††institutetext: University of Texas, Austin, Physics Department, Austin TX
78712, USA

# Spectral Form Factor in Sparse SYK Models

Elena Cáceres  Anderson Misobuchi  Amir Raz<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the spectral form factor of the sparse Sachdev-Ye-Kitaev model.
We use numerical methods to establish that at intermediate times the connected
part of the spectral form factor is the dominant one. These connected
contributions arise from fluctuations around the disconnected geometry, not
from a new saddle point. A similar effect was previously conjectured in SYK
but required a value of $N$ out of reach of current numerical simulations.

††preprint: UTTG-06-2022

##  1 Introduction

One of the hurdles in understanding foundational aspects of holography is the
scarcity of solvable strongly coupled quantum models that exhibit black hole
physics. The Sachdev-Ye-Kitaev model (SYK) Kitaev_talk ; MaldecenaStanford  is
one such model. The SYK model is a quantum mechanical system of $N$ Majorana
fermions coupled by a random all-to-all $q$-body interaction. In the large $N$
and low-energy limit, an approximate conformal symmetry emerges and the system
saturates the chaos bound Chaos . This unique combination of properties make
it a remarkable toy model for a holographic dual of a two-dimensional black
hole. The SYK model has spurred many developments in strongly correlated
systems and in holography.

One interesting feature of SYK is that it exhibits chaos. The Hamiltonian of a
chaotic system, when considering small energy windows where the density of
states is constant, is believed to resemble a random matrix. This insight due
to Wigner Wigner , implies that by studying statistical properties of random
matrices subject to the symmetries of the Hamiltonian, we can understand the
statistical properties of energy levels and eigenstates of the system. One
observable that captures the statistics of energy levels is the spectral form
factor (SFF).

In Cotler:2016fpe  the authors showed that the late time behavior of the
spectral form factor in the all-to-all SYK is governed by random matrix theory
(RMT) just as expected for a chaotic system. At early times, the SFF
decreases; this behavior is referred to as “the slope”. After certain time
$T_{d}$ it reaches a minimum or dip. After $T_{d}$ the spectral form factor
grows linearly along a “ramp”, until it saturates and reaches a plateau at
time $T_{p}.$ Beyond $T_{p}$ the value of the SFF is constant. Unlike the
early time behavior, the behavior after $T_{d}$ is believed to be universal,
valid for a large class of quantum mechanical interacting systems and has
received much attention in the literature. In Cotler:2016fpe  the authors
argued that before the dip the disconnected part of the spectral form factor
dominates. This matches the holographic intuition where at early times we
expect the gravitational path integral to be dominated by two disconnected
geometries. At late times, after the dip, the connected contributions are the
dominant ones.

Recently, in Berkooz2021  the authors showed that at intermediate time scales
there are modifications to the connected components that come from global
fluctuations of the spectrum. These fluctuations were conjectured to be
dominant at intermediate times for sufficiently large $N$. However, to
numerically observe this behavior in the all-to-all would require very large
values of $N$ out reach of current numerical simulations.

In this work we focus on the intermediate time behavior of the spectral form
factor in the _sparse SYK_ which allows for efficient numerical simulations.
The sparse SYK model was proposed by Xu, Susskind, Su and Swingle (XSSS)
xu2020 , see also Garcia2021 . In the sparse SYK is the interactions grow
linearly with $N$, as opposed to $N^{q}$, this leads to a significant
reduction in computational complexity. It has been shown that, for appropriate
sparsity, the sparse SYK presents the same features of the original all-to-all
SYK model such as fast scrambling and maximal chaos at low temperatures.

We investigate numerically the SFF of the sparse SYK model. We establish the
existence of an intermediate time regime, before the dip, where the connected
part dominates the SFF. The nature of these connected contributions is
different form the ones at late times.

At late times, the contribution of other saddle points that connect the two
replicas dominate. This can be interpreted, holographically, as wormholes
joining the geometries. The intermediate time contributions we find in this
work, do not arise from a new saddle point, they are due to fluctuations
around the naive, disconnected, saddle geometry. Thus, it’s interpretation
from the holographic point of view is not clear but it indicates that in the
large, but finite, $N$ limit, even two disconnected geometries do not fully
factorize.

##  2 The sparse SYK model

We recall that the standard Sachdev-Ye-Kitaev (SYK) model consists of a system
of $N$ Majorana fermions with $q$-body all-to-all random interactions, whose
Hamiltonian is given by

$H_{\text{all-to-all}}=\sum_{1\leq j_{1}<\ldots<j_{q}\leq N}J_{j_{1}\ldots
j_{q}}\chi^{j_{1}}\ldots\chi^{j_{q}},$ (1)

where $\chi^{j}$, $j=1,\ldots,N$ denote the Majorana fermions, and the
couplings $J_{j_{1}\ldots j_{q}}$ are sampled from a normal distribution with
zero mean and variance

$\langle\left(J_{j_{1}\ldots j_{q}}\right)^{2}\rangle_{\text{all-to-
all}}=\frac{(q-1)!J^{2}}{N^{q-1}}.$ (2)

The parameter $J$ above sets the typical energy scale. The advantage of this
model is that it can be solved analytically in the large $N$ limit. On the
other hand, the study of this model at finite $N$ relies on numerical
simulations that quickly become intractable as we increase the number of
fermions due to the increasing dimension of the Hilbert space and also due to
the all-to-all nature of this model, in which the total number of terms grows
as $\binom{N}{q}\sim N^{q}$. A variation of the SYK model, dubbed sparse SYK
xu2020 , has been proposed as a more computationally tractable system that
preserves the same features of the all-to-all SYK model, such as the quantum
chaotic behavior xu2020 ; Garcia2021  and its connection to traversable
wormholes caceres2021 . The sparse SYK model displays these properties with a
Hamiltonian that contains only $\mathcal{O}(N)$ interaction terms.

The Hamiltonian of the sparse SYK model is

$\displaystyle H$
$\displaystyle=i^{q/2}\sum_{j_{1}<\ldots<j_{q}}J_{j_{1}\ldots
j_{q}}x_{j_{1}\ldots j_{q}}\chi^{j_{1}}\ldots\chi^{j_{q}}.$ (3)

where we introduce the parameter $x_{j_{1}\ldots j_{q}}$ that can be either 0
or 1, such that different choices of $x_{j_{1}\ldots j_{q}}$ lead to different
sparse models. In the literature, two implementations of sparsity have been
considered. In the random pruning approach, one chooses $x_{j_{1}\ldots
j_{q}}=1$ with probability $p$ and $x_{j_{1}\ldots j_{q}}=0$ with probability
$1-p$. Another approach is to consider regular hypergraphs, in which the
Majorana fermions are identified as vertices of a hypergraph whose hyperedges
contain $q$ vertices. The regularity condition imposes that each fermion
appears exactly the same number of times in the list of hyperedges. The
hypergraph is also uniform, meaning that all hyperedges contain the same
number, $q$, of vertices.

The random couplings associated to the sparse SYK model are still sampled from
a Gaussian distribution of zero mean, but its variance is modified to be

$\langle\left(J_{j_{1}\ldots
j_{q}}\right)^{2}\rangle=\frac{(q-1)!J^{2}}{pN^{q-1}}=\frac{2^{q-1}\mathcal{J}^{2}(q-1)!}{pqN^{q-1}}$
(4)

In the regular hypergraph construction, the amount of sparseness is controlled
by the parameter

$k\equiv\frac{p}{N}\binom{N}{q},$ (5)

which makes the Hamiltonian to be a sum of exactly $k\,N$ interaction terms.

###  2.1 Analytical spectral density

The spectrum of the all-to-all SYK model at finite $N$ and $q$ is very well
approximated by a $Q$–Gaussian distribution Garc_a_Garc_a_2017 ;
Garc_a_Garc_a_2018_2 , and the spectrum converges to this distribution in the
double scaled limit of taking $N,q\rightarrow\infty$ but keeping $N/q^{2}$
fixed Erdos14 ; feng2018spectrum ; Micha2018 ; Berkooz_2019 . This
$Q$–Gaussian distribution is

$\rho_{\eta}(\theta)=\frac{1}{2\pi}\left(\eta,e^{\pm
2i\theta};\eta\right)_{\infty},$ (6)

where $(a;q)_{n}$ is the $Q$-Pochammer symbol

$(a;q)_{n}=\prod_{j=0}^{n-1}\left(1-aq^{j}\right),$ (7)

with the shorthand notation $(a,b;q)_{n}\equiv(a;q)_{n}(b;q)_{n}$.

In (6) the angle $\theta\in[0,\pi]$ is related to the energies by

$E(\theta)=\frac{2{\cal{J}}\cos(\theta)}{\sqrt{1-\eta}},$ (8)

and the parameter $\eta$ depends on $q$ and $N$, and is given by
Garc_a_Garc_a_2017

$\eta(q,N)={\binom{N}{q}}^{-1}\sum_{j=0}^{q}(-1)^{j}\binom{q}{j}\binom{N-q}{q-j}.$
(9)

Figure 1: Histograms of the spectral density of the sparse SYK Hamiltonian for
$N=30$ and various values of $k$. The spectral density is averaged over 100
disorder realizations. For comparison the solid red line is the $Q$–Gaussian
approximation to the spectral density (6).

In the double scaled limit $\eta\rightarrow e^{-2q^{2}/N}$. Furthermore, the
convergence of the analytical spectral density in this limit is independent of
the precise distribution of the random couplings Erdos14 ; feng2018spectrum ,
so the same exact result also holds for the sparse SYK model. Perhaps more
surprisingly is that even at finite $N$ and $q$ the spectral density of the
sparse SYK model is also well approximated by this same $Q$-Gaussian
distribution, at least for large enough $N$ and $k$ Garcia2021 . One can add
an additional correction to $\eta$, what Garcia2021  dubbed the “renormalized”
approximation, to make the agreement with spectral density even closer

$\eta(q,N;k)=\eta(q,N)+\frac{3}{kN}.$ (10)

Using this analytical spectral density the partition function can be
approximated as

$\mathcal{Z}(\beta)=\text{tr}\left(e^{-\beta H}\right)\approx\int
d\theta~{}\rho_{\eta}(\theta)e^{-\beta E(\theta)}.$ (11)

We will use this analytical approximation to give a general idea of the
expected behavior of the spectral form factor when comparing to the numerical
results.

To give a qualitative assessment of the validity of the $Q$–Gaussian
approximation, we compare it to the average spectral density of the sparse SYK
Hamiltonian computed using exact diagonalization in Fig.1. We see that for
$q=4$ the spectrum has a $Q$-Gaussian shape with some small deviations from
the analytical approximation, which is true even for $k=2$. On the other hand,
the spectral density for $q=8$ deviates from the $Q$–Gaussian approximation at
the center of the spectrum, where for small values $k\lesssim 4$ there is a
noticeable dip. Similar observations on the validity of the $Q$–Gaussian
approximation for $q=4$ were discussed in section III. B of Garcia2021 .

To understand how the spectral density deviates from the average in each
realization of the couplings, we plotted the individual spectral densities for
each of the 100 different realizations of couplings in Fig.2 for $N=30$,
$q=4$, and $k=4$. We see that the shape of the spectral density is similar for
all the different disorder realizations, with the main deviation being a
stretching or compressing of the spectrum; this corresponds to the dominant
scaling mode described in Berkooz2021 . Additional plots of the spectral
densities for other values of $k$ and $q$ and presented in the appendix.

Figure 2: The individual spectral densities of the sparse SYK Hamiltonian for
100 different realization of the couplings.

##  3 The spectral form factor

The main focus of this paper is the spectral form factor (SFF) of the sparse
SYK model. The SFF is defined by the two replica average

$g(T;\beta)=\frac{\mathcal{Z}(\beta+iT,\beta-iT)}{[\mathcal{Z}(\beta)]^{2}}.$
(12)

The SFF contains the important information about the two replica energy
correlations in disordered systems Br_zin_1997 . As such it has played a
central role in the study of holography and quantum chaos Cotler:2016fpe ;
saad2018 ; Liu_2018 ; del_Campo_2017 ; winer2021hydrodynamic . For chaotic
systems the late time SFF has a universal ramp and plateau behavior whose
features are solely determined by the discrete symmetries (time reversal,
charge, and parity) of the Hamiltonian Cotler:2016fpe ; Liu_2018 . This
universal behavior is described by a random matrix universality class, which
typically fits into one of three RMT $\beta$-ensembles Liu_2018 .

From a holographic prospective, the spectral form factor describes the
analytical continuation of the gravitational path integral over a space with
two asymptotically AdS boundaries Cotler:2016fpe ; saad2018 ; Saad:2019lba .
At early times we expect this gravitational path integral to be dominated by a
geometry consisting of two disconnected disks, while at late times the
connected geometries become the dominant configuration. These connected
geometries are exponentially suppressed by factors of $e^{-S}$, where $S$ is
the entropy.

These same dynamics happen in the SYK model. Cotler:2016fpe  showed that the
SFF of the all-to-all SYK model behaves as the product of disconnected
partition functions at early times, and follows the RMT prediction at
exponentially late times (in $N$.)111We note that the late time RMT behavior
was first observed by Garc_a_Garc_a_2016 , where they computed the Fourier
transform of the SFF (i.e. the number variance.) Later saad2018  showed that
this late time behavior is due to the need to include new connected saddle
points to the two replica path integral. These saddle points are indeed
exponentially suppressed in $N$ (recall that the entropy of the SYK model is
proportional to $N$.)

However the SFF of the SYK model also contains connected contributions that
are only polynomially suppressed in $N$ Cotler:2016fpe ; Berkooz2021 , the
leading term being of order $N^{-q}$. These connected contribution do not
arise from new saddle points, but rather from a perturbative expansion around
the disconnected saddle point Berkooz2021 .222Essentially these contributions
arise by computing the non-diagonal 1-loop determinant around the disconnected
saddle order by order in $1/N$. Though the existence of such contributions has
been known for some time Cotler:2016fpe ; feng2018spectrum2 ; Altland_2018 ;
verbaarschot2019 , recently Berkooz2021  argued that this connected
contribution should dominate the SFF of the all-to-all SYK model at
intermediate times. These connected contributions have not been seen in the
SFF of the all-to-all SYK model up to now due to finite $N$ constraints,
though Berkooz2021  conjectured that $N=60$ would be enough to this new
intermediate time behavior.

In Berkooz2021 , the authors calculated the connected contributions for a
general multi-trace thermal expectation value in the all-to-all SYK. In this
work we are interested in the spectral form factor (12) in the sparse SYK so
let us first review the results of Berkooz2021  for the double trace case that
is the relevant one for the SFF.

Consider the thermal expectation value

$\mathcal{Z}(\beta_{1},\beta_{2})=\left<\text{tr}(e^{-\beta_{1}H)})\,\text{tr}(e^{-\beta_{2}H})\right>_{J}\,,$
(13)

where the sub index $J$ denotes the ensemble average over the random
couplings. We can expand (13) as

$\displaystyle\mathcal{Z}(\beta_{1},\beta_{2})$
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}\left<\text{tr}(H^{k_{1}})\,\text{tr}(H^{k_{2}})\right>_{J}\frac{\beta_{1}^{k_{1}}}{{k_{1}}!}\frac{\beta_{2}^{k_{2}}}{{k_{2}}!}(-1)^{k_{1}+k_{2}}$
(14)
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}M(k_{1},k_{2})\frac{\beta_{1}^{k_{1}}}{{k_{1}}!}\frac{\beta_{2}^{k_{2}}}{{k_{2}}!}(-1)^{k_{1}+k_{2}},$
(15)

where $M(k_{1},k_{2})$ denotes the joint moments of the distribution. We are
interested in the leading order $1/N$ term in the connected part of these
moments. The connected part can be obtained by subtracting all lower
disconnected moments,

$M_{c}(k_{1},k_{2})=M(k_{1},k_{2})-M(k_{1})M(k_{2})\,.$ (16)

Therefore, the connected part of $\mathcal{Z}(\beta_{1},\beta_{2})$ is simply
given by the connected part of the moments,

$\displaystyle\mathcal{Z}_{c}(\beta_{1},\beta_{2})$
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}M_{c}(k_{1},k_{2})\frac{\beta_{1}^{k_{1}}}{{k_{1}}!}\frac{\beta_{2}^{k_{2}}}{{k_{2}}!}(-1)^{k_{1}+k_{2}},$
(17)

Furthermore, in the case of the all-to-all SYK, Berkooz2021  found that
$M_{c}(k_{1},k_{2})$ is simply given by,

$M_{c}(k_{1},k_{2})=\frac{k_{1}k_{2}}{2}\binom{N}{q}^{-1}M(k_{1})M(k_{2})+O(N^{-3q/2}).$
(18)

It is not clear how to interpret these connected contributions holographically
as they are not associated to wormhole configurations in the bulk.
Nevertheless, if these connected contributions dominate the SFF at some
intermediate times then they must be important to the gravitational path
integral.

Thus, we can approximate the known contributions to the spectral form factor
as

$\displaystyle\mathcal{Z}(\beta+iT,\beta-
iT)\approx|\mathcal{Z}(\beta+iT)|^{2}$
$\displaystyle+\frac{\epsilon}{2}\left(\beta^{2}+T^{2}\right)\left|\frac{\partial\mathcal{Z}(\beta^{\prime})}{\partial(\beta^{\prime})}\right|^{2}_{\beta^{\prime}=\beta+iT}+\dots$
(19) $\displaystyle+\int dE~{}e^{-2\beta
E}\min\left\\{4\frac{T}{2\pi},2\rho(E)\right\\}.$

The first term is the disconnected contribution, the second term is the
leading connected piece (coming from (17) and (18)) and the third term is the
RMT contribution given in Cotler:2016fpe .333Here we assume the RMT
contribution comes from a GUE ensemble, which is the case for $N\mod 8=2,6$
Other values of $N$ fall into the GSE or GOE universality classes, and the
ramp behavior is more complex, and can be found in Liu_2018 . The parameter
$\epsilon$ is a small parameter convenient to organize higher order
corrections (denoted by the dots in (19)), which for the all-to-all SYK is
$\epsilon={\binom{N}{q}}^{-1}$.

For the sparse SYK model, one can use the same analysis in Berkooz2021  to
compute the leading connected contribution to the SFF assuming the random
couplings are non-Gaussian. In this case the leading connected contribution
only depends on the normalized forth moment of the couplings,444See Theorem 1
of feng2018spectrum2  for a rigorous mathematical derivation. with

$\epsilon=\frac{1}{2}\binom{N}{q}^{-1}\left(\gamma-1\right),$ (20)

where $\gamma$ is the normalized fourth moment of the couplings

$\gamma=\frac{\left<\left(J_{j_{1}\ldots j_{q}}x_{j_{1}\ldots
j_{q}}\right)^{4}\right>}{\left<\left(J_{j_{1}\ldots j_{q}}x_{j_{1}\ldots
j_{q}}\right)^{2}\right>^{2}}.$ (21)

Thus even for the sparse SYK model the SFF still takes the form (19), only
with

$\epsilon=\frac{1}{2}\binom{N}{q}^{-1}\left(\frac{3}{p}-1\right)\approx\frac{3}{2kN}.$
(22)

Note that when $p\rightarrow 1$ we recover the all-to-all result, as expected.

Assuming $k=\mathcal{O}(1)$ we see that the suppression of the leading
connected term in the sparse SYK model is only of order $1/N$, and so we
expect this contribution to dominate at earlier times. Hence we may be able to
see the intermediate time behavior conjectured in Berkooz2021  at manageable
values of $N$’s for the sparse SYK model.

Finally, we note that many in the condensed matter (and high energy) community
are primarily interested in the SFF to understand the spectral correlation.
These correlations are easier to study if the energies are re-scaled so that
the local density of states is a constant, in a process called spectral
unfolding. The spectral unfolding is typically done by renormalizing the
spectrum of each disorder realization independently555Occasionally in the
literature the spectrum of the whole ensemble is unfolded together, rather
than independently in each disorder realization. In this case the global
spectral correlations will remain and our conclusions will still be valid.
(for example by using a smooth function that is fitted to the numerical data
of each realization.) This local unfolding will remove the global correlations
in the SFF and only leave the local correlations, i.e. the universal RMT part.
This was seen in verbaarschot2019  for the all-to-all SYK model, and in
Garcia2021  for the sparse SYK model. One can further focus only on the bulk
of the spectrum by adding a Gaussian filter, as was done in verbaarschot2019 ;
Garcia2021 ; untajs_2020 . Such a filter, in combination with local spectral
unfolding, removes any edge and global effects, and is particularly powerful
in understanding the local eigenvalue statistics.

In this work we focus on the full SFF at finite temperature rather than the
unfolded SFF as it has a particularly simple holographic interpretation, and
contains the global fluctuations which we are particularly interested in. One
can add a Gaussian filter to the SFF (or any other smooth filter,) and
calculate the leading contribution to the SFF (without unfolding) using the
same moment method described above, though it is not clear to us what the
holographic interpretation of these observables is.

##  4 Numerical results

We evaluated the spectral form factor using exact diagonalization for $N=30$
Majorana fermions. Our numerical results are evaluate considering the regular
hypergraph model. Although in principle the derivation of (22) assumes the
random pruning definition of the sparse SYK model since it treats
$x_{j_{1},\ldots,j_{q}}$ as a random variable, we expect corrections to this
result when one considers a regular hypergraph instead to the moments to be of
an order of $1/N^{q-1}$ Garcia2021 . We have confirmed this numerically so we
will only display the results for the regular hypergraph model.

Figure 3: Main: Exchange of dominance in the spectral form factor of sparse
SYK model, for $N=30$, $q=4$, $k=4$, $\beta=5$, averaged over 100 disorder
realizations. Inset: the critical time $T_{c}$ dependence on the sparsity $k$
and inverse temperature $\beta$.

The average over the random couplings is taken by averaging the numerator and
denominator separately, i.e.

$\displaystyle g(T,\beta)$
$\displaystyle=\frac{\langle\mathcal{Z}(\beta,T)\,Z^{*}(\beta,T)\rangle_{J}}{\langle
Z(\beta)\rangle_{J}^{2}}\,,$ (23) $\displaystyle g_{d}(T,\beta)$
$\displaystyle=\frac{\langle\mathcal{Z}(\beta,T)\rangle_{J}\,\langle
Z^{*}(\beta,T)\rangle_{J}}{\langle\mathcal{Z}(\beta)\rangle_{J}^{2}}\,,$ (24)
$\displaystyle g_{c}(T,\beta)$ $\displaystyle=g(T,\beta)-g_{d}(T,\beta)\,.$
(25)

We find the existence of a transition at some time $t_{c}$ (before the dip)
where the connected part of the spectral form factor begins to dominate over
the disconnected part – see Fig. 3. The transition time appears to increase
linearly with the sparsity parameter $k$, and seems to slightly decrease when
the inverse temperature $\beta$ is increased.

We also find that the computed SFF from exact diagonalization is in agreement
with the analytical formula (19) up to a timescale close to the point where
the transition takes place and the agreement becomes significantly better as
the sparsity parameter $k$ increases – see Fig. 4. This is true even for the
small value of $k=1.5$, which is somewhat surprising. After the exchange of
dominance the connected SFF does not agree with the first order analytical
approximation, at least for $k\gtrsim 2$. Somewhat surprisingly, the connected
contribution seems to follow the disconnected approximation for large enough
$k\gtrsim 4$. This behavior may be because the SFF is still dominated by the
same saddle point, even though this saddle no longer factorizes into the two
disconnected saddle points. A similar behavior of the SFF occurs when $q=8$
and for other values of $\beta$; these additional plots of the SFF are given
in the appendix.

Figure 4: Full spectral form factor $g$, connected part $g_{c}$, and
disconnected part $g_{d}$, obtained by exact diagonalization of the sparse SYK
Hamiltonian define over a regular hypergraph, and averaged over 100 disorder
realizations of the random couplings. These are compared to the analytical
$Q$–Gaussian approximation $\tilde{g}_{c}$, $\tilde{g}_{d}$, and
$\tilde{g}_{\text{RMT}}$.

##  5 Discussion and holographic interpretation

We have observed an intermediate time transition in the SFF of the sparse SYK
model, where the SFF is no longer dominated by the disconnected contribution.
To understand how to interpret this transition in the dual gravitational
description, it is illuminating to consider the path integral description of
the SYK model where these connected contributions arise from a perturbative
expansion around the disconnected saddle 666See section 8 of Berkooz2021  for
the full derivation.. Thus this intermediate time regime is still described by
a disconnected saddle geometry, but the connected fluctuations around the
saddle are in fact important777This is similar to a double trace deformation
of the boundary CFT Aharony:2001pa ; Witten:2001ua ; Berkooz:2002ug .. This
raises a factorization problem in the dual holographic description which is
stronger than the usual factorization problem in quantum gravity (à la
Maldacena-Maoz MaldacenaMaoz ), as even two disconnected geometries do not
factorize.

We may expect such connected contributions to be generic in models of
holography, as these models are typically only understood perturbatively in
some large parameter $N$, but have an exponential number states which goes
like $e^{\\#N}$. As a result the connected multiple boundary saddles are
exponentially suppressed in $N$, and so are relevant only at exponentially
large times. However the disconnected saddle is only understood
holographically to leading order in $N$, and so the disconnected picture is
only valid for times of order 1. At intermediate times of order $N$ (or more
generally $N$ to some positive power), $1/N$ corrections become relevant. The
$1/N$ corrections should generically contain any terms allowed by the
symmetry, and so should contain disconnected contributions; for example via
double trace deformations.

Our results suggest several directions to explore in the future:

  * •

Holographic interpretation As mentioned above, the intermediate time dominant
contributions presented here are connected fluctuations around the
disconnected saddle geometry; They do not correspond to a new, connected
saddle. Thus, it is natural to ask what is the bulk interpretation of this
regime. In Berkooz2021 , the authors suggested that these connected
contributions correspond to the existence of light fields in the bulk with
fluctuating boundary couplings. It would be interesting to study this issue
further.

  * •

Other observables The connected contributions that dominate the SFF at time
before the dip time, should also leave an imprint in other observables. We
could consider for example, the new measure of quantum state complexity
defined by minimizing the spread of the wavefunction over all choices of basis
proposed in Balasubramanian:2022tpr . In systems with chaotic behavior this
notion of complexity displays four dynamical regimes: a linear ramp, a peak
and a slope down to a plateau. This behavior can be connected to the ramp
-dip- plateau structure of the SFF. Furthermore, the complexity slope, arises
from spectral rigidity just like the SFF ramp. It would be interesting to
understand how the dominance of connected fluctuations studied here is
reflected in this new measure complexity.

  * •

Larger $N$ simulation Our numerical results relied on the exact
diagonalization of the Hamiltonian which significantly constrains the values
of $N$ that we can achieve. This was enough to demonstrate the exchange of
dominance at intermediate times in the sparse SYK model, but the same cannot
be said for the all-to-all SYK, where one expects at least $N=60$ to observe
the same effect. Recent techniques such as Krylov subspace methods for time
evolution have been employed to study quantum chaos in the SYK model for up to
$N=60$ kobrin2020manybody . However, we found that these techniques, when
applied to the SFF, are susceptible to large errors so that it fails to
correctly capture the correlation between eigenvalues. It would be desirable
to refine or find an alternative algorithm such that the numerical simulation
could be pushed to larger values of $N$, allowing us to verify the same
statement for the all-to-all SYK.

###### Acknowledgements.

We would like to thank Micha Berkooz, Vladimir Narovlansky and Antonio Garcia-
Garcia for comments on the manuscript. We also thank Kristan Jensen for
enlightening discussions. E.C. and A.M. are supported by National Science
Foundation (NSF) Grant No. PHY-2112725. A.R. is supported by the U.S.
Department of Energy under Grant No. DE-SC0022021 and a grant from the Simons
Foundation (Grant 651440, AK). E.C also thanks the Aspen Center for Physics,
supported by National Science Foundation grant PHY-1607611, where part of this
work was performed.

##  Appendix A Appendix: Extended numerical results

In this appendix we offer additional plots of the numerical results discussed
in the text. In Fig.5 and 6 we give additional plots of the SFF taking
$\beta=5$ and $q=4$ and $q=8$. To give a qualitative picture of the variance
of the SFF we plotted the region containing the individual SFF of all 100
disorder realizations, and the region containing 90 disorder realizations in
Fig.7. In Fig.8 and 9 we offer additional histograms of the average spectral
density, and compare them to the Q–Gaussian approximation. Finally in Fig.10
and 11 we present additional plots of the individual spectral densities for
the different realization of the couplings.

Figure 5: Full spectral form factor $g$, connected part $g_{c}$, and
disconnected part $g_{d}$, obtained by exact diagonalization of the sparse SYK
Hamiltonian define over a regular hypergraph, and averaged over 100 disorder
realizations of the random couplings. These are compared to the analytical
$Q$–Gaussian approximation $\tilde{g}_{c}$, $\tilde{g}_{d}$, and
$\tilde{g}_{\text{RMT}}$. Figure 6: Full spectral form factor $g$, connected
part $g_{c}$, and disconnected part $g_{d}$, obtained by exact diagonalization
of the sparse SYK Hamiltonian define over a regular hypergraph, and averaged
over 100 disorder realizations of the random couplings. These are compared to
the analytical $Q$–Gaussian approximation $\tilde{g}_{c}$, $\tilde{g}_{d}$,
and $\tilde{g}_{\text{RMT}}$. Figure 7: The spread of the spectral form factor
over the 100 disorder realizations of the random couplings. The black line is
the disorder averaged spectral form factor, the light gray area is the maximal
spread of the SFFs, while the dark gray area contains 90% of the disorder
realizations. The disconnected part of the SFF is plotted in a dashed line to
give some indication of time scales. Figure 8: A histogram of the spectral
density of the sparse SYK Hamiltonian averaged over 100 disorder realizations
for various values of $k$. For comparison the solid red line is the
$Q$–Gaussian approximation to the spectral density (6). Figure 9: A histogram
of the spectral density of the sparse SYK Hamiltonian averaged over 100
disorder realizations for various values of $k$. For comparison the solid red
line is the $Q$–Gaussian approximation to the spectral density (6). Figure 10:
The spread of the spectral density of the sparse SYK Hamiltonian for 100
disorder realizations and for various values of $k$.  Figure 11: The spread of
the spectral density of the sparse SYK Hamiltonian for 100 disorder
realizations and for various values of $k$.

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Cyclic relative difference families with block size four and their applications

a]Chenya Zhao
a]Binwei Zhao
a]Yanxun Chang Supported by NSFC under Grant 11971053
a]Tao Feng Supported by NSFC under Grant 12271023
b]Xiaomiao Wang Supported by NSFC under Grant $11771227$
a]Menglong Zhang
[a]School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, P.R. China
[b]School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China

[<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \F\}=G\setminus H$. Let $g\Z_{gh}$ is the subgroup of order $h$ in $\mathbb Z_{gh}$ generated by $g$. A $(\Z_{gh},g\Z_{gh},k,1)$-DF is called cyclic and written as a $(gh,h,k,1)$-CDF. This paper shows that for $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$, $g\geq 4$ and $(g,h)\not\in\{(9,3),(5,6)\}$. As a corollary, it is shown that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv4\pmod{12}$ and $v\neq 28$. This solves the long-standing open problem on the existence of a 1-rotational S$(2,4,v)$. As another corollary, we establish the existence of an optimal $(v,4,1)$-optical orthogonal code with $\lfloor(v-1)/12\rfloor$ codewords for any positive integer $v\equiv 1,2,3,4,6\pmod{12}$ and $v\neq 25$. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic $3$-ary constant-weight codes with weight four and minimum distance six.

Keywords: difference family; cyclic group divisible design; 1-rotational Steiner system; cyclic constant-weight code; optical orthogonal code

§ INTRODUCTION

Throughout this paper, every union of sets will be understood as multiset union with multiplicities of elements preserved. Denote by $\Z_v$ the cyclic group of order $v$.

Let $H$ be a subgroup of a finite group $(G,+)$ and let $k$ be an integer satisfying $2\leq k\leq |G|$. A $(G,H,k,1)$-relative difference family (briefly DF) is a set $\F$ of $k$-subsets of $G$ (called base blocks) such that the list

$$\Delta \F:=\bigcup_{F\in \F}\Delta F:=\bigcup_{F\in \F}\{f-f':f,f'\in F, f\neq f'\}$$

covers each element of $G\setminus H$ exactly once and each element of $H$ zero time. There are $(|G|-|H|)/(k(k-1))$ base blocks in a $(G,H,k,1)$-DF and hence, a necessary condition for the existence of such a family is $|G|\equiv |H|\pmod{k(k-1)}$. Another obvious necessary condition is $|G|\geq k|H|$; otherwise, given any base block $F\in \F$, there exist two elements of $F$ that lie in a same right coset of $H$ in $G$, a contradiction.

If there exists a $(G,H,k,1)$-DF, then $|G|\equiv |H|\pmod{k(k-1)}$ and $|G|\geq k|H|$.

When $H$ consists only of the identity $1_G$ of $G$, a $(G,1_G,k,1)$-DF is often written as a $(G,k,1)$-DF. When $G$ is a cyclic group, a $(G,H,k,1)$-DF is said to be cyclic and written as a  $(|G|,|H|,k,1)$-CDF.

Difference families arose from the study of designs with a given group of automorphisms. We
here only briefly recall what has been done for cyclic difference families with base block size three or four. The reader is referred to [2, 3] for more topics on difference families.

As far back as 1939, Peltesohn [41] showed that for $h\in\{1,3\}$, a $(gh,h,3,1)$-CDF exists if and only if $gh\equiv h \pmod{6}$ and $(g,h)\neq (3,3)$. This gives a solution to the Heffter's difference problems [31, 32] that yield the existence of cyclic Steiner triple systems (see Chapter VII.4 in [3]). For general $h$, Wang and Chang <cit.> in 2009 proved that a $(gh,h,3,1)$-CDF exists if and only if (1) $gh\equiv h \pmod{6}$ and $g\geq 4$; (2) $g\not\equiv 2,3 \pmod{4}$ when $h\equiv 2 \pmod{4}$. This implies the existence of cyclic group divisible designs with block size three without short orbits (see <cit.>).

When $p\equiv 1\pmod{12}$ is a prime, Bose [5] provided a sufficient condition for the existence of a $(p,4,1)$-CDF admitting a multiplier of order 3, and the necessary and sufficient condition for this special kind of CDFs was established by Buratti in [8]. Improving Buratti's work in [9], Chen and Zhu [19] showed that a $(p,4,1)$-CDF exists for any prime $p\equiv 1\pmod{12}$. Even though a $(v,4,1)$-CDF of composite order $v=v_1v_2$ can be obtained by means of recursive constructions in the hypothesis that a $(v_i,4,1)$-CDF exists (cf. [10]) or by direct constructions with nice algebraic properties (cf. [13]), no infinite family of $(v,4,1)$-CDFs such that $v$ can run over a congruent class was known until quite recently. Zhang, Feng and Wang [48] recently developed an efficient direct construction method to establish the existence of $(gh,h,4,1)$-CDFs for $h\in\{1,4\}$, which solves the long-standing open problem on the existence of cyclic Steiner systems with block size four.

[48]
For $h\in\{1,4\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$ and $(g,h)\not\in\{(25,1),(4,4),(7,4)\}$.

This paper is a continuation of the study of [48]. As the main result of this paper, we are to prove the following theorem.

For $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$, $g\geq 4$ and $(g,h)\not\in\{(9,3),(5,6)\}$.

Section <ref> gives a proof of Theorem <ref>. In Section <ref>, using a recursive construction, we construct more new infinite families of cyclic difference families (see Theorem <ref>). Section <ref> is devoted to providing applications of cyclic difference families to cyclic group divisible designs (see Theorem <ref>), 1-rotational Steiner systems (see Theorem <ref>), optimal cyclic constant-weight codes (see Theorem <ref>) and optimal optical orthogonal codes (see Theorem <ref>). Concluding remarks are given in Section <ref>.

§ EXISTENCE OF $(GH,H,4,1)$-CYCLIC DIFFERENCE FAMILIES

§.§ The cases $h=2,3$ and $6$

For positive integers $a,b$ and $c$ such that $a\leq b$ and $a\equiv b\pmod{c}$, we set $[a,b]_c:=\{a+ci:0\leq i\leq (b-a)/c\}$. When $c=1$, $[a,b]_1$ is simply written as $[a,b]$.

For $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF for any $gh\equiv h\pmod{12}$ and $gh\geq 218$.

Let $gh=72t+12x+h$, where $t\geq 3$ and $0\leq x\leq 5$. Then a $(gh,h,4,1)$-CDF contains $6t+x$ base blocks. We give a direct construction for such a CDF here. All base blocks are divided into two parts. The first part consists of $6t-18$ base blocks:

$F_{1,i}=\{0$,     $43t+a_1+i$,     $31t+a_2+2i$,     $8t+a_3+3i\}$,

$F_{2,i}=\{0$,     $23t+b_1+i$,     $5t+b_2+2i$,     $8t+b_3+3i\}$,

$F_{3,i}=\{0$,     $41t+c_1+i$,     $25t+c_2+2i$,     $8t+c_3+3i\}$,

$F_{4,i}=\{0$,     $35t+d_1+i$,     $5t+d_2+2i$,     $d_3+3i\}$,

$F_{5,i}=\{0$,     $47t+e_1+i$,     $19t+e_2+2i$,     $e_3+3i\}$,

$F_{6,i}=\{0$,     $21t+f_1+i$,     $13t+f_2+2i$,     $f_3+3i\}$,

where $1\leq i\leq t-2$ and $i\neq \lfloor t/2\rfloor$, and $a_j,b_j,c_j,d_j,e_j,f_j$ are shown in Table <ref>. The remaining $18+x$ base blocks are listed in Appendix <ref> according to the parity of $t$. Write $\F_r=\{F_{r,i}: 1\leq i\leq{t-2}, i\neq \lfloor t/2\rfloor\}$ for $1\leq r\leq6$, and denote by $\F^{'}$ the set consisting of the remaining $18+x$ base blocks. It remains to check that $\F=(\bigcup_{r=1}^6{\F_r})\cup{\F^{'}}$ forms a $(gh,h,4,1)$-CDF.

$h$     $x$     $a_1$     $a_2$     $a_3$     $b_1$     $b_2$     $b_3$     $c_1$     $c_2$     $c_3$     $d_1$     $d_2$     $d_3$     $e_1$     $e_2$     $e_3$     $f_1$     $f_2$     $f_3$
$0$     $1$     $1$     $2$     $1$     $0$     $1$     $0$     $1$     $0$     $1$     $1$     $1$     $2$     $2$     $2$     $1$     $1$     $0$

$1$     $8$      $7$      $5$      $4$     $3$      $4$     $7$      $4$      $3$      $7$      $2$     $1$     $9$      $5$      $2$     $4$      $1$      $0$

2     $2$     $16$     $14$     $8$      $10$     $4$     $7$     $15$     $10$     $6$      $12$     $3$     $1$     $17$     $9$      $2$     $6$      $4$      $0$

$3$     $25$     $20$     $10$     $14$     $5$     $8$     $24$     $17$     $9$      $17$     $4$     $1$     $24$     $11$     $2$     $12$     $7$      $0$

$4$     $31$     $24$     $10$     $18$     $7$     $8$     $30$     $19$     $9$      $23$     $6$     $1$     $32$     $13$     $2$     $16$     $11$     $0$

$5$     $38$     $30$     $11$     $21$     $6$     $9$     $37$     $25$     $10$     $29$     $5$     $1$     $40$     $17$     $2$     $18$     $13$     $0$

$0$     $2$      $2$     $2$     $2$     $1$     $1$     $1$     $1$     $0$     $2$     $2$     $1$     $3$     $3$     $2$     $2$     $2$     $0$

$1$     $10$     $7$     $5$     $6$     $3$     $4$     $9$     $7$     $3$     $7$     $2$     $1$     $11$     $5$     $2$     $4$     $3$     $0$

$3$     $2$     $18$     $14$     $8$     $10$     $4$     $7$     $17$     $12$     $6$     $12$     $3$     $1$     $18$     $9$     $2$     $8$     $6$     $0$

$3$     $26$     $20$     $10$     $14$     $5$     $8$     $26$     $17$     $9$     $18$     $4$     $1$     $25$     $11$     $2$     $10$     $7$     $0$

$4$     $32$     $24$     $10$     $18$     $5$     $8$     $30$     $19$     $9$     $24$     $4$     $1$     $35$     $17$     $2$     $16$     $11$     $0$

$5$     $41$     $31$     $11$     $21$     $6$     $9$     $38$     $24$     $10$     $30$     $5$     $1$     $42$     $18$     $2$     $17$     $13$     $0$

$0$     $4$     $3$     $2$     $1$     $0$     $1$     $2$     $1$     $0$     $2$     $1$     $1$     $4$     $2$     $2$     $1$     $1$     $0$

$1$     $12$     $9$      $5$      $6$     $3$      $4$     $9$      $6$      $3$      $7$      $2$     $1$     $13$     $5$      $2$     $4$      $3$      $0$

$6$     				$2$     $19$     $14$     $8$      $10$     $4$     $7$     $17$     $12$     $6$      $15$     $3$     $1$     $19$     $9$      $2$     $8$      $4$      $0$

$3$     $26$     $20$     $10$     $14$     $5$     $8$     $26$     $18$     $9$      $20$     $4$     $1$     $27$     $11$     $2$     $10$     $7$      $0$

$4$     $32$     $24$     $10$     $18$     $5$     $8$     $30$     $20$     $9$      $24$     $4$     $1$     $35$     $17$     $2$     $16$     $11$     $0$

$5$     $41$     $31$     $11$     $21$     $6$     $9$     $38$     $25$     $10$     $30$     $5$     $1$     $42$     $18$     $2$     $17$     $13$     $0$
Coefficients in the first $6t-18$ base blocks in Lemma <ref>

We take the case $h=2$, $x=0$ and $t\equiv 1\pmod{2}$ for example to show that $\F$ forms a $(2g,2,4,1)$-CDF. For convenience, for any $F\in \F$, if $x,y\in F$ and $x>y$, we call $x-y\pmod{gh}$ a positive difference from $F$, and $y-x\pmod{gh}$ a negative difference from $F$.  The collection of all positive differences (resp. negative differences) in $\Delta F$ is denoted by $\Delta^+ F$ (resp. $\Delta^- F$). Write $\Delta^+\F_r=\bigcup_{F\in\F_r} \Delta^+ F$ and $\Delta^-\F_r=\bigcup_{F\in\F_r} \Delta^- F$ for $1\leq r\leq6$. We list all positive differences from $\F_r$ with $1\leq r\leq6$ in Table <ref>, and if a positive difference is greater than $36t+1$, then we list its corresponding negative difference. We also list all differences less than $36t+1$ from the remaining 18 base blocks, written as $B_l$ with $1\leq l\leq 18$, in Table <ref>. It is readily checked that $\Delta \F=\Z_{72t+2}\setminus\{0,36t+1\}$, and hence $\F$ is a $(2g,2,4,1)$-CDF.

$\Delta^+ F_{1,i}$     $\Delta^+{\cal F}_1$     $\Delta^-{\cal F}_1$     $\Delta^+ F_{2,i}$     $\Delta^+{\cal F}_2$
$43t+1+i$     $[43t+2,44t-1]\backslash\{\frac{87t+1}{2}\}$     $[28t+3,29t]\backslash\{\frac{57t+3}{2}\}$     $23t+1+i$     $[23t+2,24t-1]\backslash\{\frac{47t+1}{2}\}$

$31t+1+2i$     $[31t+3,33t-3]_{2}\backslash\{32t\}$         $5t+2i$     $[5t+2,7t-4]_{2}\backslash\{6t-1\}$

$8t+2+3i$     $[8t+5,11t-4]_{3}\backslash\{\frac{19t+1}{2}\}$         $8t+1+3i$     $[8t+4,11t-5]_{3}\backslash\{\frac{19t-1}{2}\}$

$12t-i$     $[11t+2,12t-1]\backslash\{\frac{23t+1}{2}\}$         $18t+1-i$     $[17t+3,18t]\backslash\{\frac{35t+3}{2}\}$

$35t-1-2i$     $[33t+3,35t-3]_{2}\backslash\{34t\}$         $15t-2i$     $[13t+4,15t-2]_{2}\backslash\{14t+1\}$

$23t-1-i$     $[22t+1,23t-2]\backslash\{\frac{45t-1}{2}\}$         $3t+1+i$     $[3t+2,4t-1]\backslash\{\frac{7t+1}{2}\}$
$\Delta F^+_{3,i}$     $\Delta{\cal F}^+_3$     $\Delta{\cal F}^-_3$     $\Delta F^+_{4,i}$     $\Delta{\cal F}^+_4$
$41t+i$     $[41t+1,42t-2]\backslash\{\frac{83t-1}{2}\}$     $[30t+4,31t+1]\backslash\{\frac{61t+5}{2}\}$     $35t+1+i$     $[35t+2,36t-1]\backslash\{\frac{71t+1}{2}\}$

$25t+1+2i$     $[25t+3,27t-3]_{2}\backslash\{26t\}$         $5t+1+2i$     $[5t+3,7t-3]_{2}\backslash\{6t\}$

$8t+3i$     $[8t+3,11t-6]_{3}\backslash\{\frac{19t-3}{2}\}$         $1+3i$     $[4,3t-5]_{3}\backslash\{\frac{3t-1}{2}\}$

$16t-1-i$     $[15t+1,16t-2]\backslash\{\frac{31t-1}{2}\}$         $30t-i$     $[29t+2,30t-1]\backslash\{\frac{59t+1}{2}\}$

$33t-2i$     $[31t+4,33t-2]_{2}\backslash\{32t+1\}$         $35t-2i$     $[33t+4,35t-2]_{2}\backslash\{34t+1\}$

$17t+1-i$     $[16t+3,17t]\backslash\{\frac{33t+3}{2}\}$         $5t-i$     $[4t+2,5t-1]\backslash\{\frac{9t+1}{2}\}$
$\Delta F^+_{5,i}$     $\Delta{\cal F}^+_5$     $\Delta{\cal F}^-_5$     $\Delta F^+_{6,i}$     $\Delta{\cal F}^+_6$
$47t+2+i$     $[47t+3,48t]\backslash\{\frac{95t+3}{2}\}$     $[24t+2,25t-1]\backslash\{\frac{49t+1}{2}\}$     $21t+1+i$     $[21t+2,22t-1]\backslash\{\frac{43t+1}{2}\}$

$19t+2+2i$     $[19t+4,21t-2]_{2}\backslash\{20t+1\}$         $13t+1+2i$     $[13t+3,15t-3]_{2}\backslash\{14t\}$

$2+3i$     $[5,3t-4]_{3}\backslash\{\frac{3t+1}{2}\}$         $3i$     $[3,3t-6]_{3}\backslash\{\frac{3t-3}{2}\}$

$28t-i$     $[27t+2,28t-1]\backslash\{\frac{55t+1}{2}\}$         $8t-i$     $[7t+2,8t-1]\backslash\{\frac{15t+1}{2}\}$

$47t-2i$     $[45t+4,47t-2]_{2}\backslash\{46t+1\}$     $[25t+4,27t-2]_2\backslash\{26t+1\}$     $21t+1-2i$     $[19t+5,21t-1]_{2}\backslash\{20t+2\}$

$19t-i$     $[18t+2,19t-1]\backslash\{\frac{37t+1}{2}\}$         $13t+1-i$     $[12t+3,13t]\backslash\{\frac{25t+3}{2}\}$

Differences from the first $6t-18$ base blocks in Lemma <ref> when $h=2$, $x=0$ and $t$ is odd

$j$     differences less than $36t+1$ from $B_{2j-1}$     differences less than $36t+1$ from $B_{2j}$

$2$     $\{\frac{3t-3}{2},23t-1,28t+2,\frac{43t+1}{2},\frac{59t+1}{2},21t+1\}$

Differences less than $36t+1$ from the remaining $18$ base blocks in Lemma <ref> when $h=2$, $x=0$ and $t$ is odd

One can examine all the other cases similarly. To facilitate the reader to check the correctness of our results, we provide a computer code written by GAP [45]. The reader can get a copy of the computer code from [50].

We remark that the structure of the first $6t-18$ base blocks in this proof is exactly the same as the structure of the first $6t-18$ base blocks in the proof for the existence of $(gh,h,4,1)$-CDFs with $h\in\{1,4\}$ in <cit.>. We just need to choose appropriate parameters $a_j,b_j,c_j,d_j,e_j,f_j$ by hand and then find the remaining $18+x$ base blocks by computer search such that $\F$ forms our required difference families. It often took us around a half day to choose an appropriate set of $a_j,b_j,c_j,d_j,e_j,f_j$ for each given $h$ and $x$, and then the time used to search for the remaining $18+x$ base blocks by an ordinary personal computer was ranged from one day to three days.

The following lemma will be used to show the existence of $(gh,h,4,1)$-CDFs with $h\in\{2,3\}$ and small $g$'s later. We outline its proof for completeness.

$(1)$ <cit.> There exists a $(2p,2,4,1)$-CDF for any prime $p\equiv1\pmod{6}$.

$(2)$ [40] There exists a $(3p,3,4,1)$-CDF for any prime $p\equiv1\pmod{4}$.

Let $\omega$ be a primitive root of unity in the finite field $\mathbb{F}_p$. Let $\varepsilon=\omega^{(p-1)/6}$ if $p\equiv1\pmod{6}$, and $\varepsilon=\omega^{(p-1)/4}$ if $p\equiv1\pmod{4}$. Then

$$\{(0,0),(1,\omega^i),(1,\varepsilon^2\cdot \omega^i),(1,\varepsilon^4\cdot \omega^i)\},\ \ \ \ 0\leq i<(p-1)/6,$$

forms a $(2p,2,4,1)$-CDF over $\Z_2\times \Z_p\cong \Z_{2p}$ for any prime $p\equiv1\pmod{6}$, and

$$\{(0,\omega^i),(0,\varepsilon^2\cdot \omega^i),(1,\varepsilon\cdot \omega^i),(1,\varepsilon^3\cdot \omega^i)\},\ \ \ \ 0\leq i<(p-1)/4,$$

forms a $(3p,3,4,1)$-CDF over $\Z_3\times \Z_p\cong \Z_{3p}$ for any prime $p\equiv1\pmod{4}$.

Applying a standard recursive construction (see Construction <ref> below) together with the use of the above lemma, Yin [47] established the following results implicitly.

$(1)$ <cit.> There exists a $(2g,2,4,1)$-CDF for any positive integer $g$ such that each prime factor of $g$ is $1$ modulo $6$.

$(2)$ <cit.> There exists a $(3g,3,4,1)$-CDF for any positive integer $g$ such that each prime factor of $g$ is $1$ modulo $4$.

There exists a $(2g,2,4,1)$-CDF for any integer $g\equiv 1\pmod{6}$ and $7\leq g\leq 103$.

For $g\in\{25,55,85\}$, we list all the base blocks below.

$g=25$:     $\{0,1,3,12\}$,   $\{0,4,18,26\}$,   $\{0,6,23,43\}$,   $\{0,10,15,31\}$.

$g=55$:     $\{0,1,4,50\}$,   $\{0,2,7,36\}$,   $\{0,6,15,100\}$,   $\{0,8,31,99\}$,   $\{0,13,37,90\}$,

$\{0,14,54,72\}$,   $\{0,17,47,82\}$,   $\{0,26,48,69\}$,   $\{0,27,59,71\}$.

$g=85$:     $\{0,1,4,62\}$,   $\{0,2,7,22\}$,   $\{0,6,14,142\}$,   $\{0,9,21,72\}$,   $\{0,10,23,55\}$,

$\{0,11,29,93\}$,   $\{0,16,52,91\}$,   $\{0,17,73,116\}$,   $\{0,19,84,122\}$,   $\{0,24,59,92\}$,

$\{0,27,74,144\}$,   $\{0,31,80,120\}$,
$\{0,44,69,110\}$,   $\{0,46,76,133\}$.

For all the other cases of $g$, apply Lemma <ref>.

There exists a $(3g,3,4,1)$-CDF for any integer $g\equiv 1\pmod{4}$, $5\leq g\leq 69$ and $g\neq 9$.

For $g\in\{21,33,45,49,57,69\}$, we list all the base blocks below.

$g=21$:     $\{0,2,18,27\}$,   $\{0,3,10,15\}$,   $\{0,8,14,37\}$,   $\{0,11,30,31\}$,   $\{0,24,28,41\}$.

$g=33$:     $\{0,1,3,40\}$,   $\{0,4,12,86\}$,   $\{0,5,16,36\}$,   $\{0,6,44,70\}$,   $\{0,10,28,85\}$,

$\{0,15,56,65\}$   $\{0,19,46,67\}$,   $\{0,22,45,52\}$.

$g=45$:     $\{0,1,5,38\}$,   $\{0,2,8,60\}$,   $\{0,3,14,21\}$,   $\{0,9,29,120\}$,   $\{0,10,59,84\}$,

$\{0,12,39,107\}$,   $\{0,16,46,87\}$,   $\{0,17,73,109\}$,   $\{0,23,65,78\}$,   $\{0,32,63,85\}$,

$g=49$:     $\{0,1,3,43\}$,   $\{0,19,73,101\}$,   $\{0,6,14,38\}$,   $\{0,7,17,68\}$,   $\{0,35,69,102\}$,

$\{0,4,9,22\}$,   $\{0,20,92,108\}$,   $\{0,21,57,87\}$,   $\{0,29,77,100\}$,   $\{0,31,58,122\}$,

$\{0,11,26,63\}$,   $\{0,50,62,103\}$.

$g=57$:     $\{0,1,3,55\}$,   $\{0,44,81,108\}$,   $\{0,5,12,105\}$,   $\{0,8,17,153\}$,   $\{0,11,24,80\}$,

$\{0,14,29,99\}$,   $\{0,16,50,83\}$,   $\{0,21,79,118\}$,   $\{0,25,48,68\}$,   $\{0,40,62,135\}$,

$\{0,4,10,42\}$,   $\{0,46,87,106\}$,   $\{0,47,77,122\}$,   $\{0,51,82,110\}$.

$g=69$:     $\{0,1,5,124\}$,   $\{0,15,34,108\}$,   $\{0,35,110,156\}$,   $\{0,55,98,137\}$,   $\{0,11,24,193\}$,

$\{0,2,8,66\}$,   $\{0,18,59,85\}$,   $\{0,45,105,135\}$,   $\{0,20,77,171\}$,   $\{0,28,89,139\}$,

$\{0,3,12,170\}$,   $\{0,44,71,175\}$,   $\{0,22,100,142\}$,   $\{0,48,79,160\}$,   $\{0,52,73,153\}$,

$\{0,7,17,191\}$,   $\{0,63,92,154\}$.

For all the other cases of $g$, apply Lemma <ref>.

The proof of Lemma 2.6 in [18] implies that if there exists a skew starter in $\Z_g$, then there exists a $(6g,6,4,1)$-CDF. Since a skew starter in $\Z_g$ exists for any $g$ such that $\gcd(g,150)=1$ or $25$ by <cit.>, and a skew starter in $\Z_{35}$ exists by <cit.>, we have the following result, (1) of which was first stated explicitly in <cit.>.

[18]

$(1)$ There exists a $(6g,6,4,1)$-CDF for any positive integer $g$ such that $\gcd(g,150)=1$ or $25$.

$(2)$ There exists a $(210,6,4,1)$-CDF.

We remark that Buratti <cit.> provided a nice explicit construction for the existence of a $(6p,6,4,1)$-CDF for any odd prime $p\neq 5$ according to whether $p$ is or is not a Fermat prime.

There exists a $(6g,6,4,1)$-CDF for any odd integer $7\leq g\leq 35$.

For $g\in\{9,15,27\}$, a $(6g,6,4,1)$-CDF exists by <cit.>. For $g\in\{21,33\}$, we list all the base blocks below.

$g=21$:     $\{0,19,62,79\}$,   $\{0,2,8,78\}$,   $\{0,3,12,35\}$,   $\{0,13,31,65\}$,   $\{0,15,22,115\}$,

$\{0,16,55,106\}$,   $\{0,1,5,46\}$,   $\{0,24,38,96\}$,   $\{0,25,53,82\}$,   $\{0,27,37,86\}$.

$g=33$:    $\{0,49,88,125\}$,   $\{0,2,9,74\}$,   $\{0,25,83,106\}$,   $\{0,17,35,69\}$,   $\{0,38,94,138\}$,

$\{0,14,29,170\}$,   $\{0,4,12,82\}$,   $\{0,21,45,130\}$,   $\{0,3,13,179\}$,   $\{0,30,105,148\}$,

$\{0,31,86,127\}$,   $\{0,1,6,54\}$,   $\{0,40,91,137\}$,   $\{0,11,27,90\}$,   $\{0,47,114,134\}$,

For all the other cases of $g$, apply Lemma <ref>.

Now we are in a position to give a proof of Theorem <ref>.

By Lemma <ref>, a $(gh,h,4,1)$-CDF exists only if $gh\equiv h\pmod{12}$ and $g\geq 4$. It is known that there is no $(gh,h,4,1)$-CDF for $(g,h)\in\{(9,3),(5,6)\}$ (cf. [11, 43]). Combining the results of Lemmas <ref>, <ref>, <ref> and <ref>, one can complete the proof.

§.§ General cases

Difference matrices have proved to be a very useful tool for constructing cyclic relative difference families. Let $(G,+)$ be a finite group of order $v$ and $k\geq 2$ be an integer. A $(v,k,1)$-difference matrix (DM) over $G$, or briefly $(G,k,1)$-DM, is a $k\times v$ matrix $D=(d_{ij})$ with entries from $G$ such that for any two distinct rows $x$ and $y$, the multiset of differences $\{d_{xj}-d_{yj}:1\leq j\leq v\}$ contains each element of $G$ exactly once. A $(\mathbb{Z}_v,k,1)$-DM is often called cyclic and denoted by a $(v,k,1)$-CDM.

The following construction is a standard recursive construction for cyclic relative difference families.

[36]
If there exist a $(gn,g,k,1)$-CDF and an $(h,k,1)$-CDM, then there exists a $(gnh,gh,k,1)$-CDF. Furthermore, if there exists a $(gh,g,k,1)$-CDF, then there exists a $(ghn,g,k,1)$-CDF.

Evans [25] constructed a $(v,4,1)$-CDM for any odd integer $v\geq 5$ and $v$ is not divisible by $9$. Zhang, Feng and Wang [49] solved the case $v\equiv 9\pmod{18}$.

[25, 49]
A $(v,4,1)$-CDM exists if and only if $v\geq 5$ is odd and $v\neq 9$.

By Lemma <ref>, a $(4h,h,4,1)$-CDF exists only if $h\equiv 0\pmod{4}$. Since a  $(4h,h,4,1)$-CDF yields a strictly cyclic group divisible design of type $h^4$ with block size four, which can produce an $(h,4,1)$-CDM (cf. <cit.>), it follows from Lemma <ref> that we have the following lemma.

There does not exist a $(4h,h,4,1)$-CDF for any positive integer $h$.

There exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$ and $g\geq 5$ except for $(g,h)\in\{(9,3),(5,6)\}$ and except possibly for

$(1)$ $h\equiv 1,5,7,11\pmod{12}$ and $g=25$,

$(2)$ $h\in\{9,27\}$ and $g\equiv1\pmod{4}$,

$(3)$ $h\equiv3,9\pmod{12}$ and $g=9$,

$(4)$ $h\equiv 8,16\pmod{24}$ and $g\equiv1\pmod{3}$,

$(5)$ $h\equiv 4,20\pmod{24}$ and $g=7$,

$(6)$ $h\equiv 6\pmod{12}$ and $g=5$,

$(7)$ $h=18$ and $g\equiv 1\pmod{2}$,

$(8)$ $h=54$ and $g\equiv 3,5\pmod{6}$,

$(9)$ $h\equiv 0\pmod{24}$ and $g\geq 5$,

$(10)$ $h\equiv 12\pmod{24}$, $g\equiv 0,2\pmod{3}$ and $g\geq 5$,

$(11)$ $h\in\{12,36\}$, $g\equiv 1\pmod{3}$ and $g\geq 10$.

By Lemmas <ref> and <ref>, a $(gh,h,4,1)$-CDF exists only if $g\geq 5$ and $h(g-1)\equiv 0\pmod{12}$, that is, (i) $h\equiv 1,5,7,11\pmod{12}$ and $g\equiv1\pmod{12}$, (ii) $h\equiv 2,10\pmod{12}$ and $g\equiv1\pmod{6}$, (iii) $h\equiv 3,9\pmod{12}$ and $g\equiv1\pmod{4}$, (iv) $h\equiv 4,8\pmod{12}$ and $g\equiv1\pmod{3}$, (v) $h\equiv 6\pmod{12}$ and $g\equiv1\pmod{2}$, (vi) $h\equiv 0\pmod{12}$ and $g\geq 5$.

For $h\equiv 1,5,7,11\pmod{12}$, $g\equiv1\pmod{12}$ and $g\neq 25$, apply Construction <ref> with a $(g,1,4,1)$-CDF (from Theorem <ref>) and an $(h,4,1)$-CDM (from Lemma <ref>) to obtain a $(gh,h,4,1)$-CDF.

For $h\equiv 2,10\pmod{12}$ and $g\equiv1\pmod{6}$, apply Construction <ref> with a $(2g,2,4,1)$-CDF (from Theorem <ref>) and an $(h/2,4,1)$-CDM.

For $h\equiv 3,9\pmod{12}$, $g\equiv1\pmod{4}$, $h\not\in\{9,27\}$ and $g\neq 9$, apply Construction <ref> with a $(3g,3,4,1)$-CDF (from Theorem <ref>) and an $(h/3,4,1)$-CDM.

For $h\equiv 4,20\pmod{24}$, $g\equiv1\pmod{3}$ and $g\geq 10$, apply Construction <ref> with a $(4g,4,4,1)$-CDF (from Theorem <ref>) and an $(h/4,4,1)$-CDM.

For $h\equiv 6\pmod{12}$, $g\equiv1\pmod{2}$, $h\not\in\{18,54\}$ and $g\geq 7$, apply Construction <ref> with a $(6g,6,4,1)$-CDF (from Theorem <ref>) and an $(h/6,4,1)$-CDM. For $h=54$ and $g\equiv1\pmod{6}$, apply Construction <ref> with a $(2g,2,4,1)$-CDF (from Theorem <ref>) and a $(27,4,1)$-CDM.

For $h\equiv 12\pmod{24}$, $g\equiv1\pmod{3}$, $h\geq 60$ and $g\geq 10$, apply Construction <ref> with a $(4g,4,4,1)$-CDF (from Theorem <ref>) and an $(h/4,4,1)$-CDM. For $h\equiv 12\pmod{24}$, $g=7$, $h\geq 60$ and $h\neq 108$, start from an $(84,12,4,1)$-CDF whose base blocks are listed below:

and then apply Construction <ref> with an $(h/12,4,1)$-CDM. By <cit.>, there exists a $(36\times 7,36,4,1)$-CDF. By <cit.>, there exists a $(108\times 7,108,4,1)$-CDF.

§ APPLICATIONS

§.§ Cyclic group divisible designs

A group divisible design (GDD) is a triple $(V,\G,\B)$, where $V$ is a set of points, $\G$ is a partition of $V$ into groups, and $\B$ is a family of subsets of $V$ (called blocks) such that every block intersects any given group in at most one point, and every two points belonging to distinct groups are contained in exactly one block. A $k$-GDD of type $h^g$ is a GDD in which the size of each block is $k$ and there are $g$ groups of size $h$. For more details on group divisible designs, the reader is referred to [4, 28].

An automorphism of a GDD ($V,{\cal G},{\cal B}$) is a permutation on $V$ leaving ${\cal G}$ and ${\cal B}$ invariant, respectively. A $k$-GDD of type $h^g$ is said to be cyclic if it admits an automorphism consisting of a cycle of length $gh$. Without loss of generality, we regard $\Z_{gh}$ as an automorphism group of a cyclic $k$-GDD of type $h^g$. If the stabilizer of any block of a cyclic GDD of type $h^g$ in $\Z_{gh}$ is trivial, then the GDD is called strictly cyclic. The blocks of a strictly cyclic $k$-GDD of type $h^g$ can be partitioned into orbits under ${\mathbb Z}_{gh}$. Choose any fixed block from each orbit and then call them base blocks. These base blocks form a $(gh,h,k,1)$-CDF. Therefore, a strictly cyclic $k$-GDD of type $h^g$ yields a $(gh,h,k,1)$-CDF, and vice versa. Combining Theorem <ref>, we have the following theorem.

There exists a strictly cyclic $4$-GDD of type $h^g$ if and only if $gh\equiv h\pmod{12}$ and $g\geq 5$ except for $(g,h)\in\{(9,3),(5,6)\}$ and except possibly for

$(1)$ $h\equiv 1,5,7,11\pmod{12}$ and $g=25$,

$(2)$ $h\in\{9,27\}$ and $g\equiv1\pmod{4}$,

$(3)$ $h\equiv3,9\pmod{12}$ and $g=9$,

$(4)$ $h\equiv 8,16\pmod{24}$ and $g\equiv1\pmod{3}$,

$(5)$ $h\equiv 4,20\pmod{24}$ and $g=7$,

$(6)$ $h\equiv 6\pmod{12}$ and $g=5$,

$(7)$ $h=18$ and $g\equiv 1\pmod{2}$,

$(8)$ $h=54$ and $g\equiv 3,5\pmod{6}$,

$(9)$ $h\equiv 0\pmod{24}$ and $g\geq 5$,

$(10)$ $h\equiv 12\pmod{24}$, $g\equiv 0,2\pmod{3}$ and $g\geq 5$,

$(11)$ $h\in\{12,36\}$, $g\equiv 1\pmod{3}$ and $g\geq 10$.

§.§ 1-rotational Steiner systems

A $k$-GDD of type $1^v$ is often called a Steiner system and denoted by an S$(2,k,v)$. An S$(2,k,v)$ is said to be $1$-rotational if it admits an automorphism consisting of one fixed point and a cycle of length $v-1$. A 1-rotational S$(2,k,v)$ defined on $X$ may be assumed to have $X=\Z_{v-1}\cup\{\infty\}$ with the 1-rotational automorphism given by $i\rightarrow i+1\pmod{v-1}$ for $i\in \Z_{v-1}$ and $\infty \rightarrow \infty$.

Phelps and Rosa [42] showed that a 1-rotational S$(2,3,v)$ exists if and only if $v\equiv 3,9\pmod{24}$. It was conjectured in <cit.> that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv4\pmod{12}$ and $v\neq 28$. Applying Theorem <ref>, we can confirm this conjecture.

There exists a 1-rotational S$(2,4,v)$ if and only if $v\equiv4\pmod{12}$ and $v\neq 28$.

The necessity follows from <cit.>. To show the sufficiency, take a $(v-1,3,4,1)$-CDF, $\F$, with $v\equiv4\pmod{12}$ and $v\neq 28$ from Theorem <ref>. Then it is readily checked that $\{\{f_1+t,f_2+t,f_3+t,f_4+t\}:F=\{f_1,f_2,f_3,f_4\}\in \F,t\in \Z_{v-1}\}\cup\{\{t,\frac{v-1}{3}+t,\frac{2(v-1)}{3}+t,\infty\}:0\leq t<\frac{v-1}{3}\}$ forms a 1-rotational S$(2,4,v)$.

For more information on 1-rotational Steiner systems, the reader is referred to [2, 33].

§.§ Optimal cyclic constant-weight codes

A $q$-ary code $\mathcal{C}$ of length $n$ is a set of vectors in $\{0,1,\ldots,q-1\}^n$. An element in $\mathcal{C}$ is called a codeword. For two codewords $u=(u_i)_{0\leq i\leq n-1}$ and $v=(v_i)_{0\leq i\leq n-1}$, the Hamming distance between $u$ and $v$ is $d_H(u,v)=|\{0\leq i\leq n-1:u_i-v_i\neq 0\}|$. The Hamming weight of $u$ is the
Hamming distance between $u$ and the zero vector 0. A code is said to be of constant-weight $w$ if the Hamming weight of each codeword is a constant $w$. A code $\mathcal{C}$ is said to have minimum distance $d$ if $d_H(u,v)\geq d$ for all distinct $u,v\in \mathcal{C}$. A $q$-ary code of length $n$, minimum distance $d$ and constant-weight $w$ is denoted by an $(n,d,w)_q$ code.

An $(n,d,w)_q$ code $\mathcal{C}$ is said to be cyclic if $(u_0,u_1,\dots,u_{n-1}) \in \mathcal{C}$ implies $(u_1,u_2,\ldots,u_{n-1},u_0)\in \mathcal{C}$. We use the notation $CA_q(n,d,w)$ to denote the largest possible number of codewords among all cyclic $(n,d,w)_q$ codes with given parameters $n,d,w$ and $q$. A cyclic $(n,d,w)_q$ code is called optimal if it has $CA_q(n,d,w)$ codewords.

Lan, Chang and Wang [37] gave an equivalent definition of a cyclic $(n,d,w)_q$ code from a set-theoretic perspective.
Let $\mathcal{C}$ be a cyclic $(n,d,w)_q$ code. For each $u=(u_i)_{0\leq i\leq n-1}\in\mathcal{C}$, construct a $w$-subset $B_u$ of $\Z_n\times \{1,2,\ldots,q-1\}$ such that $(i,u_i)\in B_u$ if and only if $u_i$ is nonzero. Let $\mathcal{C}'=\{B_{u}:u\in \mathcal{C}\}$. For any codeword $B=\{(i_1,a_1),(i_2,a_2),\ldots,(i_w,a_w)\}\in \mathcal{C^{'}}$, define the projection $p(B)$ of $B$ to be the set $p(B)=\{i_1,i_2,\ldots,i_w\}$. For any $B_u,B_v \in \mathcal{C^{'}}$, define the distance $d_H(B_u,B_v)$ to be $d_H(u,v)$. It is readily checked that $d_H(B_u,B_v)=2w-|p(B_u)\cap p(B_v)|-|B_u \cap B_v|$. Then a cyclic $(n,d,w)_q$ code $\mathcal{C}$ is a set of $w$-subsets of $\Z_n\times \{1,2,\ldots,q-1\}$ satisfying that (1) for any $B\in {\cal C}$, the cardinality of $p(B)$ is $w$; (2) $d_H(A,B)\ge d$ for any distinct codewords $A,B\in {\cal C}$; (3) if $B\in {\cal C}$, then $B+1\in {\cal C}$, where $B+1=\{(x+1\mod{n},a):(x,a)\in B\}$.

For any positive integer $n$,

CA_3(n,6,4)\leq C(n):=\left\{
\begin{array}{ll}
n\lfloor\frac{n-1}{6}\rfloor,  & \mbox{if}\ n\ \mbox{is odd or } n\equiv 2\ ({\rm mod}\ 6);\\

n\lfloor\frac{n-1}{6}\rfloor+\frac{n}{2}, & \mbox{if}\ n \equiv 0,4\ ({\rm mod}\ 6).
\end{array}\right.

By <cit.>, if there exists a $(gh,h,4,1)$-CDF with $h\in\{1,2,3\}$, then there exists an optimal cyclic $(gh,6,4)_3$ code with $2gh\lfloor(gh-1)/12\rfloor=gh(gh-h)/6$ codewords. It follows from Theorems <ref> and <ref> that an optimal cyclic $(n,6,4)_3$ code with $C(n)$ codewords exists for any $n\equiv 1,2,3\pmod{12}$ and $n\not\in\{25,27\}$. For $n\in\{25,27\}$, an optimal cyclic $(n,6,4)_3$ code with $C(n)$ codewords exists by <cit.>. By <cit.>, there exists an optimal cyclic $(n,6,4)_3$ code with $C(n)$ codewords for any $n\equiv 6\pmod{12}$ or $n\equiv 0\pmod{24}$. Therefore, we have the following result.

There exists an optimal cyclic $(n,6,4)_3$ code with $C(n)$ codewords for any positive integer $n\equiv 1,2,3,6\pmod{12}$ or $n\equiv 0\pmod{24}$.

§.§ Optimal optical orthogonal codes

The study of optical orthogonal codes was first motivated by an application in a fiber optic code-division multiple access channel which requires binary sequences with good correlation properties (cf. [20]).

Let $v$ and $k$ be positive integers. A $(v, k, 1)$ optical orthogonal code (briefly $(v, k, 1)$-OOC), ${\cal C}$, is a family of $(0, 1)$-sequences (called codewords) of length $v$ and weight $k$ satisfying the following two properties (all subscripts are reduced modulo $v$):

${\sum}_{0 \le t \le v-1}x_tx_{t+i} \le 1$ for any ${\bf x} = (x_0, x_1, \ldots, x_{v-1}) \in {\cal C}$ and any integer $i\not\equiv 0 \pmod{v}$ (the auto-correlation property);

${\sum}_{0 \le t \le v-1}x_ty_{t+i} \le 1$ for any ${\bf x} = (x_0, x_1,$ $ \ldots,$ $ x_{v-1}) \in {\cal C}$, ${\bf y} = (y_0, y_1, \ldots, y_{v-1}) \in {\cal C}$ with ${\bf x} \neq {\bf y}$, and any integer $i$ (the cross-correlation property).

A convenient way of viewing OOCs is from a set-theoretic perspective. A $(v,k,1)$-OOC can be defined as a set of $k$-subsets (corresponding to codewords) of $\mathbb{Z}_v$ whose list of differences does not contain repeated elements (cf. <cit.>). A $(v,k,1)$-OOC is said to be J-optimal if the size of the set of missing differences (in $\Z_v$) of this OOC is less than or equal to $k(k-1)$, and hence a J-optimal $(v,k,1)$-OOC contains $\lfloor \frac{v-1}{k\left(k-1\right)}\rfloor$ codewords. $\lfloor \frac{v-1}{k\left(k-1\right)}\rfloor$ is an upper bound of the number of codewords of a $(v,k,1)$-OOC, and is often referred to as the Johnson bound.

There exists a J-optimal $(v,4,1)$-OOC for any positive integer $v\equiv 1,2,3,4,6\pmod{12}$ or $v\equiv 0\pmod{24}$ with the only definite exception of $v=25$.

Since a $(gh,h,4,1)$-CDF with $h\in\{1,2,3,4,6\}$ contains $\frac{gh-h}{12}=\lfloor\frac{gh-1}{12}\rfloor$ base blocks, every CDF from Theorems <ref> and <ref> implies the existence of a J-optimal $(gh,4,1)$-OOC. Therefore, there exists a J-optimal $(v,4,1)$-OOC for any $v\equiv 1,2,3,4,6\pmod{12}$ and $v\not\in\{16,18,25,27,28,30\}$. By <cit.>, there exists a J-optimal $(v,4,1)$-OOC for $v\in\{16,18,27,28,30\}$ and there is no J-optimal $(25,4,1)$-OOC. By <cit.>, there exists a J-optimal $(v,4,1)$-OOC for any $v\equiv 0\pmod{24}$.

We remark that Ge and Yin <cit.> also established the existence of a J-optimal $(v,4,1)$-OOC for any $v\equiv 6 \pmod{12}$ by using different construction methods. For more information on J-optimal $(v,4,1)$-OOC, the reader is referred to [11, 12, 16, 17, 18, 26, 27, 29, 47].

§ CONCLUDING REMARKS

By utilizing the method presented in [48], we established the existence of $(gh,h,4,1)$-CDFs with $h\in\{2,3,6\}$ and $gh\equiv h\pmod{12}$ in Theorem <ref>. It shows that the method is flexible and powerful for constructing many kinds of cyclic designs with block size four even though it looks mysterious.

Theorem <ref> provides more new infinite families of $(gh,h,4,1)$-CDFs by using a standard recursive construction. Completing the existence of $(gh,h,4,1)$-CDFs for all admissible parameters $g$ and $h$ will be a complicated and difficult work.

In order to facilitate the reader for further study, we here summarize main known results on the existence of $(gh,h,4,1)$-CDFs not covered by Theorems <ref> and <ref> below.

$(1)$ <cit.>
There exists a $(gh,h,4,1)$-CDF for $h\in\{18,60,72,96$, $108\}$ and all $g$ such that $\gcd(g,150)=1$ or $25$.

$(2)$ <cit.> There exists a $(ph,h,4,1)$-CDF for $h\in\{36,48\}$ and any prime $p>5$.

$(3)$ <cit.>
Let $g$ be a positive integer whose prime factors are all congruent to $1$ modulo $4$ and greater than $5$. Then there exists a $(12g,12,4,1)$-CDF.

$(4)$ <cit.>
There exists a $(24\cdot3^{2t}w,24,4,1)$-CDF for any integer $t\geq 0$ and all $w$ such that $\gcd(w,6)=1$.

$(5)$ <cit.> There exists a $(2^{s+4},2^s,4,1)$-CDF for any integer $s\geq3$.

$(6)$ <cit.> Let $s\in\{3,4,5,6\}$. There exists a $(2^{4n+s},2^s,4,1)$-CDF for any positive integer $n$.

$(7)$ <cit.>  Let $s\in\{3,4,5,6\}$. There exists a $(2^sp,2^s,4,1)$-CDF for any prime $p\equiv1 \pmod{6}$ except possibly for $(p,s)=(7,6)$.

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§ THE REMAINING $18+X$ BASE BLOCKS IN LEMMA <REF>

|lll|$h=2$, $x=0$ and $t$ is odd
$\{0,1,11t-2,42t\}$   $\{0,2,3t-1,11t+1\}$
$\{0,3t+1,7t+1,19t+3\}$   $\{0,\frac{9t+1}{2},8t+1,\frac{141t+5}{2}\}$

$\{0,5t,21t,28t\}$   $\{0,5t+1,\frac{55t+1}{2},\frac{129t+3}{2}\}$
$\{0,7t-2,35t-1,52t\}$   $\{0,8t,33t,58t+2\}$

$\{0,11t,34t+1,58t+1\}$   $\{0,12t+1,15t-1,39t\}$

$\{0,\frac{33t+3}{2},26t,65t+3\}$   $\{0,19t,32t+1,36t+2\}$

$h=2$, $x=0$ and $t$ is even
$\{0,1,8t+1,11t-2\}$   $\{0,2,\frac{19t}{2}+2,28t+2\}$
$\{0,3t-1,29t+1,42t+2\}$   $\{0,4t,21t+1,53t+1\}$

$\{0,4t+1,45t+1,51t+2\}$   $\{0,5t,11t,64t\}$
$\{0,7t-1,23t,65t+1\}$   $\{0,\frac{19t}{2}+1,33t+2,57t+3\}$

$\{0,12t,26t+1,\frac{85t}{2}+2\}$   $\{0,12t+2,15t,54t+1\}$
$\{0,\frac{43t}{2}+1,25t+2,50t+2\}$   $\{0,\frac{45t}{2}-1,27t-1,34t-1\}$

$h=2$, $x=1$ and $t$ is odd
$\{0,1,3t-2,15t\}$   $\{0,2,7t+3,37t+11\}$

$\{0,3t-1,18t+1,47t+7\}$   $\{0,\frac{7t+1}{2},13t+2,59t+13\}$

$\{0,4t,19t+4,52t+10\}$   $\{0,4t+1,25t+5,52t+9\}$

$\{0,7t,\frac{71t+13}{2},53t+8\}$   $\{0,7t+4,\frac{61t+15}{2},55t+13\}$

$\{0,\frac{15t+7}{2},19t+5,61t+14\}$   $\{0,8t+3,32t+6,36t+8\}$

$\{0,14t,19t+3,42t+7\}$   $\{0,15t+3,18t+4,33t+5\}$

$h=2$, $x=1$ and $t$ is even
$\{0,1,13t+2,17t+2\}$   $\{0,\frac{3t}{2},3t+1,18t+1\}$
$\{0,\frac{7t}{2}+1,48t+11,51t+10\}$   $\{0,4t+1,24t+6,61t+14\}$

$\{0,6t+2,24t+4,50t+11\}$   $\{0,6t+3,30t+8,53t+10\}$
$\{0,\frac{15t}{2}+3,20t+4,57t+15\}$   $\{0,8t+3,35t+7,47t+9\}$

$\{0,\frac{19t}{2}+4,\frac{47t}{2}+4,\frac{61t}{2}+7\}$   $\{0,15t+2,30t+6,45t+9\}$
$\{0,23t+3,28t+4,28t+6\}$   $\{0,29t+6,32t+4,36t+6\}$

$h=2$, $x=2$ and $t$ is odd
$\{0,1,8t+8,\frac{19t+13}{2}\}$   $\{0,2,3t,35t+10\}$

$\{0,\frac{7t+5}{2},\frac{61t+23}{2},\frac{87t+31}{2}\}$   $\{0,4t+2,28t+11,56t+21\}$

$\{0,5t+2,8t+3,57t+20\}$   $\{0,5t+4,41t+15,44t+17\}$
$\{0,7t+3,24t+10,37t+15\}$   $\{0,8t+4,25t+10,37t+14\}$

$\{0,8t+6,11t+3,23t+8\}$   $\{0,12t+2,17t+5,34t+9\}$
$\{0,\frac{35t+13}{2},30t+11,53t+18\}$   $\{0,18t+6,33t+11,36t+14\}$

$\{0,\frac{37t+15}{2},26t+10,\frac{99t+39}{2}\}$   $\{0,22t+6,28t+8,47t+17\}$

$h=2$, $x=2$ and $t$ is even
$\{0,1,8t+8,\frac{19t}{2}+8\}$   $\{0,\frac{3t}{2}+1,6t+3,44t+18\}$
$\{0,3t-2,22t+5,47t+15\}$   $\{0,3t-1,7t+2,15t+4\}$

$\{0,3t,27t+9,44t+15\}$   $\{0,3t+1,\frac{31t}{2}+5,39t+15\}$
$\{0,5t+3,18t+7,42t+17\}$   $\{0,6t+4,\frac{57t}{2}+10,40t+12\}$

$\{0,7t+1,25t+9,30t+11\}$   $\{0,\frac{15t}{2}+2,\frac{55t}{2}+8,\frac{71t}{2}+12\}$
$\{0,\frac{19t}{2}+6,\frac{49t}{2}+9,\frac{101t}{2}+20\}$   $\{0,12t+2,12t+4,49t+20\}$

$\{0,13t+5,27t+8,\frac{61t}{2}+11\}$   $\{0,16t+5,19t+8,49t+18\}$

$h=2$, $x=3$ and $t$ is odd
$\{0,1,\frac{3t-1}{2},64t+29\}$   $\{0,\frac{3t+1}{2},25t+14,56t+29\}$
$\{0,4t+3,32t+16,69t+39\}$   $\{0,\frac{9t+7}{2},\frac{15t+11}{2},58t+32\}$

$\{0,5t+3,34t+17,42t+24\}$   $\{0,5t+4,21t+11,52t+25\}$
$\{0,7t+6,24t+15,54t+28\}$   $\{0,8t+5,27t+14,32t+19\}$

$\{0,8t+6,12t+8,31t+20\}$   $\{0,8t+8,19t+14,69t+41\}$
$\{0,\frac{25t+15}{2},\frac{55t+27}{2},\frac{95t+47}{2}\}$   $\{0,13t+7,31t+16,44t+24\}$

$\{0,21t+9,33t+15,36t+16\}$   $\{0,23t+11,23t+13,46t+23\}$

$h=2$, $x=3$ and $t$ is even
$\{0,1,8t+8,8t+10\}$   $\{0,\frac{3t}{2}+1,31t+14,\frac{71t}{2}+17\}$    $\{0,\frac{3t}{2}+2,\frac{19t}{2}+8,57t+32\}$

$\{0,3t-3,\frac{25t}{2}+7,35t+17\}$   $\{0,3t-1,23t+11,30t+13\}$

$\{0,3t+2,21t+11,44t+24\}$   $\{0,\frac{7t}{2}+3,15t+8,21t+12\}$

$\{0,5t+5,\frac{47t}{2}+14,52t+27\}$   $\{0,6t+5,\frac{15t}{2}+5,\frac{113t}{2}+31\}$

$\{0,7t+3,36t+16,53t+24\}$   $\{0,7t+4,19t+11,22t+11\}$

$\{0,13t+8,44t+25,47t+23\}$   $\{0,14t+6,35t+15,47t+21\}$

$\{0,14t+7,17t+10,56t+31\}$    $\{0,19t+10,24t+14,65t+32\}$

$h=2$, $x=4$ and $t$ is odd
$\{0,\frac{3t-3}{2},14t+10,44t+31\}$   $\{0,\frac{3t+1}{2},23t+16,56t+39\}$
$\{0,3t,35t+23,68t+45\}$   $\{0,3t+1,\frac{89t+61}{2},54t+38\}$

$\{0,4t,23t+14,46t+32\}$   $\{0,4t+1,12t+11,24t+18\}$
$\{0,6t+6,37t+28,37t+29\}$   $\{0,8t+7,\frac{23t+15}{2},55t+38\}$

$\{0,\frac{19t+13}{2},\frac{47t+35}{2},\frac{113t+77}{2}\}$   $\{0,12t+9,27t+19,\frac{73t+55}{2}\}$
$\{0,15t+9,23t+17,34t+22\}$   $\{0,15t+11,\frac{33t+21}{2},21t+16\}$

$\{0,17t+10,32t+22,36t+28\}$   $\{0,20t+12,27t+18,31t+20\}$

$h=2$, $x=4$ and $t$ is even
$\{0,1,\frac{19t}{2}+10,39t+27\}$   $\{0,2,3t+1,45t+33\}$
$\{0,\frac{7t}{2}+1,\frac{19t}{2}+8,\frac{89t}{2}+31\}$   $\{0,4t,28t+19,36t+24\}$

$\{0,4t+1,8t+7,12t+9\}$   $\{0,4t+5,15t+12,68t+46\}$
$\{0,7t+5,12t+12,61t+44\}$   $\{0,7t+6,31t+23,66t+44\}$

$\{0,8t+10,11t+8,49t+36\}$   $\{0,11t+5,30t+19,60t+39\}$
$\{0,14t+10,26t+20,47t+31\}$   $\{0,15t+10,19t+13,46t+31\}$

$\{0,\frac{33t}{2}+10,18t+12,\frac{109t}{2}+39\}$   $\{0,17t+10,36t+22,41t+28\}$

$h=2$, $x=5$ and $t$ is odd
$\{0,1,3t-2,15t+12\}$   $\{0,\frac{3t-3}{2},\frac{19t+17}{2},\frac{141t+123}{2}\}$
$\{0,3t+1,11t+7,53t+44\}$   $\{0,4t+5,25t+22,39t+34\}$

$\{0,5t+4,\frac{55t+47}{2},58t+49\}$   $\{0,5t+5,8t+7,64t+53\}$
$\{0,\frac{15t+11}{2},\frac{23t+17}{2},\frac{125t+105}{2}\}$   $\{0,\frac{19t+15}{2},\frac{35t+31}{2},66t+57\}$

$\{0,12t+8,29t+24,64t+51\}$   $\{0,12t+9,\frac{49t+45}{2},\frac{57t+49}{2}\}$
$\{0,12t+12,28t+24,48t+40\}$   $\{0,15t+13,31t+28,42t+36\}$

$\{0,16t+13,19t+16,43t+37\}$   $\{0,16t+14,39t+35,46t+38\}$

$h=2$, $x=5$ and $t$ is even
$\{0,1,3t-2,3t\}$   $\{0,3t+1,23t+19,56t+47\}$
$\{0,4t+5,12t+12,31t+29\}$   $\{0,5t+5,12t+10,50t+45\}$

$\{0,6t+5,28t+24,59t+49\}$   $\{0,7t+4,11t+8,43t+38\}$
$\{0,8t+8,19t+15,56t+50\}$   $\{0,11t+6,41t+32,47t+38\}$

$\{0,11t+9,37t+34,58t+49\}$   $\{0,12t+9,29t+25,56t+48\}$
$\{0,\frac{31t}{2}+12,25t+23,28t+25\}$   $\{0,16t+13,\frac{35t}{2}+15,67t+58\}$

$\{0,\frac{33t}{2}+15,18t+15,26t+24\}$   $\{0,21t+18,24t+21,57t+51\}$

$h=3$, $x=0$ and $t$ is odd
$\{0,1,3t-2,11t-2\}$   $\{0,2,7t+1,19t+4\}$
$\{0,4t+1,21t+2,44t+3\}$   $\{0,4t+2,8t+1,46t+3\}$

$\{0,\frac{9t+3}{2},\frac{33t+3}{2},\frac{87t+3}{2}\}$   $\{0,5t+1,19t+3,30t+3\}$   	$\{0,5t+2,\frac{25t+5}{2},42t+3\}$
$\{0,6t,18t+2,37t+3\}$   $\{0,6t+1,\frac{35t+3}{2},\frac{125t+7}{2}\}$

$\{0,\frac{19t-3}{2},33t,59t+1\}$   $\{0,11t-1,28t+1,56t+3\}$

$\{0,15t+1,18t+1,57t+3\}$   $\{0,\frac{37t+3}{2},20t+2,\frac{95t+5}{2}\}$
$h=3$, $x=0$ and $t$ is even
$\{0,1,8t+1,11t-2\}$   $\{0,2,7t+1,19t+4\}$
$\{0,3t-2,7t,59t\}$   $\{0,3t-1,11t+1,36t+2\}$

$\{0,3t,\frac{55t}{2},\frac{141t}{2}+2\}$,   $\{0,4t+1,18t+2,45t+2\}$
$\{0,6t+1,28t+2,45t+4\}$   $\{0,6t+2,\frac{19t}{2}+2,\frac{113t}{2}+3\}$

$\{0,\frac{15t}{2},\frac{45t}{2},\frac{119t}{2}+1\}$   $\{0,\frac{19t}{2}+1,19t+1,\frac{111t}{2}+2\}$
$\{0,12t,35t,51t+2\}$   $\{0,14t+2,18t+1,31t+3\}$

$h=3$, $x=1$ and $t$ is odd
$\{0,1,3t-2,11t+3\}$   $\{0,\frac{3t-3}{2},\frac{45t+5}{2},\frac{119t+23}{2}\}$

$\{0,3t-1,7t,52t+10\}$   $\{0,4t,\frac{43t+7}{2},31t+7\}$

$\{0,4t+2,27t+4,39t+7\}$   $\{0,\frac{9t+3}{2},\frac{23t+7}{2},\frac{73t+17}{2}\}$

$\{0,7t+1,13t+3,30t+7\}$   $\{0,\frac{15t+3}{2},\frac{31t+5}{2},\frac{47t+11}{2}\}$

$\{0,8t+4,19t+4,38t+9\}$   $\{0,13t+4,\frac{33t+9}{2},46t+10\}$

$\{0,16t+5,37t+8,49t+12\}$   $\{0,19t+3,\frac{57t+11}{2},47t+9\}$

$h=3$, $x=1$ and $t$ is even
$\{0,2,\frac{3t}{2}+2,\frac{19t}{2}+5\}$   $\{0,\frac{3t}{2}+1,43t+10,49t+12\}$
$\{0,5t+1,46t+8,54t+12\}$   $\{0,6t+3,\frac{19t}{2}+4,\frac{99t}{2}+13\}$

$\{0,7t+2,\frac{47t}{2}+6,53t+11\}$   $\{0,\frac{15t}{2}+1,25t+4,64t+14\}$
$\{0,11t+2,27t+4,67t+12\}$   $\{0,11t+3,42t+10,45t+8\}$

$\{0,11t+4,30t+7,45t+10\}$   $\{0,\frac{23t}{2}+3,33t+7,56t+12\}$
$\{0,22t+3,36t+6,49t+9\}$   $\{0,25t+6,25t+7,28t+6\}$

$h=3$, $x=2$ and $t$ is odd
$\{0,1,8t+8,\frac{19t+13}{2}\}$   $\{0,\frac{3t-1}{2},18t+6,38t+15\}$

$\{0,3t,26t+10,\frac{95t+35}{2}\}$   $\{0,3t+2,\frac{25t+13}{2},42t+16\}$

$\{0,\frac{7t+5}{2},\frac{37t+15}{2},\frac{87t+35}{2}\}$   $\{0,4t+2,11t+3,53t+20\}$
$\{0,7t+3,21t+8,25t+11\}$   $\{0,12t+4,43t+17,51t+20\}$

$\{0,15t+3,23t+7,59t+21\}$   $\{0,16t+5,22t+7,58t+23\}$
$\{0,\frac{45t+13}{2},27t+9,34t+11\}$   $\{0,23t+6,28t+10,31t+11\}$

$\{0,23t+8,\frac{61t+21}{2},54t+20\}$   $\{0,25t+9,37t+15,45t+17\}$

$h=3$, $x=2$ and $t$ is even
$\{0,1,8t+6,8t+8\}$   $\{0,\frac{3t}{2},\frac{35t}{2}+6,\frac{101t}{2}+19\}$
$\{0,3t-2,\frac{25t}{2}+6,35t+12\}$   $\{0,3t,19t+7,68t+24\}$

$\{0,3t+1,\frac{61t}{2}+10,67t+25\}$   $\{0,\frac{7t}{2}+3,\frac{19t}{2}+7,\frac{113t}{2}+22\}$
$\{0,8t+4,30t+11,49t+21\}$   $\{0,11t+3,36t+14,42t+17\}$

$\{0,11t+4,27t+9,68t+25\}$   $\{0,11t+5,23t+9,36t+15\}$
$\{0,14t+6,21t+7,58t+24\}$   $\{0,15t+4,18t+6,33t+11\}$

$\{0,17t+7,\frac{49t}{2}+9,53t+18\}$   $\{0,18t+8,41t+15,48t+17\}$

$h=3$, $x=3$ and $t$ is odd
$\{0,1,8t+10,\frac{19t+17}{2}\}$   $\{0,2,31t+15,64t+33\}$
$\{0,3t,23t+11,\frac{55t+29}{2}\}$   $\{0,3t+1,21t+10,26t+15\}$

$\{0,3t+3,36t+22,41t+25\}$   $\{0,\frac{7t+5}{2},11t+6,66t+35\}$
$\{0,7t+2,32t+19,35t+16\}$   $\{0,3t-2,34t+18,53t+27\}$

$\{0,8t+5,11t+7,31t+17\}$   $\{0,11t+5,\frac{59t+29}{2},47t+24\}$
$\{0,15t+8,19t+10,68t+36\}$   $\{0,16t+9,31t+16,55t+31\}$

$\{0,22t+9,35t+17,47t+25\}$   $\{0,22t+11,29t+14,37t+21\}$

$h=3$, $x=3$ and $t$ is even
$\{0,1,8t+10,\frac{19t}{2}+10\}$   $\{0,3t,27t+14,68t+36\}$    $\{0,3t+1,21t+10,52t+29\}$

$\{0,3t+3,8t+7,39t+20\}$   $\{0,4t+2,15t+8,72t+37\}$

$\{0,5t+5,19t+11,69t+41\}$   $\{0,7t+4,32t+20,\frac{99t}{2}+29\}$

$\{0,\frac{19t}{2}+8,\frac{89t}{2}+25,\frac{101t}{2}+29\}$   $\{0,11t+5,\frac{25t}{2}+7,17t+10\}$

$\{0,12t+8,15t+5,53t+27\}$   $\{0,13t+8,35t+18,60t+32\}$

$\{0,15t+9,38t+23,41t+25\}$   $\{0,20t+11,\frac{47t}{2}+14,59t+32\}$

$\{0,21t+9,29t+14,48t+24\}$    $\{0,\frac{57t}{2}+13,30t+14,47t+22\}$

$h=3$, $x=4$ and $t$ is odd
$\{0,\frac{3t-3}{2},3t-2,11t+7\}$   $\{0,3t+1,36t+24,39t+26\}$
$\{0,6t+4,\frac{37t+31}{2},43t+32\}$   $\{0,7t+4,33t+21,60t+39\}$

$\{0,7t+5,47t+34,51t+36\}$   $\{0,7t+6,12t+9,67t+46\}$
$\{0,8t+8,\frac{73t+55}{2},38t+28\}$   $\{0,\frac{19t+13}{2},14t+10,31t+22\}$

$\{0,\frac{19t+15}{2},21t+16,65t+48\}$   $\{0,11t+5,30t+22,56t+40\}$
$\{0,12t+10,15t+10,29t+21\}$   $\{0,13t+11,18t+15,72t+50\}$

$\{0,15t+11,23t+18,64t+45\}$   $\{0,\frac{31t+23}{2},\frac{85t+61}{2},\frac{97t+67}{2}\}$

$h=3$, $x=4$ and $t$ is even
$\{0,1,3t-2,53t+34\}$   $\{0,2,\frac{19t}{2}+10,69t+50\}$
$\{0,3t+2,21t+15,60t+42\}$   $\{0,\frac{9t}{2}+3,41t+30,49t+36\}$

$\{0,5t+4,31t+23,36t+28\}$   $\{0,7t+2,15t+10,49t+33\}$
$\{0,7t+6,11t+8,43t+32\}$   $\{0,\frac{15t}{2}+5,\frac{87t}{2}+32,55t+40\}$

$\{0,8t+5,35t+23,55t+39\}$   $\{0,11t+5,28t+19,60t+40\}$
$\{0,15t+9,\frac{33t}{2}+10,58t+40\}$   $\{0,17t+13,21t+16,28t+20\}$

$\{0,\frac{35t}{2}+13,\frac{47t}{2}+18,39t+29\}$   $\{0,\frac{45t}{2}+14,41t+29,\frac{85t}{2}+31\}$

$h=3$, $x=5$ and $t$ is odd
$\{0,1,\frac{3t-1}{2},45t+40\}$   $\{0,\frac{3t+1}{2},\frac{33t+29}{2},\frac{125t+111}{2}\}$
$\{0,3t+3,41t+35,57t+50\}$   $\{0,\frac{7t+5}{2},\frac{55t+49}{2},58t+50\}$

$\{0,4t+4,22t+21,61t+52\}$   $\{0,4t+5,12t+13,15t+11\}$
$\{0,6t+4,39t+34,56t+49\}$   $\{0,\frac{15t+9}{2},\frac{43t+33}{2},\frac{125t+107}{2}\}$

$\{0,8t+6,30t+26,49t+42\}$   $\{0,8t+10,19t+19,69t+66\}$
$\{0,12t+11,34t+30,41t+34\}$   $\{0,12t+14,19t+17,39t+35\}$

$\{0,\frac{25t+27}{2},31t+30,36t+34\}$   $\{0,13t+13,21t+17,66t+58\}$

$h=3$, $x=5$ and $t$ is even
$\{0,1,33t+29,41t+33\}$   $\{0,\frac{3t}{2},11t+11,45t+41\}$
$\{0,3t+1,35t+29,\frac{119t}{2}+50\}$   $\{0,3t+2,39t+32,52t+45\}$

$\{0,4t+4,31t+29,34t+29\}$   $\{0,4t+5,15t+15,64t+57\}$
$\{0,7t+4,26t+23,29t+22\}$   $\{0,7t+5,43t+37,48t+43\}$

$\{0,\frac{15t}{2}+4,17t+14,58t+50\}$   $\{0,8t+5,20t+17,25t+21\}$
$\{0,13t+14,32t+31,50t+47\}$   $\{0,15t+13,\frac{37t}{2}+16,54t+46\}$

$\{0,16t+15,27t+21,27t+23\}$   $\{0,17t+15,21t+17,29t+25\}$

$h=6$, $x=0$ and $t$ is odd
$\{0,1,8t+1,11t-2\}$   $\{0,2,\frac{19t+1}{2},\frac{141t+15}{2}\}$
$\{0,3t,\frac{43t+1}{2},\frac{119t+9}{2}\}$   $\{0,3t+1,22t+2,33t+3\}$

$\{0,\frac{7t+1}{2},\frac{57t+5}{2},\frac{73t+9}{2}\}$   $\{0,4t+1,\frac{23t+3}{2},59t+5\}$
$\{0,6t-1,\frac{47t+1}{2},54t+5\}$   $\{0,7t,19t+2,49t+5\}$   $\{0,7t+1,37t+5,57t+7\}$

$\{0,\frac{19t-1}{2},14t,59t+4\}$   $\{0,11t+2,28t+3,43t+4\}$

$\{0,21t+1,27t+1,57t+6\}$   $\{0,21t,26t,45t+3\}$

$h=6$, $x=0$ and $t$ is even
$\{0,1,3t-2,11t-2\}$   $\{0,2,27t+1,42t+3\}$
$\{0,\frac{3t}{2}+2,11t+2,24t+3\}$   $\{0,\frac{7t}{2}+1,\frac{37t}{2},\frac{113t}{2}+5\}$

$\{0,4t,7t-1,56t+4\}$   $\{0,4t+1,11t-1,39t+2\}$
$\{0,6t,37t+4,\frac{97t}{2}+5\}$   $\{0,7t+1,28t+2,60t+4\}$

$\{0,\frac{15t}{2},20t+1,\frac{83t}{2}+2\}$   $\{0,11,30t+1,66t+5\}$
$\{0,16t+1,33t+3,46t+5\}$   $\{0,18t+1,\frac{55t}{2}+2,\frac{109t}{2}+5\}$

$h=6$, $x=1$ and $t$ is odd
$\{0,1,5t+2,54t+15\}$   $\{0,\frac{3t-3}{2},\frac{19t+3}{2},\frac{83t+17}{2}\}$

$\{0,3t-2,8t+1,53t+12\}$   $\{0,3t+1,39t+12,42t+11\}$

$\{0,\frac{7t+1}{2},16t+4,68t+17\}$   $\{0,\frac{9t+3}{2},\frac{85t+25}{2},58t+16\}$

$\{0,7t+1,26t+5,56t+15\}$   $\{0,7t+2,\frac{49t+11}{2},34t+8\}$

$\{0,8t+5,30t+8,43t+12\}$   $\{0,11t,11t+2,15t+2\}$

$\{0,15t+3,26t+6,45t+9\}$   $\{0,17t+4,23t+6,48t+13\}$

$h=6$, $x=1$ and $t$ is even
$\{0,1,\frac{19t}{2}+5,39t+10\}$   $\{0,\frac{3t}{2},\frac{3t}{2}+2,\frac{25t}{2}+3\}$
$\{0,4t,49t+12,54t+14\}$   $\{0,6t+2,45t+14,48t+13\}$

$\{0,\frac{15t}{2}+1,11t+2,58t+15\}$   $\{0,8t+1,16t+4,30t+6\}$
$\{0,13t+4,30t+8,45t+10\}$   $\{0,\frac{31t}{2}+3,\frac{45t}{2}+4,\frac{83t}{2}+9\}$

$\{0,16t+3,21t+4,41t+9\}$   $\{0,17t+3,20t+4,25t+7\}$
$\{0,19t+6,22t+3,59t+15\}$   $\{0,\frac{49t}{2}+5,29t+6,36t+8\}$

$h=6$, $x=2$ and $t$ is odd
$\{0,1,3t-2,64t+23\}$   $\{0,2,35t+15,\frac{125t+51}{2}\}$

$\{0,3t+3,17t+6,56t+25\}$   $\{0,4t+2,\frac{33t+13}{2},18t+6\}$

$\{0,5t+2,16t+6,33t+13\}$   $\{0,5t+3,28t+12,39t+18\}$
$\{0,\frac{15t+9}{2},\frac{57t+23}{2},\frac{97t+41}{2}\}$   $\{0,\frac{19t+11}{2},\frac{49t+23}{2},\frac{101t+45}{2}\}$

$\{0,\frac{23t+11}{2},27t+11,34t+12\}$   $\{0,13t+4,20t+8,35t+11\}$
$\{0,18t+8,25t+11,\frac{109t+47}{2}\}$   $\{0,23t+7,30t+12,53t+20\}$

$\{0,25t+12,33t+16,48t+18\}$   $\{0,28t+10,32t+13,47t+17\}$

$h=6$, $x=2$ and $t$ is even
$\{0,1,15t+3,23t+7\}$   $\{0,2,3t-1,64t+24\}$
$\{0,3t-2,22t+7,28t+11\}$   $\{0,3t,33t+12,36t+14\}$

$\{0,3t+1,19t+8,67t+26\}$   $\{0,3t+3,\frac{49t}{2}+11,26t+13\}$
$\{0,4t+3,17t+7,57t+26\}$   $\{0,5t+2,29t+13,56t+24\}$

$\{0,5t+3,23t+9,39t+14\}$   $\{0,7t+1,14t+3,33t+13\}$
$\{0,\frac{15t}{2}+4,\frac{37t}{2}+7,\frac{97t}{2}+20\}$   $\{0,\frac{19t}{2}+7,25t+12,\frac{85t}{2}+18\}$

$\{0,11t+4,29t+11,54t+22\}$   $\{0,\frac{23t}{2}+5,24t+9,\frac{57t}{2}+11\}$

$h=6$, $x=3$ and $t$ is odd
$\{0,1,23t+11,64t+33\}$   $\{0,\frac{3t-3}{2},11t+5,11t+7\}$
$\{0,3t-3,15t+5,53t+30\}$   $\{0,3t-2,28t+16,36t+19\}$

$\{0,3t+1,16t+8,43t+25\}$   $\{0,4t+3,31t+19,39t+23\}$
$\{0,5t+5,25t+15,49t+30\}$   $\{0,6t+3,\frac{35t+19}{2},\frac{85t+51}{2}\}$

$\{0,7t+2,15t+8,\frac{113t+67}{2}\}$   $\{0,\frac{15t+7}{2},20t+11,\frac{47t+27}{2}\}$
$\{0,\frac{19t+15}{2},26t+17,\frac{101t+65}{2}\}$   $\{0,13t+8,37t+24,40t+23\}$

$\{0,14t+6,21t+9,35t+16\}$   $\{0,15t+7,18t+9,68t+40\}$

$h=6$, $x=3$ and $t$ is even
$\{0,1,41t+23,64t+33\}$   $\{0,2,3t-1,22+11\}$    $\{0,\frac{3t}{2},\frac{87t}{2}+26,56t+33\}$

$\{0,\frac{3t}{2}+1,\frac{45t}{2}+10,\frac{121t}{2}+36\}$   $\{0,3t+1,22t+10,35t+18\}$

$\{0,4t+3,7t+3,68t+38\}$   $\{0,5t+3,19t+10,35t+20\}$

$\{0,6t+4,24t+13,49t+29\}$   $\{0,6t+5,24t+15,51t+32\}$

$\{0,8t+3,23t+11,35t+19\}$   $\{0,8t+5,23t+12,55t+32\}$

$\{0,11t+5,36t+23,39t+21\}$   $\{0,15t+5,\frac{49t}{2}+15,52t+31\}$

$\{0,\frac{31t}{2}+8,25t+17,\frac{83t}{2}+26\}$    $\{0,23t+14,26t+17,57t+33\}$

$h=6$, $x=4$ and $t$ is odd
$\{0,1,34t+24,\frac{141t+109}{2}\}$   $\{0,2,37t+32,65t+51\}$
$\{0,3t-2,42t+29,60t+42\}$   $\{0,3t-1,15t+9,23t+16\}$

$\{0,3t,11t+5,\frac{107t+77}{2}\}$   $\{0,3t+2,11t+8,28t+22\}$
$\{0,5t+4,22t+15,61t+48\}$   $\{0,6t+4,\frac{47t+35}{2},\frac{125t+95}{2}\}$

$\{0,7t+2,26t+19,56t+42\}$   $\{0,7t+4,26t+20,30t+22\}$
$\{0,\frac{19t+17}{2},13t+11,\frac{49t+39}{2}\}$   $\{0,11t+9,19t+18,55t+42\}$

$\{0,12t+11,39t+29,44t+34\}$   $\{0,13t+12,18t+15,36t+29\}$

$h=6$, $x=4$ and $t$ is even
$\{0,1,34t+23,\frac{125t}{2}+45\}$   $\{0,\frac{3t}{2}+2,18t+13,\frac{137t}{2}+51\}$
$\{0,3t,21t+15,54t+38\}$   $\{0,3t+2,\frac{15t}{2}+5,35t+23\}$

$\{0,4t+2,11t+6,\frac{121t}{2}+46\}$   $\{0,4t+3,18t+14,48t+34\}$
$\{0,6t+4,\frac{37t}{2}+15,\frac{85t}{2}+34\}$   $\{0,6t+5,\frac{47t}{2}+18,\frac{61t}{2}+24\}$

$\{0,7t+5,30t+21,35t+24\}$   $\{0,8t+8,\frac{19t}{2}+8,11t+9\}$
$\{0,12t+10,12t+12,23t+17\}$   $\{0,13t+11,39t+32,55t+42\}$

$\{0,14t+10,31t+24,53t+39\}$   $\{0,15t+9,23t+18,59t+42\}$

$h=6$, $x=5$ and $t$ is odd
$\{0,1,\frac{3t-1}{2},52t+49\}$   $\{0,2,4t+4,22t+20\}$
$\{0,3t+1,8t+6,19t+16\}$   $\{0,3t+3,27t+24,51t+49\}$

$\{0,\frac{7t+5}{2},11t+7,53t+48\}$   $\{0,\frac{9t+9}{2},\frac{61t+57}{2},\frac{125t+117}{2}\}$
$\{0,6t+4,42t+40,56t+53\}$   $\{0,7t+4,35t+29,60t+53\}$

$\{0,8t+10,29t+26,65t+63\}$   $\{0,8t+11,16t+16,\frac{87t+81}{2}\}$
$\{0,12t+12,30t+27,59t+52\}$   $\{0,12t+14,15t+11,65t+61\}$

$\{0,\frac{25t+27}{2},24t+24,\frac{107t+99}{2}\}$   $\{0,\frac{31t+27}{2},\frac{47t+41}{2},33t+29\}$

$h=6$, $x=5$ and $t$ is even
$\{0,\frac{3t}{2},24t+20,30t+25\}$   $\{0,3t-3,3t-1,53t+47\}$
$\{0,3t+1,\frac{43t}{2}+17,34t+30\}$   $\{0,3t+2,8t+7,57t+52\}$

$\{0,\frac{7t}{2}+3,\frac{31t}{2}+13,\frac{97t}{2}+45\}$   $\{0,4t+2,30t+27,59t+53\}$
$\{0,7t+5,36t+30,60t+53\}$   $\{0,8t+8,\frac{19t}{2}+10,\frac{141t}{2}+65\}$

$\{0,8t+10,12t+14,31t+30\}$   $\{0,11t+7,22t+17,67t+60\}$
$\{0,16t+14,19t+17,41t+38\}$   $\{0,17t+16,22t+20,61t+58\}$

$\{0,21t+16,36t+29,48t+41\}$   $\{0,22t+19,30t+28,48t+45\}$

Cyclic group divisible designs are a kind of generalization of cyclic Steiner systems.

Let $X$ be a set of $v$ points and $\B$ be a set of $k$-subsets of $X$ called blocks. A pair $(X,\B)$ is called a $(v,k,1)$-design or a Steiner system S$(2,k,v)$ if every pair of distinct elements of $X$ is contained in exactly one block of $\B$. An automorphism of a $(v,k,1)$-design $(X,\B)$ is a permutation on $X$ leaving $\B$ invariant. A $(v,k,1)$-design is said to be cyclic if it admits an automorphism consisting of a cycle of length $v$.

Difference families have a considerable importance in combinatorics for the following reason:

[24]

$(1)$ Let $\F$ be a $(v,k,1)$-CDF. Then $\F$ forms the set of base blocks of a cyclic $(v,k,1)$-design.

$(2)$ Let $\F$ be a $(v,k,k,1)$-CDF. Then $\F\cup \{\{0,v/k,2v/k,\ldots,(k-1)v/k\}\}$ forms the set of base blocks of a cyclic $(v,k,1)$-design.

A $(v,k,1)$-CDF $\F_1$ can generate a cyclic $(v,k,1)$-design with the block set $\dev \F_1:=\{F+t:F\in \F_1,t\in \Z_v\}$, and a $(v,k,k,1)$-CDF $\F_2$ can generate a cyclic $(v,k,1)$-design with the block set $\F_2\cup \{\{0,v/k,2v/k,\ldots,(k-1)v/k\}\}$ (cf. [24]).

Let $\F$ be a $(v,k,1)$-CDF. Then $\F$ forms the set of base blocks of a cyclic $(v,k,1)$-design.
Let $\F$ be a $(v,k,k,1)$-CDF. Then $\F\cup \{\{0,v/k,2v/k,\ldots,(k-1)v/k\}\}$ forms the set of base blocks of a cyclic $(v,k,1)$-design.

A method for the direct construction of designs (the method of differences)
arises from the hypothetical investigation of the structure of a
Steiner system with respect to a given group of automorphisms. This technique was
first described and applied in 1939 in a classic paper of R. C. Bose [l]
(also see [6]). We do not go very deeply into this but only consider a
special case.

Let $p$ be a prime:
[26]
If $p=12t+1$ is a prime number, then there exists a $(5p,5,4,1)$ cyclic difference family.
4) [11]$p\equiv1\pmod{6}$, then a $(8p,8,4,1)$ cyclic difference family exists ;5) <cit.>$p\equiv1\pmod{4}$ and $p>5$, then a $(9p,9,4,1)$ cyclic difference family exists ;6) <cit.>$p\equiv1\pmod{4}$, then a $(27p,27,4,1)$ cyclic difference family exists ;7) <cit.>$p\equiv7\pmod{12}$, then a $(32p,32,4,1)$ cyclic difference family exists for $p>7$ ;8) <cit.>$p\equiv7\pmod{12}$. If $p\geq223$ and $p\neq271,379$, then a $(16p,16,4,1)$ cyclic difference family exists ;9) <cit.>$p\equiv7\pmod{12}$, then a $(64p,64,4,1)$ cyclic difference family exists for $p>7$.

[30]
There exist a $(120,24,4,1)$-CDF and a $(1944,24,4,1)$-CDF.

[30]
There exists a $(gn,g,4,1)$-CDF for any $(g,n)\in\{(12,6),(12,25),(24,9)\}$.