library specifically aimed at users who want to
use but also develop inference methods. In Python, AeMCMC automatically
constructs MCMC samplers for probabilistic models by exploiting the symbolic
graphs structure of programs written in Aesara111The project is on indefinite
hiatus as we write this article.. In Julia, Mamba.jl provides a platform for
implementing and applying MCMC methods to perform Bayesian analysis. Other
libraries implement domain-specific algorithms, such as EMCEE (Foreman-Mackey
et al., 2019), Dynesty (Speagle, 2020), and pocoMC (Karamanis et al., 2022),
or are directly tied to a PPL (Bingham et al., 2019; Tran et al., 2019; Oriol
et al., 2023; Carpenter et al., 2017; Lao et al., 2020).

##  3 Past impact of BlackJAX on the practice of Bayesian inference

BlackJAX joins the already existing rich ecosystem of JAX-powered scientific
libraries (Schoenholz and Cubuk, 2020; Wilkinson et al., 2023; Bonnet et al.,
2023) and is directly compatible with several of them either as a client:
consuming outputs from these (Pinder and Dodd, 2022; DeepMind et al., 2020),
or as a component, used within the libraries (Phan et al., 2019; Kumar et al.,
2019).

BlackJAX contains a comprehensive implementation of state-of-the-art HMC
algorithms, including vanilla HMC with various integrators, the no-U-turn
sampling (NUTS) to choose the number of integration steps at each iteration
dynamically (Hoffman et al., 2014), and the generalized HMC algorithm
(Horowitz, 1991). It also contains adaptation schemes for the algorithms’
hyper-parameters: window adaptation222https://mc-stan.org/docs/reference-
manual/hmc-algorithm-parameters.html and sophisticated calibration methods
such as Hoffman et al. (2021); Hoffman and Sountsov (2022). Consequently,
several papers have leveraged BlackJAX to conduct research in various fields
(see, e.g., Galan et al., 2022; Price-Whelan et al., 2024; Balkenhol et al.,
2024). Moreover, BlackJAX has made contributions to the methodological
development of Bayesian inference: it has been adopted in a range of papers to
develop new Bayesian sampling methods (Staber and Da Veiga, 2022; Cabezas and
Nemeth, 2023; Cooper et al., 2023).

Beyond research publications, BlackJAX has found a place in courses and
tutorials, for example by Darren Wilkinson333https://github.com/darrenjw/fp-
ssc-course and its use444https://github.com/probml/pyprobml in Kevin Murphy’s
authoritative manuscript (Murphy, 2022). This usage attests to the library’s
recognition as a practical resource for teaching Bayesian concepts, making it
accessible to a broader audience of learners and practitioners.

##  4 The future of BlackJAX

#### Enhanced portfolio.

We first aim to diversify BlackJAX’s Bayesian computation methods,
particularly “meta-algorithms” which consume base MCMC and VI methods to
produce enhanced samplers. New additions include parallel and sequential
tempering (Marinari and Parisi, 1992; Geyer and Thompson, 1995; Syed et al.,
2022); debiasing methods (Jacob et al., 2020) which allow for parallelising
MCMC samplers; structured VI techniques, like the Integrated Nested Laplace
Approximation (INLA, Rue et al., 2009). Alongside these, BlackJAX will
introduce more performance diagnostic tools.

#### Documentation and tutorials.

These will be expanded to cater to users at all levels, enhancing BlackJAX’s
usability and educational value. We plan to develop a comprehensive suite of
tutorials and documentation (a “sampling book”) detailing the integration of
BlackJAX with popular probabilistic programming languages, streamlining the
development process for complex probabilistic models. Additionally, the
introduction of an inference database feature similar to Stan’s posteriordb
will allow users to access a collection of posteriors for testing and
benchmarking. This feature aligns with the principles of transparency and
reproducibility in Bayesian inference. It will include reference
implementations in probabilistic programming languages and reference posterior
inferences in the form of posterior samples.

##  5 Project openness and development

BlackJAX’s source code is available under the Apache License v2.0, allowing
commercial use. BlackJAX is hosted at https://github.com/blackjax-
devs/blackjax. Anyone is welcome to contribute to the BlackJAX project.
Contributions can be made in code, documentation, expert reviews of open pull
requests, or other forms of support, such as case studies using BlackJAX.
BlackJAX follows the self-appointing council or board open-source governance
model555https://www.redhat.com/en/blog/understanding-open-source-governance-
models, that is, contributions are reviewed by core contributors, and breaking
decisions about the BlackJAX project are made by consensus among these. A
comprehensive test suite is run automatically by a continuous integration
service before code can be merged into the code base. As of the writing of
this article, the test coverage of the library is 99%.

## References

  * Abril-Pla et al. (2023) Oriol Abril-Pla, Virgile Andreani, Colin Carroll, Larry Dong, Christopher J Fonnesbeck, Maxim Kochurov, Ravin Kumar, Junpeng Lao, Christian C Luhmann, Osvaldo A Martin, et al.  PyMC: a modern, and comprehensive probabilistic programming framework in Python.  _PeerJ Computer Science_ , 9:e1516, 2023.
  * Balkenhol et al. (2024) L Balkenhol, C Trendafilova, K Benabed, and S Galli.  CANDL: Cosmic microwave background analysis with a differentiable likelihood.  _arXiv preprint arXiv:2401.13433_ , 2024.
  * Besag (1994) Julian Besag.  Comments on “Representations of knowledge in complex systems” by U. Grenander and M. I. Miller.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 56(591-592):4, 1994.
  * Betancourt (2016) Michael Betancourt.  Identifying the optimal integration time in Hamiltonian Monte Carlo, 2016.
  * Bingham et al. (2019) Eli Bingham, Jonathan P Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, and Noah D Goodman.  Pyro: Deep universal probabilistic programming.  _The Journal of Machine Learning Research_ , 20(1):973–978, 2019.
  * Bonnet et al. (2023) Clément Bonnet, Daniel Luo, Donal Byrne, Shikha Surana, Vincent Coyette, Paul Duckworth, Laurence I. Midgley, Tristan Kalloniatis, Sasha Abramowitz, Cemlyn N. Waters, Andries P. Smit, Nathan Grinsztajn, Ulrich A. Mbou Sob, Omayma Mahjoub, Elshadai Tegegn, Mohamed A. Mimouni, Raphael Boige, Ruan de Kock, Daniel Furelos-Blanco, Victor Le, Arnu Pretorius, and Alexandre Laterre.  Jumanji: a diverse suite of scalable reinforcement learning environments in JAX, 2023.  URL https://arxiv.org/abs/2306.09884.
  * Bradbury et al. (2018) James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang.  JAX: composable transformations of Python+NumPy programs, 2018.  URL http://github.com/google/jax.
  * Cabezas and Nemeth (2023) Alberto Cabezas and Christopher Nemeth.  Transport elliptical slice sampling.  In _International Conference on Artificial Intelligence and Statistics_ , pages 3664–3676. PMLR, 2023.
  * Carpenter et al. (2017) Bob Carpenter, Andrew Gelman, Matthew D Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus A Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell.  Stan: A probabilistic programming language.  _Journal of statistical software_ , 76, 2017.
  * Cooper et al. (2023) Alex Cooper, Aki Vehtari, Catherine Forbes, Lauren Kennedy, and Dan Simpson.  Bayesian cross-validation by parallel Markov chain Monte Carlo.  _arXiv preprint arXiv:2310.07002_ , 2023.
  * DeepMind et al. (2020) DeepMind, Igor Babuschkin, Kate Baumli, Alison Bell, Surya Bhupatiraju, Jake Bruce, Peter Buchlovsky, David Budden, Trevor Cai, Aidan Clark, Ivo Danihelka, Antoine Dedieu, Claudio Fantacci, Jonathan Godwin, Chris Jones, Ross Hemsley, Tom Hennigan, Matteo Hessel, Shaobo Hou, Steven Kapturowski, Thomas Keck, Iurii Kemaev, Michael King, Markus Kunesch, Lena Martens, Hamza Merzic, Vladimir Mikulik, Tamara Norman, George Papamakarios, John Quan, Roman Ring, Francisco Ruiz, Alvaro Sanchez, Laurent Sartran, Rosalia Schneider, Eren Sezener, Stephen Spencer, Srivatsan Srinivasan, Miloš Stanojević, Wojciech Stokowiec, Luyu Wang, Guangyao Zhou, and Fabio Viola.  The DeepMind JAX Ecosystem, 2020.  URL http://github.com/google-deepmind.
  * Del Moral et al. (2006) Pierre Del Moral, Arnaud Doucet, and Ajay Jasra.  Sequential Monte Carlo samplers.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 68(3):411–436, 2006.  doi: https://doi.org/10.1111/j.1467-9868.2006.00553.x.  URL https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2006.00553.x.
  * Depaoli et al. (2016) Sarah Depaoli, James P. Clifton, and Patrice R. Cobb.  Just Another Gibbs Sampler (JAGS): Flexible software for MCMC implementation.  _Journal of Educational and Behavioral Statistics_ , 41(6):628–649, 2016.  doi: 10.3102/1076998616664876.  URL https://doi.org/10.3102/1076998616664876.
  * Duane et al. (1987) Simon Duane, Anthony D Kennedy, Brian J Pendleton, and Duncan Roweth.  Hybrid Monte Carlo.  _Physics letters B_ , 195(2):216–222, 1987.
  * Foreman-Mackey et al. (2019) Daniel Foreman-Mackey, Will M Farr, Manodeep Sinha, Anne M Archibald, David W Hogg, Jeremy S Sanders, Joe Zuntz, Peter KG Williams, Andrew RJ Nelson, Miguel de Val-Borro, et al.  EMCEE v3: A Python ensemble sampling toolkit for affine-invariant MCMC.  _arXiv preprint arXiv:1911.07688_ , 2019.
  * Galan et al. (2022) Aymeric Galan, Georgios Vernardos, Austin Peel, Frédéric Courbin, and J-L Starck.  Using wavelets to capture deviations from smoothness in galaxy-scale strong lenses.  _Astronomy & Astrophysics_, 668:A155, 2022.
  * Geyer and Thompson (1995) Charles J Geyer and Elizabeth A Thompson.  Annealing Markov chain Monte Carlo with applications to ancestral inference.  _Journal of the American Statistical Association_ , 90(431):909–920, 1995.
  * Hastings (1970) W. K. Hastings.  Monte Carlo sampling methods using Markov chains and their applications.  _Biometrika_ , 57(1):97–109, 04 1970.  ISSN 0006-3444.  doi: 10.1093/biomet/57.1.97.  URL https://doi.org/10.1093/biomet/57.1.97.
  * Hoffman et al. (2021) Matthew Hoffman, Alexey Radul, and Pavel Sountsov.  An adaptive-MCMC scheme for setting trajectory lengths in Hamiltonian Monte Carlo.  In _International Conference on Artificial Intelligence and Statistics_ , pages 3907–3915. PMLR, 2021.
  * Hoffman and Sountsov (2022) Matthew D Hoffman and Pavel Sountsov.  Tuning-free generalized Hamiltonian Monte Carlo.  In _International conference on artificial intelligence and statistics_ , pages 7799–7813. PMLR, 2022.
  * Hoffman et al. (2014) Matthew D Hoffman, Andrew Gelman, et al.  The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo.  _J. Mach. Learn. Res._ , 15(1):1593–1623, 2014.
  * Horowitz (1991) Alan M Horowitz.  A generalized guided Monte Carlo algorithm.  _Physics Letters B_ , 268(2):247–252, 1991.
  * Jacob et al. (2020) Pierre E Jacob, John O’Leary, and Yves F Atchadé.  Unbiased Markov chain Monte Carlo methods with couplings.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 82(3):543–600, 2020.
  * Jordan et al. (1999) Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul.  An introduction to variational methods for graphical models.  _Machine learning_ , 37:183–233, 1999.
  * Karamanis et al. (2022) Minas Karamanis, David Nabergoj, Florian Beutler, John A Peacock, and Uros Seljak.  pocoMC: A Python package for accelerated Bayesian inference in astronomy and cosmology.  _arXiv preprint arXiv:2207.05660_ , 2022.
  * Kumar et al. (2019) Ravin Kumar, Colin Carroll, Ari Hartikainen, and Osvaldo Martin.  Arviz a unified library for exploratory analysis of bayesian models in python.  _Journal of Open Source Software_ , 4(33):1143, 2019.  doi: 10.21105/joss.01143.  URL https://doi.org/10.21105/joss.01143.
  * Lao et al. (2020) Junpeng Lao, Christopher Suter, Ian Langmore, Cyril Chimisov, Ashish Saxena, Pavel Sountsov, Dave Moore, Rif A. Saurous, Matthew D. Hoffman, and Joshua V. Dillon.  tfp.mcmc: Modern Markov chain Monte Carlo tools built for modern hardware, 2020.
  * Lonsdorf (2020) Brian Lonsdorf.  Pure Happiness with Pure Functions, 2020.  URL https://github.com/MostlyAdequate/mostly-adequate-guide/blob/master/ch03.md.
  * Lunn et al. (2000) David J Lunn, Andrew Thomas, Nicky Best, and David Spiegelhalter.  WinBUGS-a Bayesian modelling framework: concepts, structure, and extensibility.  _Statistics and computing_ , 10:325–337, 2000.
  * Marinari and Parisi (1992) E. Marinari and G. Parisi.  Simulated tempering: A new Monte Carlo scheme.  _Europhysics Letters_ , 19(6):451, jul 1992.  doi: 10.1209/0295-5075/19/6/002.  URL https://dx.doi.org/10.1209/0295-5075/19/6/002.
  * Metropolis et al. (1953) Nicholas Metropolis, Arianna W Rosenbluth, Marshall N Rosenbluth, Augusta H Teller, and Edward Teller.  Equation of state calculations by fast computing machines.  _The journal of chemical physics_ , 21(6):1087–1092, 1953.
  * Meyer and Yu (2000) Renate Meyer and Jun Yu.  BUGS for a Bayesian analysis of stochastic volatility models.  _The Econometrics Journal_ , 3(2):198–215, 2000.
  * Murphy (2022) Kevin P. Murphy.  _Probabilistic Machine Learning: An introduction_.  MIT Press, 2022.  URL probml.ai.
  * Neal (2020) Radford M Neal.  Non-reversibly updating a uniform [0, 1] value for Metropolis accept/reject decisions.  _arXiv preprint arXiv:2001.11950_ , 2020.
  * Oriol et al. (2023) Abril-Pla Oriol, Andreani Virgile, Carroll Colin, Dong Larry, Fonnesbeck Christopher J., Kochurov Maxim, Kumar Ravin, Lao Jupeng, Luhmann Christian C., Martin Osvaldo A., Osthege Michael, Vieira Ricardo, Wiecki Thomas, and Zinkov Robert.  PyMC: A modern and comprehensive probabilistic programming framework in Python.  _PeerJ Computer Science_ , 9:e1516, 2023.  doi: 10.7717/peerj-cs.1516.
  * Phan et al. (2019) Du Phan, Neeraj Pradhan, and Martin Jankowiak.  Composable effects for flexible and accelerated probabilistic programming in NumPyro.  _arXiv preprint arXiv:1912.11554_ , 2019.
  * Pinder and Dodd (2022) Thomas Pinder and Daniel Dodd.  GPJax: A Gaussian process framework in JAX.  _Journal of Open Source Software_ , 7(75):4455, 2022.  doi: 10.21105/joss.04455.  URL https://doi.org/10.21105/joss.04455.
  * Price-Whelan et al. (2024) Adrian M Price-Whelan, Jason AS Hunt, Danny Horta, Micah Oeur, David W Hogg, Kathryn V Johnston, and Lawrence Widrow.  Data-driven dynamics with orbital torus imaging: A flexible model of the vertical phase space of the galaxy.  _arXiv preprint arXiv:2401.07903_ , 2024.
  * Robert (2016) Christian P. Robert.  The Metropolis–Hastings algorithm, 2016.
  * Rue et al. (2009) Håvard Rue, Sara Martino, and Nicolas Chopin.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 71(2):319–392, 2009.
  * Schoenholz and Cubuk (2020) Samuel S. Schoenholz and Ekin D. Cubuk.  JAX M.D. a framework for differentiable physics.  In _Advances in Neural Information Processing Systems_ , volume 33. Curran Associates, Inc., 2020.  URL https://papers.nips.cc/paper/2020/file/83d3d4b6c9579515e1679aca8cbc8033-Paper.pdf.
  * Speagle (2020) Joshua S Speagle.  DYNESTY: a dynamic nested sampling package for estimating Bayesian posteriors and evidences.  _Monthly Notices of the Royal Astronomical Society_ , 493(3):3132–3158, 2020.
  * Staber and Da Veiga (2022) Brian Staber and Sébastien Da Veiga.  Benchmarking Bayesian neural networks and evaluation metrics for regression tasks.  _arXiv preprint arXiv:2206.06779_ , 2022.
  * Syed et al. (2022) Saifuddin Syed, Alexandre Bouchard-Côté, George Deligiannidis, and Arnaud Doucet.  Non-reversible parallel tempering: a scalable highly parallel MCMC scheme.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 84(2):321–350, 2022.
  * Tran et al. (2019) Dustin Tran, Michael W. Dusenberry, Danijar Hafner, and Mark van der Wilk.  Bayesian Layers: A module for neural network uncertainty.  In _Neural Information Processing Systems_ , 2019.
  * Wilkinson et al. (2023) William J Wilkinson, Simo Särkkä, and Arno Solin.  Bayes–Newton methods for approximate Bayesian inference with PSD guarantees.  _Journal of Machine Learning Research_ , 24(83):1–50, 2023.


# On the $p$-converse to a theorem of Gross–Zagier and Kolyvagin

Chan-Ho Kim  Center for Mathematical Challenges, Korea Institute for Advanced
Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We give a simple proof of a $p$-converse theorem to a theorem of Gross–Zagier
and Kolyvagin for non-CM elliptic curves with good ordinary reduction at $p>3$
under mild hypotheses. In particular, no condition on the conductor is
imposed. Combining with the known results, we obtain the equivalence

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1,\\#\textrm{{SH}}(E/\mathbb{Q})<\infty\Leftrightarrow\mathrm{ord}_{s=1}L(E,s)=1$

for _every_ elliptic curve $E$ over $\mathbb{Q}$.

###### Key words and phrases:

Birch and Swinnerton-Dyer conjecture, elliptic curves, Iwasawa theory, Iwasawa
main conjecture, Kato’s Euler systems

###### 2010 Mathematics Subject Classification:

11F67, 11G40, 11R23

##  1\. Statement of the main result

The aim of this short article is to give a succinct proof of the following
_$p$ -converse_ result to the theorem of Gross–Zagier and Kolyvagin on
elliptic curves of analytic rank one.

######  Theorem 1.1.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ and $p>2$ a non-additive
reduction prime for $E$. Denote by $V$ the two dimensional Galois
representation over $\mathbb{Q}_{p}$ associated to $E$. Assume that $E[p]$ is
an irreducible mod $p$ Galois representation. If

  1. (cork1)

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1$,

  2. (IMC[$1/p$])

the Iwasawa main conjecture (inverting $p$) holds, and

  3. (res)

the restriction map $\mathrm{res}_{p}:\mathrm{Sel}(\mathbb{Q},V)\to
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$ is an isomorphism,

then $\mathrm{ord}_{s=1}L(E,s)=1$. In particular,
$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1$ and
$\\#\textrm{{SH}}(E/\mathbb{Q})<\infty$.

By the work of Kato, Skinner–Urban, and X. Wan [Kat04, SU14, Wan15] on the
Iwasawa main conjecture for elliptic curves, the following result is
immediate.

######  Corollary 1.2.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ and $p>3$ a good ordinary
prime for $E$. Assume that $E[p]$ is an irreducible mod $p$ Galois
representation. If

  1. (cork1)

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1$, and

  2. (res)

the restriction map $\mathrm{res}_{p}:\mathrm{Sel}(\mathbb{Q},V)\to
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$ is an isomorphism,

then $\mathrm{ord}_{s=1}L(E,s)=1$. In particular,
$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1$ and
$\\#\textrm{{SH}}(E/\mathbb{Q})<\infty$.

######  Remark 1.3.

Assumption (res) also appears in [Ski20, Theorem B] in a slightly different
context. We explain why this assumption appears in our approach in Remark
2.11. When $p=3$, Corollary 1.2 still holds if we further assume that there
exists a prime $\ell$ exactly dividing the conductor of $E$ such that $E[p]$
is ramified at $\ell$ [SU14]. For a non-CM elliptic curve $E$, the density of
good ordinary primes for $E$ is one [Ser81] and $E[p]$ is an irreducible
Galois representation for a sufficiently large prime $p\gg 0$ [Ser89].

Since Assumption (res) holds if we further assume
$\\#\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]<\infty$ in Corollary 1.2, we
obtain the following statement.

######  Corollary 1.4.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. If
$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1$ and
$\\#\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]<\infty$ for at least one good
ordinary prime $p>3$ such that $E[p]$ is irreducible, then
$\mathrm{ord}_{s=1}L(E,s)=1$. In particular,
$\\#\textrm{{SH}}(E/\mathbb{Q})<\infty$.

Indeed, Corollary 1.4 itself is stronger than the converse to the theorem of
Gross–Zagier and Kolyvagin on elliptic curves of analytic rank one [GZ86],
[Kol90]. Especially, if $\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1$ and
$\textrm{{SH}}(E/\mathbb{Q})$ is infinite, then
$\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ is infinite for every good ordinary
prime $p$ such that $E[p]$ is irreducible.

This type of the $p$-converse theorem is first proved by Skinner [Ski20] and
W. Zhang [Zha14], independently. In both work, there are certain technical
assumptions on the conductor of elliptic curves. In [Ski20], the conductor is
square-free and satisfies a certain existence condition on split/non-split
reduction primes. In [Zha14], the conductor satisfies a certain ramification
condition arising from the arithmetic of Shimura curves. We completely remove
these assumptions on the conductor.

The key ingredients are the Iwasawa main conjecture (inverting $p$) and
Perrin-Riou’s conjecture [PR93] recently settled by Bertolini–Darmon–Venerucci
[BDV21]. We are informed that Perrin-Riou’s conjecture and Corollaries 1.2 and
1.4 also appear in [BST] when $p>3$. See also Remark 2.2. We refer the reader
to [Ven16a, Büy20, BDV21, BPS, BST] for the recent developments of Perrin-
Riou’s conjecture.

It is remarkable that the argument itself is quite straightforward and
extremely simple: the Selmer corank is one $\Rightarrow$ the $p$-strict Selmer
corank is zero $\Rightarrow$ Kato’s zeta element is non-trivial $\Rightarrow$
the analytic rank is one. The simplicity of the argument would be interesting
to the reader.

Together with the work of Rubin [Rub94] and Burungale–Tian [BT20] on the CM
case, Corollary 1.4 completes the converse to the theorem of Gross–Zagier and
Kolyvagin for elliptic curves over $\mathbb{Q}$. More precisely, for _every_
elliptic curve $E$ over $\mathbb{Q}$, we have the equivalence

---
$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1,$
$\\#\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]<\infty$ Corollary 1.4 for non-CM
$E$[Rub94, BT20] for CM
$E$$\textstyle{\mathrm{ord}_{s=1}L(E,s)=1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Gross–ZagierKolyvagin
$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=1,$
$\\#\textrm{{SH}}(E/\mathbb{Q})<\infty$

where $p>3$ is a good ordinary prime for $E$ such that $E[p]$ is irreducible
when $E$ is non-CM.

On the other hand, the $p$-converse theorem

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1\Rightarrow\mathrm{ord}_{s=1}L(E,s)=1$

_without_ Assumption (res) can be obtained from the Heegner point main
conjecture (inverting $p$) [PR87], the control theorem, the Gross–Zagier
formula, and a choice of a suitable imaginary quadratic field as explained in
[Wan21] and [BCK21]. See also [CW]. One can use the Kolyvagin conjecture
[Kol91, Zha14] instead of the Heegner point main conjecture. In our context,
the Heegner point main conjecture can also be used to remove Assumption (res)
(Remark 2.11). We avoid using the Heegner point main conjecture here since it
is unclear for us whether the Heegner point main conjecture (inverting $p$) is
established in full generality yet. In Remark 2.12, we compare our strategy
with this Heegner point strategy in one simple diagram.

The $p$-converse theorem itself is extensively studied recently in various
forms. We list some of them, but we do not specify the precise conditions and
statements here:

  * •

[Ven16b] when $p$ is split-multiplicative for $E$,

  * •

[BT20] when $E$ has CM and $p>3$ is good ordinary for $E$,

  * •

[Wan21] when $p$ is good ordinary for $E$,

  * •

[CGLS21] when $E[p]$ is reducible,

  * •

[BCST] when $E$ has CM and $p$ is good ordinary for $E$,

  * •

[SZ] when $p$ is multiplicative for $E$,

  * •

[CW] when $p$ is supersingular for $E$,

  * •

[Swe] when $p$ is good for $E$, and

  * •

[BST] when $p$ is good ordinary for $E$.

##  2\. Proof of Theorem 1.1

Let $T$ be the Tate module of an elliptic curve $E$ and
$V=T\otimes\mathbb{Q}_{p}$. Kato’s explicit reciprocity law [Kat04, Theorem
12.5] implies that

$\mathrm{exp}^{*}\circ\mathrm{res}_{p}(z_{\mathrm{Kato}})=\dfrac{L(E,1)}{\Omega^{+}_{E}}\cdot\omega_{E}$

up to a non-zero rational factor where

  * •

$z_{\mathrm{Kato}}\in\mathrm{H}^{1}(\mathbb{Q},V)$ is Kato’s zeta element over
$\mathbb{Q}$,

  * •

$\mathrm{res}_{p}:\mathrm{H}^{1}(\mathbb{Q},V)\to\mathrm{H}^{1}(\mathbb{Q}_{p},V)$
is the restriction map,

  * •

$\mathrm{exp}^{*}$ is Bloch–Kato’s dual exponential map,

  * •

$\omega_{E}$ is the Néron differential of a global minimal Weierstrass
equation for $E$, and

  * •

$\Omega^{+}_{E}$ is the Néron period for $E$.

We use the following result of Bertolini–Darmon–Venerucci [BDV21].

######  Theorem 2.1 (Bertolini–Darmon–Venerucci).

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ and $p$ an odd
prime with $p^{2}\nmid N$. If $L(E,1)=0$, then there exists a global point
$P\in E(\mathbb{Q})$ satisfying the following properties.

  1. (1)

The point $P$ has infinite order if and only if $\mathrm{ord}_{s=1}L(E,s)=1$.

  2. (2)

The following equality holds in $\mathbb{Q}_{p}$ up to multiplication by a
non-zero rational number:

$\mathrm{log}_{\omega_{E}}(\mathrm{res}_{p}(z_{\mathrm{Kato}}))=\left(\mathrm{log}_{\omega_{E}}(P)\right)^{2}$

where $\mathrm{log}_{\omega_{E}}$ is the $p$-adic Lie group logarithm
corresponding to $\omega_{E}$.

######  Remark 2.2.

In [BST], Theorem 2.1 is also obtained when $p\nmid 6N$.

We extract the statement we need from Theorem 2.1.

######  Corollary 2.3.

Under the setting of Theorem 2.1, the following statements are equivalent.

  1. (1)

$\mathrm{ord}_{s=1}L(E,s)=1$.

  2. (2)

$P$ has infinite order.

  3. (3)

$\mathrm{res}_{p}(z_{\mathrm{Kato}})$ is non-zero.

Although the $L(E,1)=0$ assumption is incorporated in Theorem 2.1, it can be
removed in our main result thanks to the following theorem.

######  Theorem 2.4 (Kato).

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. If $L(E,1)\neq 0$, then
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ is finite.

###### Proof.

See [Rub98, Theorem 8.1] and [Kat04, Theorem 14.2]. ∎

We quickly review the Iwasawa main conjecture without $p$-adic $L$-functions à
la Kato [Kur02, $\S$6], [Kat04, Conjecture 12.10]. Let $\mathbb{Q}_{\infty}$
be the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and
$\Lambda=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket$
the Iwasawa algebra. Write
$\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)=\varprojlim_{n}\mathrm{H}^{1}(\mathbb{Q}_{n},T)$
where $\mathbb{Q}_{n}$ is the cyclic subextension of $\mathbb{Q}$ of order
$p^{n}$ in $\mathbb{Q}_{\infty}$ and $\varprojlim_{n}$ is taken with respect
to the corestriction. Let
$z^{\infty}_{\mathrm{Kato}}\in\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)$ be
Kato’s zeta element over $\mathbb{Q}_{\infty}$ (with respect to $T$). Let
$\mathrm{Sel}_{0}(\mathbb{Q}_{n},E[p^{\infty}])\subseteq\mathrm{Sel}(\mathbb{Q}_{n},E[p^{\infty}])$
be the $p$-strict Selmer group of $E[p^{\infty}]$ over $\mathbb{Q}_{n}$, and
$\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])=\varinjlim_{n}\mathrm{Sel}_{0}(\mathbb{Q}_{n},E[p^{\infty}])$.
Write
$(-)^{\vee}=\mathrm{Hom}_{\mathbb{Z}_{p}}(-,\mathbb{Q}_{p}/\mathbb{Z}_{p})$.

######  Assumption 2.5 (IMC[$1/p$]).

We assume the Iwasawa main conjecture inverting $p$. In other words, we assume
equality

(2.1)
$\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z^{\infty}_{\mathrm{Kato}}}\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right)$

in $\Lambda\otimes\mathbb{Q}_{p}$.

######  Remark 2.6.

When $p\geq 5$ is good ordinary for $E$ and $E[p]$ is an irreducible Galois
representation, Assumption 2.5 follows from the work of Kato [Kat04, Theorem
12.5 and Theorem 17.4], Skinner–Urban [SU14, Theorem 3.33], and X. Wan [Wan15,
Theorem 4].

######  Proposition 2.7.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ and $p$ an odd prime.
Assume that $E[p]$ is an irreducible Galois representation and the Iwasawa
main conjecture inverting $p$ holds (Assumption 2.5). Then the following
statements are equivalent.

  1. (1)

$z_{\mathrm{Kato}}$ is non-zero.

  2. (2)

$\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$ is finite.

###### Proof.

Write $f_{z},f_{0}\in\Lambda$ to be the distinguished polynomials such that

$\textstyle{\left(f_{z}\right)=\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda
z^{\infty}_{\mathrm{Kato}}}\right),}$$\textstyle{\left(f_{0}\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right),}$

respectively. Then (2.1) implies that $\mathbf{1}(f_{0})=0$ if and only if
$\mathbf{1}(f_{z})=0$ where $\mathbf{1}$ is the trivial character. By using
the control theorem for $\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$
[Kur02, Lemma 4.2 and Remark 4.4], the finiteness of
$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is equivalent to
$z_{\mathrm{Kato}}\neq 0$. ∎

######  Proposition 2.8.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ and $p$ an odd prime.
Assume that $E[p]$ is an irreducible Galois representation, the Iwasawa main
conjecture inverting $p$ holds (Assumption 2.5), and
$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1$. Then
the following statements are equivalent.

  1. (1)

$z_{\mathrm{Kato}}$ is non-zero.

  2. (2)

$\mathrm{res}_{p}(z_{\mathrm{Kato}})$ is non-zero.

###### Proof.

$(2)\Rightarrow(1)$ is trivial. We now assume $z_{\mathrm{Kato}}$ is non-zero.
Consider the exact sequence

(2.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{0}(\mathbb{Q},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}(\mathbb{Q},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}_{p}}$$\textstyle{E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}.}$

By Proposition 2.7, $\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is finite, so
$\mathrm{Sel}_{0}(\mathbb{Q},V)=0$. Thus, the restriction map
$\mathrm{res}_{p}$ is an isomorphism of one-dimensional
$\mathbb{Q}_{p}$-vector spaces, so $\mathrm{res}_{p}(z_{\mathrm{Kato}})$ is
non-zero. ∎

######  Remark 2.9.

Although Proposition 2.8 is trivial if we just assume
$\mathrm{res}_{p}:\mathrm{Sel}(\mathbb{Q},V)\simeq
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$, we would like to use Assumption (res)
minimally.

The following proposition completes the proof.

######  Proposition 2.10.

If $\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1$
and $\mathrm{res}_{p}:\mathrm{Sel}(\mathbb{Q},V)\simeq
E(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}$, then
$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is finite.

###### Proof.

By the assumptions, we have $\mathrm{Sel}_{0}(\mathbb{Q},V)=0$ by (2.2). Thus,
$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is finite. ∎

Theorem 1.1 follows from the combination of Corollary 2.3, Theorem 2.4,
Proposition 2.7, Proposition 2.8, and Proposition 2.10. The last “In
particular” part follows from the work of Gross–Zagier and Kolyvagin
[GZ86],[Kol90] with a choice of a suitable imaginary quadratic field. The
existence of such an imaginary quadratic field is ensured by using [BFH90],
[MM91].

######  Remark 2.11.

Assumption (res) is crucially and only used in Proposition 2.10. If we assume
the Heegner point main conjecture, then, by the control theorem, we are able
to deduce that $\mathrm{Sel}(\mathbb{Q},V)\simeq\mathbb{Q}_{p}$ if and only if
there exists a non-torsion point $P$ in $E(\mathbb{Q})$ obtained from Heegner
points. In other words, the existence of such a non-torsion global point
becomes an input, not an output (cf. Theorem 2.1). Thus, we have
$\mathrm{Sel}(\mathbb{Q},V)\simeq E(\mathbb{Q})\otimes\mathbb{Q}_{p}$, so the
restriction map becomes an isomorphism. The advantage of Heegner points here
is that they are genuinely global points on elliptic curves. See also [Ski20,
Remark 2.9.1.(x)].

######  Remark 2.12.

The comparison between our strategy and the Heegner point strategy is
summarized in the following diagram

$\textstyle{z_{\mathrm{Kato}}\neq
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textrm{PRC}+\textrm{IMC}}$IMC$\textstyle{\\#\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])<\infty\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L(E,1)=0}$$\textstyle{\mathrm{ord}_{s=1}L(E,s)=1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$GZ
formula$\textstyle{P\neq
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$HPMCKolyvagin$\textstyle{\textrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$Prop.
2.10

## Acknowledgement

The discussion with Ashay Burungale and Francesc Castella leads us to improve
the main result and the exposition in an earlier version significantly. We
deeply thank to both. This research was partially supported by a KIAS
Individual Grant (SP054102) via the Center for Mathematical Challenges at
Korea Institute for Advanced Study and by the National Research Foundation of
Korea(NRF) grant funded by the Korea government(MSIT) (No. 2018R1C1B6007009).

## References

  * [BCK21] Ashay A. Burungale, Francesc Castella, and Chan-Ho Kim, _A proof of Perrin-Riou’s Heegner point main conjecture_ , Algebra Number Theory (2021), to appear.
  * [BCST] Ashay A. Burungale, Francesc Castella, Christopher Skinner, and Ye Tian, _$p^{\infty}$ -Selmer groups and rational points on CM elliptic curves_, preprint.
  * [BDV21] Massimo Bertolini, Henri Darmon, and Rodolfo Venerucci, _Heegner points and Beilinson–Kato elements: a conjecture of Perrin-Riou_ , Adv. Math. (2021), to appear.
  * [BFH90] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, _Nonvanishing theorems for $L$-functions of modular forms and their derivatives_, Invent. Math. 102 (1990), no. 3, 543–618.
  * [BPS] Kâzim Büyükboduk, Robert Pollack, and Shu Sasaki, _$p$ -adic Gross–Zagier formula at critical slope and a conjecture of Perrin-Riou_, preprint, arXiv:1811.08216.
  * [BST] Ashay A. Burungale, Christopher Skinner, and Ye Tian, _Elliptic curves and Beilinson–Kato elements: rank one aspects_ , preprint.
  * [BT20] Ashay A. Burungale and Ye Tian, _$p$ -converse to a theorem of Gross–Zagier, Kolyvagin and Rubin_, Invent. Math. 220 (2020), no. 1, 211–253.
  * [Büy20] Kâzim Büyükboduk, _Beilinson–Kato and Beilinson–Flach elements, Coleman–Rubin–Stark classes, Heegner points and a conjecture of Perrin-Riou_ , Development of Iwasawa Theory – the Centennial of K. Iwasawa’s Birth (Tokyo) (Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, and Takeshi Tsuji, eds.), Adv. Stud. Pure Math., vol. 86, Mathematical Society of Japan, 2020, pp. 141–193.
  * [CGLS21] Francesc Castella, Giada Grossi, Jaehoon Lee, and Christopher Skinner, _On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes_ , Invent. Math. (2021), to appear.
  * [CW] Francesc Castella and Xin Wan, _Perrin-Riou’s Heegner point main conjecture and $p$-converse to Gross–Zagier–Kolyvagin for supersingular primes_, preprint.
  * [GZ86] Benedict Gross and Don Zagier, _Heegner points and derivatives of $L$-series_, Invent. Math. 84 (1986), no. 2, 225–320.
  * [Kat04] Kazuya Kato, _$p$ -adic Hodge theory and values of zeta functions of modular forms_, Astérisque 295 (2004), 117–290.
  * [Kol90] Victor Kolyvagin, _Euler systems_ , The Grothendieck Festschrift Volume II (Pierre Cartier, Luc Illusie, Nicholas M. Katz, Gerard Laumon, Yuri Manin, and Kenneth A. Ribet, eds.), Progr. Math., vol. 87, Birkhäuser Boston, 1990, pp. 435–483.
  * [Kol91] by same author, _On the structure of Selmer groups_ , Math. Ann. 291 (1991), no. 2, 253–259.
  * [Kur02] Masato Kurihara, _On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I_ , Invent. Math. 149 (2002), 195–224.
  * [MM91] M. Ram Murty and V. Kumar Murty, _Mean values of derivatives of modular $L$-series_, Ann. of Math. (2) 133 (1991), no. 3, 447–475.
  * [PR87] Bernadette Perrin-Riou, _Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner_, Bull. Soc. Math. France 115 (1987), no. 4, 399––456.
  * [PR93] by same author, _Fonctions $L$ $p$-adiques d’une courbe elliptique et points rationnels_, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945–995.
  * [Rub94] Karl Rubin, _$p$ -adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication_, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) (Providence, RI) (Barry Mazur and Glenn Stevens, eds.), Contemp. Math., vol. 165, American Mathematical Society, 1994, pp. 71–80.
  * [Rub98] by same author, _Euler systems and modular elliptic curves_ , Galois representations in arithmetic algebraic geometry (Durham, 1996) (Anthony Scholl and Richard Taylor, eds.), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge University Press, 1998, pp. 351–367.
  * [Ser81] Jean-Pierre Serre, _Quelques applications du théorème de densité de Chebotarev_ , Publ. Math. Inst. Hautes Études Sci. 54 (1981), 323–401.
  * [Ser89] by same author, _Abelian $\ell$-adic representations and elliptic curves_, Advanced Book Classics, Addison-Wesley Publishing Company, 1989, With the collaboration of Willem Kuyk and John Labute.
  * [Ski20] Christopher Skinner, _A converse to a theorem of Gross, Zagier, and Kolyvagin_ , Ann. of Math. 191 (2020), no. 2, 329–354.
  * [SU14] Christopher Skinner and Eric Urban, _The Iwasawa main conjectures for $\mathrm{GL}_{2}$_, Invent. Math. 195 (2014), no. 1, 1–277.
  * [Swe] Naomi Sweeting, _Kolyvagin’s conjecture and patched Euler systems in anticyclotomic Iwasawa theory_ , preprint, arXiv:2012.11771.
  * [SZ] Christopher Skinner and Wei Zhang, _Indivisibility of Heegner points in the multiplicative case_ , preprint, arXiv:1407.1099.
  * [Ven16a] Rodolfo Venerucci, _Exceptional zero formulae and a conjecture of Perrin-Riou_ , Invent. Math. 203 (2016), no. 3, 923–972.
  * [Ven16b] by same author, _On the $p$-converse of the Kolyvagin–Gross–Zagier theorem_, Comment. Math. Helv. 91 (2016), no. 3, 397–444.
  * [Wan15] Xin Wan, _The Iwasawa main conjecture for Hilbert modular forms_ , Forum Math. Sigma 3 (2015), e18 (95 pages).
  * [Wan21] by same author, _Heegner point Kolyvagin system and Iwasawa main conjecture_ , Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 1, 104–120.
  * [Zha14] Wei Zhang, _Selmer groups and the indivisibility of Heegner points_ , Camb. J. Math. 2 (2014), no. 2, 191–253.


[table]capposition=top

# A Direct Construction of Cross Z-Complementary Sets with Flexible Lengths
and Large Zero Correlation Zone

Praveen Kumar, Sudhan Majhi, Subhabrata Paul

###### Abstract

This letter proposes a direct construction for cross Z-complementary sets
(CZCSs) with flexible lengths and a large zero correlation zone (ZCZ). CZCS is
an extension of the cross Z-complementary pair (CZCP). The maximum possible
ZCZ width of a CZCP is half of its sequence length. In this letter, for the
first time, a generalized Boolean function based construction of CZCSs with a
large number of constituent sequences and a ZCZ ratio of $2/3$ is presented.
For integers $m$ and $\delta$, the proposed construction produces CZCS with
length expressed as $2^{m-1}+2^{\delta}$ ($0\leq\delta<m-1,m\geq 4$), where
both odd and even lengths CZCS can be obtained. Additionally, the constructed
CZCS also feature a complementary set of the same length. Finally, the
proposed construction is compared with the existing works.

###### Index Terms:

Cross Z-complementary set (CZCS), complementary set (CS), generalized Boolean
functions (GBFs), zero correlation zone (ZCZ).

##  I Introduction

Golay was the first to introduce Golay complementary pairs (GCPs) [1]. The
aperiodic auto-correlation sum (AACS) of a GCP becomes zero for all shifts
other than zero. Binary GCPs can only have lengths of the type
$2^{\alpha}10^{\beta}26^{\gamma}$, where $\alpha,\beta,$ and $\gamma$ are all
non-negative integers [2]. Using generalized Boolean functions (GBFs) over
$\mathbb{Z}_{2^{h}}$, for any positive integer $h$, Davis and Jedwab provide a
simple construction of GCP [3]. The findings of Davis and Jedwab were further
expanded by Paterson to include $q$-ary GCPs for even $q$ [4]. Recently, GCP
of non-powers-of-two lengths using GBFs is provided in [5]. When the number of
sequences is more than two it is called complementary set (CS) and its direct
constructions are provided in [4, 6]. Fan et al. introduced the concept of
Z-complementary pairs (ZCPs) as a generalization of GCP [7]. Inside a certain
zone known as the zero correlation zone (ZCZ), AACS of ZCP is zero. ZCPs, in
contrast to GCPs, come in various ZCZ widths and arbitrary lengths [8, 9, 10,
11, 12, 13].

Cross Z-complementary pair (CZCP) was produced as a consequence of Liu et al.
investigation into the training sequence design for the spatial modulation
(SM) system [14]. AACS of CZCPs have both front-end and tail-end ZCZ, and
aperiodic cross-correlation sum (ACCS) of CZCPs have only tail-end ZCZ. For a
perfect CZCP of length $N$, the largest ZCZ width is $N/2$, i.e.,
$Z_{max}=N/2$ [14]. The length of perfect CZCPs is extremely constrained,
though, binary perfect CZCPs exist with lengths
$2^{\alpha+1}10^{\beta}26^{\gamma}$ where $\alpha+\beta+\gamma\geq 0$ [14].
The $q$-ary CZCPs with lengths of the type
$10^{\beta}26^{\gamma},2^{\alpha+1}10^{\beta}26^{\gamma}+2$, and $12\times
2^{\alpha+1}10^{\beta}26^{\gamma}$ have recently been proposed in [15, 16].
The majority of CZCP constructions are built on existing GCPs or Barker
sequences [15, 14]. CZCPs can also be constructed b͡y using existing ZCPs as
suggested in [17, 18]. The direct constructions of CZCPs based on GBFs are
also given in [15, 14, 18]. These CZCPs have lengths of the following types:
$2^{m}~{}(m\geq 2)$, or $2^{m-1}+2^{v+1}~{}(0\leq v\leq m-3,m\geq 4)$. CZCPs
exist only for even length and maximum ZCZ width is restricted only to half of
the sequence length [14].

To overcome these shortcomings of CZCPs, recently Huang et al. proposed many
indirect constructions of the CZCSs having four sequences, acknowledging the
constraints on the ZCZ width of the CZCPs [19]. In [14], identical CZCPs are
combined to create cross Z-complementary sets (CZCSs) with the same length and
same ZCZ ratio as CZCP. The construction of CZCSs containing a large number of
constituent sequences based on GBF is proposed in [20], but their ZCZ ratio is
less than $1/2$.

In this letter, we present a GBF based construction of CZCSs of length of the
form $2^{m-1}+2^{\delta}$, where $m\geq 4$ and $0\leq\delta<m-1$. The proposed
CZCSs have a large number of constituent sequences, flexible lengths, and
large ZCZ widths. The proposed construction is capable of producing CZCSs of
odd and even lengths with ZCZ ratios up to $2/3$. The proposed CZCSs have
larger flock size as compared to [19], cover more length and have better ZCZ
ratio than [14, 20]. Therefore the proposed direct construction is
advantageous over [14, 19, 20]. Moreover, the constructed CZCS also provides a
CS of the same length.

The remainder of the letter is organised as follows. The preliminaries are
covered in section II. The proposed CZCS construction based on GBF is covered
in section III. Final conclusions are included in section IV.

##  II Preliminaries

This section explains the fundamental definitions, notations, and previously
known results required for the proposed construction.

###### Definition 1

Let $\mathbf{u}$ $=(u_{0},u_{1},\ldots,u_{N-1})$ and $\mathbf{v}$
$=(v_{0},v_{1},\ldots,v_{N-1})$ be two $N$ length sequences over
$\mathbb{Z}_{q}$. The aperiodic cross-correlation function (ACCF) is defined
at a shift $\tau$ by

$\mathcal{C}\left({\mathbf{u},\mathbf{v}}\right)(\tau)=\begin{cases}\sum_{i=0}^{N-1-\tau}\omega^{u_{i}-v_{i+\tau}},&0\leq\tau\leq
N-1,\\\ \sum_{i=0}^{N-1+\tau}\omega^{u_{i-\tau}-v_{i}},&-N+1\leq\tau\leq-1,\\\
0,&|\tau|\geq N,\end{cases}$ (1)

where $q~{}(\geq 2)$ is an integer and $\omega=\exp(2\pi\sqrt{-1}/q)$. When
$\mathbf{u}=\mathbf{v},~{}\mathcal{C}(\mathbf{u},\mathbf{v})(\tau)$ is known
as aperiodic auto-correlation function (AACF) of $\mathbf{u}$ and is
represented by $\mathcal{A}(\mathbf{u})(\tau)$.

###### Definition 2

A pair of sequences $\mathbf{u}$ and $\mathbf{v}$ of length $N$, whose AACS is
zero for all non-zero shifts inside a zone, constitutes a ZCP with ZCZ width
$Z$, i.e.,

$\mathcal{A}({\mathbf{u}})(\tau)+\mathcal{A}({\mathbf{v}})(\tau)=0,\text{ for
all }0<\tau<Z.$ (2)

The pair $(\mathbf{u},\mathbf{v})$ is referred to as a GCP when $Z=N$.

###### Definition 3

A set $A=\\{\mathbf{a}_{0},\mathbf{a}_{1},\ldots,\mathbf{a}_{M-1}\\}$,
containing $M$ sequences each of length $N$ is known as an $(M,N,Z)$-CZCS if
it satisfies the following two conditions

$\displaystyle{C1}:\sum_{i=0}^{M-1}\mathcal{A}({\mathbf{a}_{i}})(\tau)=0\text{,
for all }|\tau|\in\mathcal{T}_{1}\cup\mathcal{T}_{2};$ (3)
$\displaystyle{C2}:\sum_{i=0}^{M-1}\mathcal{C}({\mathbf{a}_{i},\mathbf{a}_{i+1}})(\tau)=0,\text{
for all }|\tau|\in\mathcal{T}_{2},$

where $\mathbf{a}_{M}=\mathbf{a}_{0}$ and $\mathcal{T}_{1}=\\{1,2,\ldots,Z\\}$
and $\mathcal{T}_{2}=\\{N-Z,N-Z+1,\ldots,N-1\\}$. For an $(M,N,Z)$-CZCS, $M$
is the flock size, $N$ is the sequence length, and $Z$ is the ZCZ width. For
the case when $M=2$, it is known as a CZCP.

Corresponding to a GBF $f:\\{0,1\\}^{m}\rightarrow\mathbb{Z}_{q}$ of $m$
variables $x_{1},x_{2},\ldots,x_{m}$, the $\mathbb{Z}_{q}$-valued sequence is
written as $\mathbf{f}=\left({f_{0}},{f_{1}},\ldots,{f_{2^{m}-1}}\right)$ ,
where $f_{r}=f({r_{1}},{r_{2}},\ldots,r_{m})$, and
$r=({r_{1}},{r_{2}},\ldots,r_{m})$ is the binary vector representation of the
integer $r$ [4]. Corresponding to $m$ variables GBF $f$, the sequence
$\mathbf{f}$ is of length $2^{m}$. The remainder of this letter concentrates
on $q$-ary CZCSs of length in the form of non-power-of-two, where $q$ is a
positive even integer. Here, the final $2^{m}-L$ components of the sequence
$\mathbf{f}$ are removed to create the truncated $L$ length sequence
$\mathbf{f}^{(L)}$ corresponding to the GBF $f$. For simplicity, when it is
clear from the context, we will use $f$ to represent all three
$(f,\mathbf{f},\mathbf{f}^{(L)})$. For a GBF $f$ in $m$ variables
$x_{1},x_{2},\ldots,x_{m}$,
$\bar{f}(x_{1},x_{2},\ldots,x_{m})=f(\bar{x}_{1},\bar{x}_{2},\ldots,\bar{x}_{m})$,
where $\bar{x}_{i}=1-x_{i},1\leq i\leq m$.

##  III Proposed Construction

This section describes a direct CZCS construction based on GBF with flexible
lengths of the type $2^{m-1}+2^{\delta}$ ($0\leq\delta<m-1,m\geq 4$).

###### Theorem 1

For any positive integer $m\geq 4$ and even integer $q$, let
$S_{1},S_{2},\ldots,S_{k}$ be a partition of the set $\\{1,2,\ldots,m-1\\}$,
$1\leq k\leq m-1$. Let $S_{\beta}$ contains $m_{\beta}$ elements and
$\pi_{\beta}$ be a bijective mapping from $\\{1,2,\ldots,m_{\beta}\\}$ to
$S_{\beta}$ for $\beta=1,2,\ldots,k.$ Let us define a GBF
$f:\\{0,1\\}^{m}\rightarrow\mathbb{Z}_{q}$ as

$f(x_{1},x_{2},\ldots,x_{m})=\bar{x}_{m}\bar{g}+x_{m}g+\sum_{l=1}^{m-1}\lambda_{l}x_{l}+\lambda_{0},$
(4)

where $\lambda_{l}\in\mathbb{Z}_{q}$, $0\leq l\leq m-1$ and
$g:\\{0,1\\}^{m-1}\rightarrow\mathbb{Z}_{q}$ is defined as

$g(x_{1},x_{2},\ldots,x_{m-1})=\frac{q}{2}\sum_{\beta=1}^{k}\sum_{\gamma=1}^{m_{\beta}-1}x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}.$
(5)

If sets $\\{\pi_{1}(1),\pi_{1}(2),\ldots,\pi_{1}(\delta)\\}$ and
$\\{1,2,\ldots,\delta\\}$ are equal for a non-negative integer $\delta<m-1$,
then the ordered set
$A=\\{\mathbf{a}_{0},\mathbf{a}_{1},\ldots,\mathbf{a}_{2^{k+1}-1}\\}$ with the
natural order induced by the vector
$(\sigma_{1},\sigma_{2},\ldots,\sigma_{k+1})$ forms a
$\left(2^{k+1},2^{m-1}+2^{\delta},2^{\pi_{k}(1)-1}+2^{\delta}\right)$-CZCS by
deleting the last $2^{m-1}-2^{\delta}$ elements of every sequence, where

$\mathbf{a}_{\sigma}=f+\frac{q}{2}\left(\sum_{\beta=1}^{k}\sigma_{\beta}x_{\pi_{\beta}(1)}+\sigma_{k+1}x_{m}\right).$
(6)

###### Proof:

The Appendix contains the proof. ∎

###### Remark 1

The ZCZ width of the proposed CZCS is maximum when $\pi_{k}(1)=m-1$. For
$\delta=m-2$ and $\pi_{k}(1)=m-1$, the
$ZCZ_{ratio}=\frac{Z}{N}=\frac{2^{m-2}+2^{m-2}}{2^{m-1}+2^{m-2}}=\frac{2}{3}$.

Now, we provide an example to illustrate how the proposed construction is
utilized to generate CZCSs with flexible length and large ZCZ width.

###### Example 1

Let $m=5,q=4$, $S_{1}=\\{1,2,3\\}$ and $S_{2}=\\{4\\}$ such that
$\pi_{1}(1)=1,\pi_{1}(2)=2,\pi_{1}(3)=3$ and $\pi_{2}(1)=4$. Let the GBF $f$
be given by
$f=2\bar{x}_{5}\left(\bar{x}_{1}\bar{x}_{2}+\bar{x}_{2}\bar{x}_{3}\right)+2x_{5}\left(x_{1}x_{2}+x_{2}x_{3}\right)$.
Consider the ordered set

$\displaystyle A=\\{$ $\displaystyle
f,f+2x_{1},f+2x_{4},f+2(x_{1}+x_{4}),f+2x_{5}$
$\displaystyle,f+2(x_{1}\\!+x_{5}),f+2(x_{4}\\!+x_{5}),f+2(x_{1}\\!+x_{4}\\!+x_{5})\\}.$

Since $\\{\pi_{1}(1)\\}=\\{1\\}$, by taking $\delta=1$ from Theorem 1 the set
$A$ is a $2^{4}+2^{1}=18$ length CZCS with ZCZ width $Z=2^{3}+2^{1}=10$ and
flock size $8$. So, the ZCZ ratio in this case is $10/18=5/9>1/2$. If we take
$\delta=2$, then from Theorem 1, $A$ becomes a $(8,20,12)$-CZCS, and the ZCZ
ratio in this case is $3/5>1/2$. Also, since
$\\{\pi_{1}(1),\pi_{1}(2),\pi_{1}(3)\\}=\\{1,2,3\\}$, so by taking $\delta=3$,
the set $A$ is a $(8,24,16)$-CZCS from Theorem 1. In this case, the CZCS
achieves its maximum ZCZ ratio of $2/3$. For $\delta=3$, the AACF and ACCF
values of sequences of the set $A$ at all the positive shifts including zero
are listed in Table I and Table II, respectively. By setting $\delta=0$, we
can get an odd length CZCS of length $2^{4}+2^{0}=17$ and ZCZ width
$Z=2^{3}+2^{0}=9$, hence the ZCZ ratio in this case is $9/17>1/2$.

TABLE I: AACF of the sequences of set $A$ in Example 1 for $\delta=3$.

$\mathcal{A}(a_{0})(\tau)$ | (24,-1,2,13,0,7,2,5,8,-1,8,3,0,5,0,5,0,-1,6,-3,0,3,-2,1)
---|---
$\mathcal{A}(a_{1})(\tau)$ | (24,1,2,-13,0,-7,2,-5,8,1,8,-3,0,-5,0,-5,0,1,6,3,0,-3,-2,-1)
$\mathcal{A}(a_{2})(\tau)$ | (24,-5,-2,5,0,-1,-2,1,-8,3,-4,3,0,-3,4,1,0,-1,6,-3,0,3,-2,1)
$\mathcal{A}(a_{3})(\tau)$ | (24,5,-2,-5,0,1,-2,-1,-8,-3,-4,-3,0,3,4,-1,0,1,6,3,0,-3,-2,-1)
$\mathcal{A}(a_{4})(\tau)$ | (24,-3,-2,7,0,5,-2,-1,8,-1,-8,3,0,-3,0,-3,0,1,-6,3,0,-3,2,-1)
$\mathcal{A}(a_{5})(\tau)$ | (24,3,-2,-7,0,-5,-2,1,8,1,-8,-3,0,3,0,3,0,-1,-6,-3,0,3,2,1)
$\mathcal{A}(a_{6})(\tau)$ | (24,-3,2,11,0,1,2,7,-8,-1,4,-9,0,1,-4,-3,0,1,-6,3,0,-3,2,-1)
$\mathcal{A}(a_{7})(\tau)$ | (24,3,2,-11,0,-1,2,-7,-8,1,4,9,0,-1,-4,3,0,-1,-6,-3,0,3,2,1)
$\sum_{\sigma=0}^{7}\mathcal{A}(a_{\sigma})(\tau)$ | (192,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

TABLE II: ACCF of the sequences of set $A$ in Example 1 for $\delta=3$.

$\mathcal{C}(a_{0},a_{1})(\tau)$ | (0,1,0,-7,0,5,0,5,0,1,-2,-1,0,-1,0,-1,2,5,0,1,0,1,0,1,0,1)
---|---
$\mathcal{C}(a_{1},a_{2})(\tau)$ | (0,-1,4,3,0,-9,-4,3,0,1,-2,3,0,3,2,-3,0,-1,0,-1,0,-1,0,-1)
$\mathcal{C}(a_{2},a_{3})(\tau)$ | (0,5,-4,1,0,13,4,1,0,-3,2,-1,0,-9,-2,1,0,1,0,1,0,1,0,1)
$\mathcal{C}(a_{3},a_{4})(\tau)$ | (0,1,-4,-3,0,9,4,-3,0,1,2,-1,0,7,-2,-3,0,-1,0,-1,0,-1,0,-1)
$\mathcal{C}(a_{4},a_{5})(\tau)$ | (0,3,0,-5,0,15,0,-1,0,1,-2,-1,0,7,2,-3,0,-1,0,-1,0,-1,0,-1)
$\mathcal{C}(a_{5},a_{6})(\tau)$ | (0,1,4,5,0,1,-4,-3,0,-3,-2,-1,0,-9,2,1,0,1,0,1,0,1,0,1)
$\mathcal{C}(a_{6},a_{7})(\tau)$ | (0,3,-4,-1,0,3,4,7,0,1,2,3,0,3,-2,-3,0,-1,0,-1,0,-1,0,-1)
$\mathcal{C}(a_{7},a_{0})(\tau)$ | (0,-1,-4,-5,0,-1,4,3,0,1,2,-1,0,-1,-2,5,0,1,0,1,0,1,0,1)
$\sum_{\sigma=0}^{7}\mathcal{C}(a_{\sigma},a_{\sigma+1})(\tau)$ | (0,12,-8,-12,0,36,8,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

The proposed construction of CZCS is compared with existing work and given in
Table III. The best case scenarios have been considered in the comparison.

###### Remark 2

For $\delta=0$, the proposed construction can generate odd length CZCS.

###### Remark 3

The set $A=\\{\mathbf{a}_{0},\mathbf{a}_{1},\ldots,\mathbf{a}_{2^{k+1}-1}\\}$
of Theorem 1 is a CS of length $2^{m-1}+2^{\delta}$, so the CSs of [21, 22]
are the special cases of the proposed construction.

###### Remark 4

By taking $\delta=1$, we can get $2^{m-1}+2$ length CZCSs, which is also
obtained in [20]. So CZCSs of [20] are the special case of the proposed
construction.

TABLE III: A comparison between the proposed and existing construction.

[t]  Ref. Method Flock size Length Constraint ZCZ Ratio 1 [14] Indirect $M$
$L$* $M\geq 2$ ${{1}/{2}}$ [19] Indirect $4$ $2^{\alpha}10^{\beta}26^{\gamma}$
$\alpha,\beta,\gamma\in\mathbb{N}$ $1$ [20] Direct $2^{n+1}$ $2^{m-1}+2$
$m\geq 4$,$n>0$ $>{1}/{4}$ Theorem 1 Direct $2^{k+1}$ $2^{m-1}+2^{\delta}$
$m\geq 4,0\leq\delta\leq m-1$ ${2}/{3}$

  * 1

maximum ZCZ ratio is considered.

  * *

$L$ is the length of a GCP.

##  IV Conclusion

We proposed a direct construction of CZCS with variable lengths and large ZCZ
width. The proposed GBF based construction generates CZCSs with a ZCZ ratio
upto $2/3$. The proposed CZCSs contain a large number of constituent sequences
and generate both even and odd length sequences. Compared to the existing
construction, the proposed construction is direct, generates CZCSs with larger
ZCZ width and flexible lengths, and generalizes many existing works.

## Appendix
Proof of Theorem 1

Before we begin the proof of Theorem 1, we introduce a few lemmas for proving
our main theorem. We utilize the notations stated in Theorem 1 and define some
basic notations as follows. Let the binary representations of two non-negative
integers $r,s<2^{m}$ be $(r_{1},r_{2},\ldots,r_{m})$ and
$(s_{1},s_{2},\ldots,s_{m})$, respectively.

###### Lemma 1

[21] If $r_{i}=s_{i}$ for $i=1,2,\ldots,\delta$, where $\delta$ is a positive
integer, then $s\geq r+2^{\delta}$ for integers $r$ and $s$ with $0\leq
r<s<2^{m}$ and $m\geq 2$.

###### Lemma 2

[21] For an integer $r$ with $2^{m-1}\leq r<2^{m-1}+2^{\delta}$ where
$1\leq\delta\leq m-1$ and $m\geq 2$. If the integer $r^{\prime}$ has a binary
vector representation that varies from $r$ solely at the $\alpha$-th place for
$\alpha\leq\delta$, then $2^{m-1}\leq r^{\prime}<2^{m-1}+2^{\delta}.$

###### Lemma 3

[23] Using the same notations and definitions as in Theorem 1. If
$r_{\pi_{\beta}(1)}\neq$ $s_{\pi_{\beta}(1)}$ for some
$\beta\in\\{1,2,\ldots,k\\}$, then for any sequences
$\mathbf{a}_{\sigma}=\left(a_{\sigma,0},a_{\sigma,1},\ldots,a_{\sigma,N-1}\right)\in
A$, there exists $\mathbf{a}^{\prime}_{\sigma}=$
$\left(a_{\sigma,0}^{\prime},a_{\sigma,1}^{\prime},\ldots,a_{\sigma,N-1}^{\prime}\right)=\mathbf{a}_{\sigma}+(q/2)x_{\pi_{\beta(1)}}\in
A$ such that
$\omega^{a_{\sigma,r}-a_{\sigma,s}}+\omega^{a_{\sigma,r}^{\prime}+a_{\sigma,s}^{\prime}}=0$.
Similarly, if $r_{m}\neq s_{m}$, then for any sequence $\mathbf{a}_{\sigma}\in
A$, there exists
$\mathbf{a}^{\prime}_{\sigma}=\mathbf{a}_{\sigma}+(q/2){x}_{m}\in A$ such that
$\omega^{a_{\sigma,r}-a_{\sigma,s}}+\omega^{a_{\sigma,r}^{\prime}+a_{\sigma,s}^{\prime}}=0$.

###### Lemma 4

[23] Suppose $r_{\pi_{\beta}(1)}=s_{\pi_{\beta}(1)}$ for $\beta=1,2,\ldots,k$.
Let us consider three conditions as
C1: $\hat{\beta}$ is the largest integer satisfying
$r_{\pi_{\beta}(\gamma)}=s_{\pi_{\beta}(\gamma)}$ for
$\beta=1,2,\ldots,\hat{\beta}-1$ and $\gamma=1,2,\ldots,m_{\beta}$.
C2: $\hat{\gamma}$ is the smallest integer such that
$r_{\pi_{\hat{\beta}}(\hat{\gamma})}\neq s_{\pi_{\hat{\beta}}(\hat{\gamma})}$.
C3: Let $r^{\prime}$ and $s^{\prime}$ be integers that differ from $r$ and
$s$, respectively, at only one position $\pi_{\hat{\beta}}(\hat{\gamma}-1)$,
i.e.,
$r_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}^{\prime}=1-r_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}$
and
$s_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}^{\prime}=1-s_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}$.
If all the above conditions are satisfied, then we have

$h_{r}-h_{s}-h_{r^{\prime}}+h_{s^{\prime}}\equiv\frac{q}{2}\ (\mathrm{mod}\
q),$ (7)

where

$h=g+\sum_{l=1}^{m-1}\lambda_{l}x_{l}+\lambda_{0}.$ (8)

###### Proof:

In order to prove that the set $A$ constructed from Theorem 1 is a
$(2^{k+1},2^{m-1}+2^{\delta},2^{\pi_{k}(1)-1}+2^{\delta})$-CZCS, we need to
prove the two conditions of Definition 3, with
$M=2^{k+1},N=2^{m-1}+2^{\delta},$ and $Z=2^{\pi_{k}(1)-1}+2^{\delta}$.

First, we prove that the set $A$ is a CS, i.e., for
$0<\tau<2^{m-1}+2^{\delta}$, the AACS of the sequences in set $A$ is zero,
i.e.,

$\sum_{\sigma=0}^{2^{k+1}-1}\mathcal{A}(\mathbf{a}_{\sigma})(\tau)=\sum_{r=0}^{2^{m-1}-\tau-1}\sum_{\sigma=0}^{2^{k+1}-1}\omega^{\left(a_{\sigma,r}-a_{\sigma,r+\tau}\right)}=0.$
(9)

Let $s=r+\tau$ for every integer $r$. Then we prove that (9) holds true in
each of the three scenarios listed below.

Case I: $r_{\pi_{\beta}(1)}\neq s_{\pi_{\beta}(1)}$ for some $1\leq\beta\leq
k$, or $r_{m}\neq s_{m}$.

Then from Lemma 3, we get

$\sum_{\sigma=0}^{2^{k+1}-1}\omega^{\left(a_{\sigma,r}-a_{\sigma,s}\right)}=0.$
(10)

Case II: $r_{\pi_{\beta}(1)}=s_{\pi_{\beta}(1)}$ for all $1\leq\beta\leq k$
and $r_{m}=s_{m}=0$.

There exists $\hat{\beta}$ and $\hat{\gamma}$ that meet the requirements (C1)
and (C2) of Lemma 4. Let $r^{\prime}$ and $s^{\prime}$ be numbers that satisfy
condition (C3) of Lemma 4. According to Lemma 4, $s^{\prime}=r^{\prime}+\tau$
and $h_{r}-h_{s}-h_{r^{\prime}}+h_{s^{\prime}}\equiv q/2\ (\mathrm{mod}\ q)$.
In this case, as $r_{m}=s_{m}=0$, so $f_{r^{\prime}}-f_{r}\equiv
h_{r^{\prime}}-h_{r}$ and $f_{s^{\prime}}-f_{s}\equiv h_{s^{\prime}}-h_{s}$
and hence $f_{r}-f_{s}-f_{r^{\prime}}+f_{s^{\prime}}\equiv q/2\mod q$.

$a_{\sigma,r}-a_{\sigma,s}-a_{\sigma,r^{\prime}}-a_{\sigma,s^{\prime}}=f_{r}-f_{s}-f_{r^{\prime}}+f_{s^{\prime}}\equiv
q/2\ (\mathrm{mod}\ q).$ (11)

So, from the above equation, we get
$\omega^{\left(a_{\sigma,r}-a_{\sigma,s}\right)}+\omega^{\left(a_{\sigma,r^{\prime}}+a_{\sigma,s^{\prime}}\right)}=0$.

Case III: $r_{\pi_{\beta}(1)}=s_{\pi_{\beta}(1)}$ for all $1\leq\beta\leq k$
and $r_{m}=s_{m}=1$.

In this case, $2^{m-1}\leq r,s<2^{m-1}+2^{\delta}$. We claim the existence of
an integer $v\leq\delta$ such that $r_{\pi_{1}(v)}\neq s_{\pi_{1}(v)}$. On the
contrary, let us assume that $v>\delta$. Since
$\left\\{\pi_{1}(1),\pi_{1}(2),\ldots,\pi_{1}(\delta)\right\\}=\\{1,2,\ldots,\delta\\}$
we can write $r_{i}=s_{i}$ for $i=1,2,\ldots,\delta$, thus we can have $s\geq
r+2^{\delta}\geq 2^{m-1}+$ $2^{\delta}$ according to Lemma 1. This contradicts
the condition that $s<2^{m-1}+2^{\delta}$. Now, we have $v\leq\delta$, and we
let $v$ be the smallest integer such that $r_{\pi_{1}(v)}\neq s_{\pi_{1}(v)}$.
Let $r^{\prime}$ and $s^{\prime}$ be integers that are different from $r$ and
$s$, respectively only at one position $\pi_{1}(v-1)$, i.e.,
$r_{\pi_{1}(v-1)}^{\prime}=1-$ $r_{\pi_{1}(v-1)}$ and
$s_{\pi_{1}(v-1)}^{\prime}=1-s_{\pi_{1}(v-1)}$; so
$s^{\prime}=r^{\prime}+\tau$. Since
$\left\\{\pi_{1}(1),\pi_{1}(2),\ldots,\pi_{1}(\delta)\right\\}=\\{1,2,\ldots,\delta\\}$
and $v-1<\delta$, we have $\pi_{1}(v-1)\leq\delta$. Therefore, according to
Lemma 2, $2^{m-1}\leq r^{\prime},s^{\prime}<$ $2^{m-1}+2^{\delta}$. Following
a similar approach as in Case III, it is possible to conclude that there
exists $r^{\prime}$ and $s^{\prime}$ such that
$\omega^{\left(a_{\sigma,r}-a_{\sigma,s}\right)}+\omega^{\left(a_{\sigma,r^{\prime}}+a_{\sigma,s^{\prime}}\right)}=0$.

Next, we prove that for $|\tau|\geq 2^{m-1}-2^{\pi_{k}(1)-1}$, the ACCS of the
sequences in set $A$ is zero, i.e.,

$\sum_{\sigma=0}^{2^{k+1}-1}\mathcal{C}({\mathbf{a}_{\sigma},\mathbf{a}_{\sigma+1}})(\tau)\\!=\\!\sum_{r=0}^{2^{m-1}-\tau-1}\sum_{\sigma=0}^{2^{k+1}-1}\omega^{\left(a_{\sigma,r}-a_{\sigma+1,r+\tau}\right)}\\!=\\!0,$
(12)

where $\mathbf{a}_{2^{k+1}-1}=\mathbf{a}_{0}$. The proof of the same is split
into three cases.

Case-I: $r_{\pi_{k}(1)}\neq s_{\pi_{k}(1)}$.

In this case, there exists
$\mathbf{a}^{\prime}_{\sigma}=\mathbf{a}_{\sigma}+(q/2)x_{\pi_{k}(1)}\in A$,
i.e.,

$a^{\prime}_{\sigma,r}=a_{\sigma,r}+\frac{q}{2}r_{\pi_{k}(1)},~{}a^{\prime}_{\sigma+1,s}=a_{\sigma+1,s}+\frac{q}{2}s_{\pi_{k}(1)}.$

So, $a^{\prime}_{\sigma,r}-a^{\prime}_{\sigma+1,s}\equiv
a_{\sigma,r}-a_{\sigma+1,s}+\frac{q}{2}\ (\mathrm{mod}\ q)$ as
$r_{\pi_{k}(1)}\neq s_{\pi_{k}(1)}$. Therefore,
$\omega^{\left(a_{\sigma,r}-a_{\sigma+1,s}\right)}+\omega^{\left(a^{\prime}_{\sigma,r}+a^{\prime}_{\sigma+1,s}\right)}=0$.

Case-II: $r_{m}\neq s_{m}$.

Similar to Case I, there exist
$\mathbf{a}^{\prime}_{\sigma}=\mathbf{a}_{\sigma}+(q/2)x_{m}\in A$, such that
$\omega^{\left(a_{\sigma,r}-a_{\sigma+1,s}\right)}+\omega^{\left(a^{\prime}_{\sigma,r}+a^{\prime}_{\sigma+1,s}\right)}=0$.

Case III: $r_{\pi_{k}(1)}=s_{\pi_{k}(1)}$ and $r_{m}=s_{m}$.

In this case, the value of $\tau$ is

$\displaystyle\tau=s-r=\sum_{i=1}^{m}(s_{i}-r_{i})2^{i-1}\leq
2^{m-1}-2^{\pi_{k}(1)-1}.$

So, $\tau=2^{m-1}-2^{\pi_{k}(1)-1}$, and so the binary vector representation
of $r=(0,0,\ldots,r_{\pi_{k}(1)},0,\ldots,r_{m})$ and
$s=(1,1,\ldots,r_{\pi_{k}(1)},1,\ldots,r_{m})$, i.e., all the components of
$r$ are zero except possibly at $r_{\pi_{k}(1)},r_{m}$ and all the components
of $s$ are $1$, except possibly at two positions where
$r_{\pi_{k}(1)}=s_{\pi_{k}(1)}$ and $r_{m}=s_{m}$. Let
$(\sigma_{1},\sigma_{2},\ldots,\sigma_{k+1})$ and
$(\sigma_{1}^{\prime},\sigma_{2}^{\prime},\ldots,\sigma_{k+1}^{\prime})$ be
the binary vector representation of integers $\sigma$ and $\sigma+1$,
respectively. Now, depending upon the values of $r_{\pi_{k}(1)}$ and $r_{m}$,
we have the following four sub-cases.

Sub-case I: $r_{\pi_{k}(1)}=0$ and $r_{m}=0$.

In this case, $r=0$, which has binary representation as $(0,0,\ldots,0)$ and
$s=r+\tau=2^{m-1}-2^{\pi_{k}(1)-1}$ whose binary vector representation is
$(1,1,\ldots,0,1,\ldots,0)$. The difference between $r$-th and $s$-th term of
the sequences $\mathbf{a}_{\sigma}$ and $\mathbf{a}_{\sigma+1}$, respectively
is calculated as

$\displaystyle a_{\sigma,r}-a_{\sigma+1,s}$
$\displaystyle=\frac{q}{2}(m-k-1)+\lambda_{0}$ (13)
$\displaystyle-\left(\sum_{l=1}^{m-1}\lambda_{l}+\lambda_{0}+\frac{q}{2}\left(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k-1}^{\prime}\right)\right)$
$\displaystyle=\lambda-\frac{q}{2}\left(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k-1}^{\prime}\right),$

where
$\frac{q}{2}(m-k-1)-\sum_{l=1}^{m-1}\lambda_{l}=\lambda\in\mathbb{Z}_{q}$. The
ACCS between $\mathbf{a}_{\sigma}$ and $\mathbf{a}_{\sigma+1}$ can be written
as

$\displaystyle\sum_{\sigma=0}^{2^{k+1}-1}\mathcal{C}({\mathbf{a}_{\sigma},\mathbf{a}_{\sigma+1}})(\tau)$
$\displaystyle=\sum_{\sigma=0}^{2^{k+1}-1}\omega^{a_{\sigma,r}-a_{\sigma+1,s}}$
(14)
$\displaystyle=\omega^{\lambda}\sum_{\sigma=0}^{2^{k+1}-1}(-1)^{(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k-1}^{\prime})}.$

The above equation is zero as the sum
$(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k-1}^{\prime})$ is
$0$ as many times as it is $1$.

Sub-case II: $r_{\pi_{k}(1)}=1$ and $r_{m}=0$.

In this case, the difference between $r$-th and $s$-th term of the sequences
$\mathbf{a}_{\sigma}$ and $\mathbf{a}_{\sigma+1}$, respectively is given by

$\displaystyle a_{\sigma,r}-a_{\sigma+1,s}$
$\displaystyle=\frac{q}{2}(m-k-3)+\lambda_{\pi_{k}(1)}+\lambda_{0}+\frac{q}{2}\sigma_{k}$
(15)
$\displaystyle-\left(\sum_{l=1}^{m-1}\lambda_{l}+\lambda_{0}+\frac{q}{2}\left(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k}^{\prime}\right)\right)$
$\displaystyle=\lambda^{\prime}-\frac{q}{2}\left(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\cdots+\sigma_{k}^{\prime}-\sigma_{k}\right),$

where
$\frac{q}{2}(m-k-3)+\lambda_{\pi_{k}(1)}-\sum_{l=1}^{m-1}\lambda_{l}=\lambda^{\prime}\in\mathbb{Z}_{q}$.
So, similar to Sub-case I, the ACCS

$\sum_{\sigma=0}^{2^{k+1}-1}C({\mathbf{a}_{\sigma},\mathbf{a}_{\sigma+1}})(\tau)=0.$

Sub-case III: $r_{\pi_{k}(1)}=0$ and $r_{m}=1$.

In this case, the difference is given by

$\displaystyle a_{\sigma,r}-a_{\sigma+1,s}$
$\displaystyle=\lambda_{0}+\frac{q}{2}\sigma_{k+1}-\lambda_{0}$ (16)
$\displaystyle-\left(\frac{q}{2}(m-k-3)+\sum_{\begin{subarray}{c}l=1\\\
l\neq\pi_{k}(1)\end{subarray}}^{m-1}\lambda_{l}+\frac{q}{2}\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\pi_{k}(1)\end{subarray}}^{k+1}\sigma_{\beta}^{\prime}\right)$
$\displaystyle=\lambda^{\prime\prime}\\!-\\!\frac{q}{2}\left(\sigma_{1}^{\prime}\\!+\sigma_{2}^{\prime}\\!+\cdots+\\!\sigma_{k+1}^{\prime}\\!-\\!\sigma_{\pi_{k}(1)}^{\prime}\\!-\\!\sigma_{k+1}\right),$

where $-\frac{q}{2}(m-k-3)-\sum_{\begin{subarray}{c}l=1\\\
l\neq\pi_{k}(1)\end{subarray}}^{m-1}\lambda_{l}=\lambda^{\prime\prime}\in\mathbb{Z}_{q}$.
So, in a similar manner to Sub-case I the ACCS of the sequences is zero, i.e.,

$\sum_{\sigma=0}^{2^{k+1}-1}\mathcal{C}({\mathbf{a}_{\sigma},\mathbf{a}_{\sigma+1}})(\tau)=0.$

Sub-case IV: $r_{\pi_{k}(1)}=1$ and $r_{m}=1$.

In this case $r=2^{\pi_{k}(1)-1}+2^{m-1}$, and
$s=r+\tau=2^{\pi_{k}(1)-1}+2^{m-1}+2^{m-1}-2^{\pi_{k}(1)-1}=2^{m}$, which is
not possible as the sequence length $N=2^{m-1}+2^{\delta}\leq
2^{m-1}+2^{m-2}<2^{m}$. So, this case is not feasible. So, the result follows.
∎

## References

  * [1] M. Golay, “Complementary series,” _IRE Trans. Inf. Theory_ , vol. 7, no. 2, pp. 82–87, Apr. 1961.
  * [2] J. Jedwab and M. G. Parker, “Golay complementary array pairs,” _Des. Codes Cryptogr._ , vol. 44, pp. 209–216, Sep. 2007.
  * [3] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” _IEEE Trans. Inf. Theory_ , vol. 45, no. 7, pp. 2397–2417, Nov. 1999.
  * [4] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” _IEEE Trans. Inf. Theory_ , vol. 46, no. 1, pp. 104–120, Jan. 2000.
  * [5] P. Kumar, S. Majhi, and S. Paul, “A direct construction of GCP and binary CCC of length non power of two,” 2021. [Online]. Available: https://arxiv.org/abs/2109.08567
  * [6] P. Sarkar, S. Majhi, and Z. Liu, “A direct and generalized construction of polyphase complementary sets with low PMEPR and high code-rate for OFDM system,” _IEEE Trans. Commun._ , vol. 68, no. 10, pp. 6245–6262, 2020.
  * [7] P. Fan, W. Yuan, and Y. Tu, “Z-complementary binary sequences,” _IEEE Signal Process. Lett._ , vol. 14, no. 8, pp. 509–512, Aug. 2007.
  * [8] C.-Y. Pai, S.-W. Wu, and C.-Y. Chen, “Z-complementary pairs with flexible lengths from generalized Boolean functions,” _IEEE Commun. Lett._ , vol. 24, no. 6, pp. 1183–1187, 2020.
  * [9] C.-Y. Chen, “A novel construction of Z-complementary pairs based on generalized Boolean functions,” _IEEE Signal Process. Lett._ , vol. 24, no. 7, pp. 987–990, 2017.
  * [10] A. R. Adhikary, P. Sarkar, and S. Majhi, “A direct construction of $q$-ary even length Z-complementary pairs using generalized Boolean functions,” _IEEE Signal Process. Lett._ , vol. 27, pp. 146–150, 2020.
  * [11] A. R. Adhikary, S. Majhi, Z. Liu, and Y. L. Guan, “New sets of optimal odd-length binary Z-complementary pairs,” _IEEE Trans. Inf. Theory_ , vol. 66, no. 1, pp. 669–678, 2020.
  * [12] ——, “New sets of even-length binary Z-complementary pairs with asymptotic ZCZ ratio of $3/4$,” _IEEE Signal Process. Lett._ , vol. 25, no. 7, pp. 970–973, 2018.
  * [13] P. Kumar, P. Sarkar, S. Majhi, and S. Paul, “A direct construction of even length ZCPs with large ZCZ ratio,” _Cryptogr. Commun._ , 2022.
  * [14] Z. Liu, P. Yang, Y. L. Guan, and P. Xiao, “Cross Z-complementary pairs for optimal training in spatial modulation over frequency selective channels,” _IEEE Trans. Signal Process._ , vol. 68, pp. 1529–1543, 2020.
  * [15] A. R. Adhikary, Z. Zhou, Y. Yang, and P. Fan, “Constructions of cross Z-complementary pairs with new lengths,” _IEEE Trans. Signal Process._ , vol. 68, pp. 4700–4712, 2020.
  * [16] C. Fan, D. Zhang, and A. R. Adhikary, “New sets of binary cross Z-complementary sequence pairs,” _IEEE Commun. Lett._ , vol. 24, no. 8, pp. 1616–1620, 2020.
  * [17] M. Yang, S. Tian, N. Li, and A. R. Adhikary, “New sets of quadriphase cross Z-complementary pairs for preamble design in spatial modulation,” _IEEE Signal Process. Lett._ , vol. 28, pp. 1240–1244, 2021.
  * [18] Z.-M. Huang, C.-Y. Pai, and C.-Y. Chen, “Binary cross Z-complementary pairs with flexible lengths from Boolean functions,” _IEEE Commun. Lett._ , vol. 25, no. 4, pp. 1057–1061, 2021.
  * [19] ——, “Cross Z-complementary sets for training design in spatial modulation,” _IEEE Trans. Commun._ , pp. 1–1, 2022.
  * [20] P. Kumar, S. Majhi, and S. Paul, “A direct construction of cross Z-complementary sequence sets with large set size,” 2022. [Online]. Available: https://arxiv.org/abs/2207.02551
  * [21] C.-Y. Chen, “Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM,” _IEEE Trans. Inf. Theory_ , vol. 62, no. 12, pp. 7538–7545, 2016.
  * [22] ——, “A new construction of Golay complementary sets of non-power-of-two length based on Boolean functions,” in _IEEE Wirel. Commun. Netw. Conf. (WCNC)_ , 2017, pp. 1–6.
  * [23] S.-W. Wu, C.-Y. Chen, and Z. Liu, “How to construct mutually orthogonal complementary sets with non-power-of-two lengths?” _IEEE Trans. Inf. Theory_ , vol. 67, no. 6, pp. 3464–3472, 2021.