# Zero-field microwave-free magnetometry with ensembles of nitrogen-vacancy
centers in diamond

Omkar Dhungel  Helmholtz-Institut Mainz, GSI Helmholtzzentrum für
Schwerionenforschung GmbH, 55128 Mainz, Germany  Johannes Gutenberg-
Universität Mainz, 55128 Mainz, Germany  Till Lenz  Helmholtz-Institut Mainz,
GSI Helmholtzzentrum für Schwerionenforschung GmbH, 55128 Mainz, Germany
Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany  Muhib Omar
Helmholtz-Institut Mainz, GSI Helmholtzzentrum für Schwerionenforschung GmbH,
55128 Mainz, Germany  Johannes Gutenberg-Universität Mainz, 55128 Mainz,
Germany  Joseph Shaji Rebeirro  Helmholtz-Institut Mainz, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, 55128 Mainz, Germany  Johannes Gutenberg-
Universität Mainz, 55128 Mainz, Germany  Viktor Ivády  Department of Physics
of Complex Systems, ELTE Eötvös Loránd University, Egyetem tér 1-3, H-1053
Budapest, Hungary MTA-ELTE Lendület “Momentum” NewQubit Research Group,
Pázmány Péter, Sétány 1/A, 1117 Budapest, Hungary  Department of Physics,
Chemistry and Biology, Linköping University, 581 83 Linköping, Sweden  Adam
Gali  Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest,
Hungary  Budapest University of Technology and Economics, Institute of
Physics, Department of Atomic Physics, Műegyetem rakpart 3., 1111 Budapest,
Hungary  Arne Wickenbrock  Helmholtz-Institut Mainz, GSI Helmholtzzentrum für
Schwerionenforschung GmbH, 55128 Mainz, Germany  Johannes Gutenberg-
Universität Mainz, 55128 Mainz, Germany  Dmitry Budker<EMAIL_ADDRESS>Helmholtz-Institut Mainz, GSI Helmholtzzentrum für Schwerionenforschung GmbH,
55128 Mainz, Germany  Johannes Gutenberg-Universität Mainz, 55128 Mainz,
Germany  Department of Physics, University of California, Berkeley, California
94720-300, USA

###### Abstract

We study cross-relaxation features near zero magnetic field with ensembles of
nitrogen-vacancy (NV) centers in diamond and examine their properties in
samples with a range ($0.9$ ppm - $16.0$ ppm) of NV concentrations. The
observed NV-NV cross-relaxation features between differently oriented NV
centers in high ($\gtrsim 0.9$ ppm)-NV-density samples hold promise for a
variety of magnetometry applications where microwave fields (or any bias
field) disturb the system under study. We theoretically determine the values
of the bias magnetic fields corresponding to cross-relaxations between
different axes and experimentally validate them. The behavior of zero-field
cross-relaxation features as a function of temperature is also investigated.
The sensitivity of magnetometry based on this method is determined to be in
the 10 nT/$\sqrt{\textrm{Hz}}$ range.

##  I Introduction

Nitrogen-vacancy (NV) centers in diamond are used in applications as magnetic
field Rondin2014 , electric field dolde2011electric  and temperature sensors
doi:10.1021/nl401216y  over a wide range of environmental conditions, e.g.,
pressure and temperature CHOE20181066 ; PhysRevLett.112.047601 . In most
applications as magnetometers, the magnetic field is determined by using
optically detected magnetic resonance (ODMR) PhysRevB.42.8605 . In this
measurement protocol, the diamond is illuminated with green laser light and a
microwave field of tunable frequency is applied to measure the energy
separation between the magnetically-sensitive spin 1 ground-state levels to
obtain the magnetic field value. For measuring weak magnetic fields where
Zeeman shifts of sublevels may be nonlinear in field Rondin2014 , this method
usually requires a bias field to lift the degeneracy between the $m_{s}=\pm 1$
sublevels. However, the use of microwave and external bias magnetic fields may
disrupt the system of interest in some applications. To overcome this
limitation, different strategies emerged. Recently, microwave-free
magnetometry doi:10.1063/1.4960171 , as well as vector magnetometry
PhysRevApplied.13.044023  based on the level anti-crossing in the triplet
ground state at 102.4 mT have been demonstrated. Zero-field magnetometry was
realized for both ensemble and single NV centers by using circularly polarized
microwave fields to individually address transitions to the $m_{s}=+1$ or
$m_{s}=-1$ states PhysRevApplied.11.064068 ; lenz2021 . Still, due to the high
external magnetic field or the application of microwave fields, these
techniques may be problematic for studying systems where both external
magnetic field and the microwaves might disturb the system. Examples of such
systems are high-Tc superconductors (Tc stands for the superconducting
transition temperature) PhysRevApplied.10.034032 , zero- to ultra-low field
NMR BLANCHARD2021106886 , biological samples and various magnetic
materialssimpson2016magneto .

Recently, ODMR experiments with an additional radiofrequency field were used
to study features occurring in fluorescence measurements at zero and low
fields Holliday1989 ; dmitriev2019high ; PhysRevA.105.043509 . In addition,
microwave-free magnetometry at low field was proposed. The technique consists
in measuring the position of the cross-relaxation resonances
PhysRevA.94.021401 ; Anishchik2015 ; PhysRevB.40.6509  and allows for vector
magnetometry PhysRevA.96.013806 ; PhysRevA.100.043844 . This new magnetometry
technique therefore overcomes the limitations of microwave-based protocols.

In this work, we perform a detailed study of the cross-relaxation features
with respect to the sample cut, NV density, and the sample temperature. In
addition, we study the cross-relaxations under a transverse field upto
$\approx 2.0$ mT as a function of the azimuthal angle. We numerically predict
and experimentally verify the observed cross-relaxation patterns.

##  II NV-center ground state

The fluorescence rate is reduced due to the ${T_{1}}$ relaxation between
bright and dark states. The coupling mediated by dipolar coupling leads to
faster ${T_{1}}$ relaxation and such coupling is enhanced when NV states of
different axes are degenerate Anishchik2015 . In the following, we are
studying the Hamiltonian to determine at what external fields this might be
observable and compare it to the experiment. The NV ground state in the
presence of an external magnetic field can be described by the Hamiltonian:

$H=D(S_{z}^{2}-\frac{1}{3}\vec{S}^{2})+E(S_{x}^{2}-S_{y}^{2})+g_{s}\mu_{B}\vec{B}\cdot\vec{S}\,,$
(1)

where $D$= 2.87 GHz is the axial and $E$ is the transverse zero-field-
splitting (ZFS) parameters, $\vec{B}$ is the magnetic field, $\vec{S}$ is the
electronic spin with components $S_{x}$, $S_{y}$ and $S_{z}$. The electron $g$
factor is $g_{s}$ = 2.003 PhysRevB.79.075203  and $\mu_{B}$ is the Bohr
magneton. The Larmor frequency of the NV center is $\Omega$ =
$\frac{g_{s}\mu_{B}}{{\hbar}}\,B$. Here, the $z$-axis is chosen along the NV
axis. The spin operators $S_{x}$, $S_{y}$, $S_{z}$ can be written as:

$\begin{pmatrix}0&\frac{1}{{\sqrt{2}}}&0\\\
\frac{1}{{\sqrt{2}}}&0&\frac{1}{{\sqrt{2}}}\\\
0&\frac{1}{{\sqrt{2}}}&0\end{pmatrix},\begin{pmatrix}0&\frac{-i}{\sqrt{2}}&0\\\
\frac{i}{{\sqrt{2}}}&0&\frac{-i}{{\sqrt{2}}}\\\
0&\frac{1}{{\sqrt{2}}}&0\end{pmatrix},\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}\,.$ (2)

A unit vector with an angle of $\beta$ to the $z$ axis can be written as:

$\vec{n}=(\cos\phi\sin\beta,\sin\beta\sin\phi,\cos\beta),$ (3)

where $\phi$ is the azimuthal angle. Then eq. (1) can be written as:

$H=\begin{pmatrix}D+\Omega\cos\beta&\frac{e^{-i\phi}\Omega\sin\beta}{{\sqrt{2}}}&E\\\
\frac{e^{i\phi}\Omega\sin\beta}{\sqrt{2}}&0&\frac{e^{-i\phi}\Omega\sin\beta}{{\sqrt{2}}}\\\
E&\frac{e^{i\phi}\Omega\sin\beta}{\sqrt{2}}&D-\Omega\cos\beta\,\end{pmatrix}.$
(4)

A schematic of the energy levels obtained by diagonalization of the
Hamiltonian (4) is shown in Fig. 1. The ground state $\ket{g}$ is split by the
dipole-dipole interaction described by the parameter $D$, lifting the
degeneracy between the $m_{s}$= $\pm$1 and $m_{s}$= 0 sublevels. Application
of a magnetic field along NV-axis further lifts the $m_{s}=\pm 1$ degeneracy
level by 2$\gamma_{e}B_{\textrm{NV}}$ above the $m_{s}$= -1 when
$B_{\textrm{NV}}$ is applied.

Diamond Name | Cut | [N] density (ppm) | NV density (ppm) | Laser intensity (mW) |  $\gamma$ (mT) | Optimum C (%) |  $\gamma$ (mT)/ C (%)
---|---|---|---|---|---|---|---
Super diamond | 111 |  $\sim$3 | 0.9 | 5 | 0.12 | 0.04 | 3.00
George | 100 |  $\sim$13 | 3.7 | 5.93 | 0.20 | 0.91 | 0.21
1970608-29 | 111 | - | 3.8 | 2.98 | 0.78 | 2.85 | 0.27
S2 | 111 |  $<$100 | 16.0 | 20.2 | 1.42 | 4.01 | 0.35
sumi_300 kev* | 110 |  $\sim$150 | 14.0 | 37.00 | 1.42 | 1.21 | 1.17
2170612-13* | 100 |  $\sim$14 | 4.5 | 37.00 | 0.17 | 0.29 | 0.59
Table 1: Characterization of the zero-field feature in different diamond
samples where $B_{z}$ is along the cut surface. Here, $\gamma$ and C are the
linewidth and contrast, respectively. The samples with * in their names are
thin-layer samples. Figure 1: (a) Energy-level diagram of the NV center in the
presence of magnetic field. (b) Energy separation (transition energy) between
the ground $m_{s}=0$ and $\pm 1$ states for all four crystallographic axes as
a function of $B_{z}$ when a 2 mT transverse field is applied along $\hat{x}$.
Blue stars are the positions where the crossings happen between transition
energies for different NV orientations. Fifteen transition-energy crossing
between different orientations are expected in total.

Consider the following four possible crystallographic axes of diamond lattice
for NV centers: [111], [1$\overline{1}\overline{1}$],
[$\overline{1}1\overline{1}$] and [$\overline{1}\overline{1}$1]. In the
diamond samples used in this work, the NV centers are uniformly distributed
over the possible orientations. Since NVs of all orientations have equal
ground state splitting, the transition energies of all orientations cross at
$\vec{B}=0$. When the splittings of orientationally inequivalent NV centers
match, a cross-relaxation feature occurs and we observe a decrease in
fluorescence. A recently developed model PhysRevLett.118.093601  suggests that
local energy relaxation occurs when a randomly distributed portion of NV
centers rapidly incoherently depolarizes. Through dipolar interactions, these
spins can depolarize the entire ensemble at zero-field. It was also observed
that there is a significant contribution of local electric fields and the
interaction between same and differently oriented NVs on depolarization at
zero-field for high-density samples pellet2022spin .

Let us concentrate on the ground $m_{s}$= $\pm 1$ states. Only $m_{s}$= $\pm
1$ states are taken into account since these are the states for which the
transition energies to $m_{s}=0$ intersect/cross at zero-field. The ground
state splitting for transition energies of different axes differs when a
transverse field is present. As a result, there are multiple crossings between
the transition energies at various values of $B_{z}$. The magnitude of the
transverse field and the azimuth angle $\phi$ affect positions of crossings.
When the field is scanned along [111] in the presence of 2 mT of transverse
field along $\hat{x}$ (this direction is defined so that one of the carbons
associated with the NV center is in the x-z plane), the transition energies
behave as shown in Fig. 1 (b) for ground $\pm 1$ states for all possible NV
axes. There are fifteen transition-energy crossings, however some of them
occur exactly at the same $B_{z}$ such that only five cross-relaxation
features are expected. Note that for the calculations shown in Fig. 1 (for 2
mT of transverse field), we neglected the parameter $E$. While in general, up
to six crossing positions can be observed when the [111] axis of the diamond
is aligned with the z-axis of the setup.

##  III Experimental setup

To study the cross-relaxation features, we use a home-built wide-field
fluorescence microscope (Fig. 2), where a laser (Toptica iBeam smart) with the
output wavelength of 515 nm is employed. The diamond is mounted on a
rotational mount, and the laser beam is reflected via a dichroic mirror. The
beam is then focused into the diamond using a microscope objective (Olympus
PLN Plan Achromat 10x Microscope Objective, 0.25 NA, 10.6 mm WD). The NV-
center fluorescence is collected using the same objective, passing the
dichroic mirror and a long-pass filter to remove the green reflection of the
laser light from diamond. The fluorescence is collected by utilizing lenses of
various focal lengths and detected with a photo-diode (APD120A/M - Si
Avalanche Photodetector, 400 - 1000 nm). Electromagnetic coils are used to
apply magnetic field in three different directions.

Figure 2: Schematic diagram of experimental setup.  Figure 3: Zero-fields
features recorded on two different diamond samples while magnetic field along
$\hat{z}$. For sample S2 (16 ppm NV density), full width at half maximum
(FWHM) and the contrast are 1.42(3) mT and 3.95(4) %, respectively. For sample
1970608-29 (3.8 ppm NV density), these are 0.78(4) mT and 1.79(5) %.

##  IV Results

###  IV.1 Characterization of zero-field feature

We explored the properties of the cross-relaxation feature at zero field in
different diamond samples with varying NV density. During these experiments
neither microwave nor radiofrequency fields were applied. We only observed the
cross-relaxation feature at zero-field in diamonds with NV density $\gtrsim
0.9$ ppm. Moreover, linewidth and contrast of the zero-field feature depends
on the NV-center density. In the examples in Fig. 3, both linewidth and
contrast increase with NV-center density. The contrast of the feature also
depends on the laser intensity. Table 1 summarizes the characteristics of the
observed zero-field feature in different samples. Characterization of the the
zero-field feature was performed with a laser spot diameter of 50 $\mu$m. For
the bulk NVs, the optimum contrast is observed at different values of laser
intensity for each sample. Higher laser intensity is used in the case of NV
layers because the signal-to-noise ratio for layer samples is low at reduced
power and the contrast increases with laser intensity. The NV density of the
Sumi_300 kev sample is estimated based on a comparative fluorescence
measurement with respect to the 2170612-13 sample. The linewidth and contrast
are slightly position dependent because of the variation of NV density over
the sample. The lowest (i.e, most favorable) ratio of linewidth and contrast
is observed for 3.7 ppm-NV-concentration bulk sample (“George”).

Figure 4: Simulation and experimental measurement of the cross-relaxation
features in the presence of a transverse field of 2 mT. (a) simulation, (b)
measurement of the transition energy difference between $m_{s}$= +1 and -1
states for all four NV axes. Here $B_{z}$ is along [111] and the azimuthal
angle of the transverse field is $\phi$. (c), (d) Varying the transverse field
and $B_{z}$ at an angle of 5∘. (e), (f) Transition energy difference between
$m_{s}$= +1 and -1 states when $B_{z}$ is along [100] at varying angles when
16 mT of transverse field is applied.

###  IV.2 NV-NV cross-relaxation in the presence of a transverse field

This section describes calculations and experimentally measured additional
cross-relaxation features observed when $B_{z}$ is along [111] (and [100]
later on) when applying 2 mT (1.6 mT for [100]) transverse field.

Figure 4(a) shows a density plot of the simulated contrast based on the
numerically evaluated transition frequency differences as a function of
$B_{z}$ along [111] and the azimuthal angle of the transverse magnetic field
(2 mT). Here, we assume that the cross-relaxation features have a Lorentzian
line shape. The parameters of the Lorentzian function are adjusted to reflect
the measured data. There are multiple crossings between the transition
energies for differently oriented NVs at different values of $B_{z}$. The
position of the crossings and the overlap between the transition energies
depend on $\phi$ when the diamond is rotated around $\hat{z}$ with a
transverse field present. When $B_{z}$ is along [111] (Fig. 4 (a)), for every
multiple of 60∘, starting at 30∘, transitions for NVs along two of three non-
aligned crystallographic axes overlap each another. These are the orientations
where the crossings between the transition energies at the three values of
$B_{z}$ occur. Also, two out of three inequivalent transition energies
successively cross each other at $B_{z}$ = 0 for every multiple of 60∘,
starting from 0∘. For each of these values of $\phi$, crossings occur at five
distinct values of $B_{z}$, Fig. 1 (b) shows one of those cases when $\phi$ =
0∘.

Figure 4 (b) shows experimentally measured cross-relaxation patterns between
NVs oriented along different axes. For every multiple of 60∘, starting from
30∘ there is an overlap of three separate cross-relaxation features at -1.4 mT
and +1.4 mT and of two separate features at + 0.7 mT and - 0.7 mT. As a
result, cross-relaxation features are observed at only three values of $B_{z}$
at these angles. For multiples of 60∘ starting from 0∘, there is an overlap of
two separate features at $B_{z}$ = 0 so that crossings are observed at these
angles. There are always six cross-relaxation values of $B_{z}$ for the rest
of the angles but these values depend on the angle $\phi$. The observed cross-
relaxation features when we apply transverse field have narrower linewidth
than the zero-field feature, which can be attributed to the effect of electric
field, strain, and additional cross-relaxation channels in the sample. We also
note the different widths of the lines in the figure. This is related to the
difference in the derivatives of the crossing transition energies as a
function of $B_{z}$. Note also that one can estimate the direction of the
magnetic field from the positions of the crossings, which is useful for vector
magnetometry applications PhysRevA.100.043844 .

Figure 5: Dominant central cross-relaxation feature observed while applying
1.6 mT of transverse field at an azimuthal angle of $\approx$ 90∘. The FWHM
and the contrast for this feature are 0.20(2) mT and 1.85(10)%, respectively.

When $B_{z}$ is scanned for each value of the transverse field, the splitting
of the cross-relaxation features linearly depends upon the transverse field
i.e. all of these cross-relaxation features are magnetically sensitive. (An
exception are crossings occurring at $B_{z}$ = 0 for certain values of the
azimuthal angle.) Figure 4(c), (d) shows the simulated and experimentally
measured cross-relaxation features for $\phi\approx$ 5∘ angle. It shows the
linear dependence on the transverse magnetic field. A limitation of these
extra cross-relaxation features (other than the zero-field feature) for
magnetometry applications requiring operation near-zero field is that
application of a 0.5-1.0 mT transverse field is necessary to clearly resolve
these features.

Similarly, in the case of $B_{z}$ along [100] (Fig. 4 (e)), due to the crystal
symmetry, transverse fields are equal for two groups of NVs that have pairs of
transition energies crossing each other at $B_{z}$ = 0 for every multiple of
90∘, starting at 0∘. Since, these two pairs of transition energies intersect
at $B_{z}$ = 0 there is just one crossing between the transition energies at
$B_{z}$ = 0. Figure 5 shows the dominant single cross-relaxation feature at
one of such angles, $\approx$ 90∘. At this angle, the contrast is larger than
that of the zero-field feature of the same sample (“George”) without
transverse field. Other small features are still observed, likely because of a
small misalignment with respect to the transverse field. Additionally, two out
of every four transition energies successively overlap at every multiple of
45∘, resulting in crossings at three different values of $B_{z}$. Figure 4(e)
shows the density plot of transition-frequency differences as a function of
$B_{z}$ along [100] and the azimuthal angle of the transverse magnetic field
at 1.6 mT. Figure 4 (f) shows the experimentally measured cross-relaxations
features. Since the transition energies for each angle cross at $B_{z}$ = 0 a
cross-relaxation feature is always visible at zero-field. There are five
cross-relaxation positions for the remaining angles. In our setup, while
rotating the diamond, there is a slight translation. Therefore, if we compare
the experimentally measured data in Fig. 4(f) with the calculations in Fig.
4(e), we see a slight displacement of the features which, however, does not
obscure the overall good agreement.

The cross-relaxation features in the presence of a transverse field are
measured on diamond sample “S2” for [111] and on sample “George” for [100]
(see Tab. 1).

The photon shot noise limited sensitivity is approximated as
$\eta\approx\Delta B/C/\sqrt{I_{0}}$, where $\Delta B$ = 0.20 mT is the
linewidth and $C$ = 1.85% is the contrast for the narrowest linewidth we
observed at $\phi\,=90^{\circ}$ in the presence of a 1.6 mT transverse field
for the “George” sample. In the present work, we did not fully optimize the
sensitivity as a function of light power and instead estimated the sensitivity
for a benchmark value of the photon detection rate (achieved with a few
milliwatt of green light power on the diamond) of $I_{0}$ = 6
$\times\,10^{12}$ /s. For these values, the estimated photon shot noise
limited sensitivity is around 4.5 nT/$\sqrt{\textrm{Hz}}$ with a spot diameter
on the diamond of $\approx 50\,\mu$m, which is sufficient to study
superconducting vortices and magnetic properties of magnetic materials
PhysRevApplied.10.034032 ; PhysRevApplied.15.024040 . We estimate that the
sensitivity can be improved by at least an order of magnitude by optimizing
the light power and improving the light-collection efficiency
RevModPhys.92.015004 .

Figure 6: Temperature dependence of zero-field feature. a) zero-field feature
at selected temperatures for sample S2. zero-field feature only start
observing at $\approx$ 20 K. b), c) linewidth and contrast dependence on
temperature for sample S2 and 1970608-29. The features are Lorentzian fitted
with 95 % confidence interval for each temperature to extract the linewidth
and contrast.

###  IV.3 Temperature dependence of zero-field feature

We observed that the sample temperature affects the contrast of the zero-field
feature. The temperature dependence was investigated on two samples cut along
(111) with the results shown in Fig. 6.

There are several effects that are of note. The zero-field signal is only
observed above $\approx$ 20 K. The contrast increases with temperature up to
60 K. Above 60 K, the contrast saturates and then declines above $\approx 250$
K, retaining a sizable value at room temperature (Fig. 6 b). The highest
contrast is observed in the range of 180-200 K. The linewidth is roughly
constant over the entire temperature range (Fig. 6 c). The high-temperature
behavior is likely due to the increase of the longitudinal relaxation rate
with temperature.

In order to understand the observed temperature dependence of the contrast of
the zero-field feature at cryogenic temperatures, we need to identify the
factors affecting the contrast. To do so we first study the energy-level
structure of coupled NV centers, see Fig. 7(a). When hyperfine, strain and
electric-field-induced splittings of the spin states are neglected, the energy
levels of a many-NV system fall into branches with a ladder-like structure,
where the spacing of the steps is equal to the zero-field splitting (ZFS)
parameter $D$. It is important to note the degeneracy of the branches. The
lowest-lying $\left|0,0,…,0\right\rangle$ state is non-degenerate, even for NV
centers of different orientations. The higher-lying states are, however,
degenerate and include mixed $\left|0\right\rangle$ and $\left|\pm
1\right\rangle$ states, see Fig. 7(a) illustrating this for for three NV
centers. When all NV centers are completely polarized to $\ket{0}$, the many-
spin system populates the non-degenerate (lowest energy) state. Spin-
relaxation effects may connect the lower-lying state with-higher lying states;
however, such processes are suppressed by the large value of the ZFS compared
to the typical strength of dipole-dipole interactions. Drop of the NV
polarization, i.e., population of the higher-lying energy states, enables spin
flip-flop among the NV enters. However, the averaged probability of finding
the NV centers in $\left|0\right\rangle$ does not change even in this case.
Dipolar spin relaxation alone cannot account for the zero-field feature.

To understand depolarization in a dense NV ensemble, we need to utilize the
concept of spin-fluctuators as explained in Ref. PhysRevLett.118.093601 .
Spin-fluctuators are NV centers with a short lived ground-state electron spin,
through which a polarized, dipolar coupled NV bath can dissipate its energy
faster than through the conventional spin-lattice relaxation. At non-zero
magnetic field, only NV centers and fluctuators along the same axes couple. At
zero magnetic field, however, all NV centers couple to each other, which
increases the effective density of the NV centers and the spin-fluctuator
bath. In turn, this gives rise to the zero-field feature. The contrast of the
zero-field feature thus depends on the density of the NV centers as well as
the ratio of the short lived spin fluctuators and the “normal” long-lived NV
centers. Since the former is not temperature but sample dependent, the latter
as a possible temperature-dependent factor.

Figure 7: a) Energy levels of three coupled NV centers and b) temperature
dependence of the lifetime of the ${}^{1}E$ NV shelving state
thiering_theory_2018 ; PhysRevB.82.201202 .

As discussed in Ref. PhysRevLett.118.093601, , spin-fluctuators can be close
NV centers of different charge states, where electron (charge) tunneling
between the centers can shorten the spin-state lifetime in the ground state of
the negatively charged NV center. Therefore, the fluctuator per normal NV
center ratio and, correspondingly, the contrast of the zero-field feature
depend on the NV($0$)/NV($-$) ratio, which is temperature, excitation-power,
and sample dependent. Considering the applied 515 nm excitation, which is
energetic enough to ionize the NV center from the ${}^{1}E$ shelving state
(see, for example, Ref. Bockstedte2018, ), we attribute the temperature
dependence of the contrast to the temperature dependence of the lifetime of
the ${}^{1}E$ electronic state of the NV center. The ${}^{1}E$ electronic
state couples to the higher-lying ${}^{1}A_{1}$ state through the pseudo Jahn-
Teller effect and to the even higher lying ${}^{1}E^{\prime}$ state through
the dynamics Jahn-Teller effect thiering_theory_2018 . The first excited
vibronic level can be found 16 meV above the lowest vibronic level of
${}^{1}E$ state, which gives rise to the characteristic temperature
dependence, depicted for example for the lifetime of ${}^{1}E$ in Fig. 7 (b).
Boltzmann occupation of the excited vibronic levels significantly increases
beyond 20 K, which on one hand shortens the lifetime of the ${}^{1}E$ state,
see Fig. 7 (b), and on the other hand increases the photo-ionization rate from
${}^{1}E$ state towards the conduction band. We attribute the onset of the
zero-field contrast at around 20 K to the increasing ionization rate due to
the thermal occupation of the vibronic excited energy levels of the ${}^{1}E$
state (e.g., such an effect is discussed for silicon carbide divacancy centers
in Ref. Csore2022, ), whereas the shortening of the ${}^{1}E$ lifetime, i.e.,
intersystem crossing towards the ground state of NV($-$), will compete with
this photo-ionization process at elevated temperatures. Quantitative
simulation of the contrast requires additional information on the dynamics of
the spin fluctuators, which is the subject of further investigations.

From the study of temperature dependence and the above discussion we conclude
that the NV(0)/NV(-) ratio governs the temperature dependence of the zero-
field feature contrast. Sensitive dependence of this parameter on excitation
power and wavelength gives pathways to engineer the contrast of zero-field
feature without the need for microwave excitations.

###  IV.4 Conclusion

We have investigated the zero-field feature with respect to the NV density and
the axis of sample cut. The linewidth and the contrast of this feature
increases with NV density. In the presence of a transverse field, various
cross-relaxation features are detected due to the dipolar interaction between
differently oriented NVs providing depolarization channels. The number and
location of cross-relaxation features are determined by the diamond cut,
azimuthal angle and the strength of the transverse field. These cross-
relaxation features follow specific pattern (well reproduced by our
theoretical calculations) when the azimuthal angle is changed with respect to
the transverse field. These features have narrower linewidth than the zero-
field feature. Moreover, higher contrast is observed at certain angles where
there is only one decay channel available to depolarize. Finally, the
temperature dependence of the zero-field feature was studied. The feature is
only observed above 20 K. The contrast increases up-to 60 K, levels off, and
gradually decreases from about 250 K.

These results will be used in scalar- and vector-magnetometry applications,
forming the basis of a practical near-zero-field microwave-free or even all-
optical technique. With optimization of excitation-light intensity,
fluorescence-collection efficiency, and sensing volume, we expect to achieve
sensitivities better than 100 pT/$\sqrt{\rm{Hz}}$.

###  IV.5 Acknowledgement

We thank Junichi Isoya for helpful discussions and providing a sample. This
work was supported by the European Commission’s Horizon Europe Framework
Program under the Research and Innovation Action MUQUABIS GA no. 101070546, by
the German Federal Ministry of Education and Research (BMBF) within the
Quantumtechnologien program (Grant No. 13N15064 and Grant No. 13N16455), and
by the DAAD/JSPS 2021-23 cooperation grant No. 57569949. A.G. and V.I.
acknowledge support from the Ministry of Culture and Innovation and the
National Research, Development and Innovation Office within the Quantum
Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004). V.I. acknowledges support from the National
Research, Development, and Innovation Office of Hungary (NKFIH) (Grant No. FK
145395) and the Knut and Alice Wallenberg Foundation through WBSQD2 project
(Grant No. 2018.0071). A.G. acknowledges the Hungarian NKFIH grant No.
KKP129866 of the National Excellence Program of Quantum-coherent materials
project, the EU HE EIC Pathfinder project QuMicro (Grant No. 101046911), and
the QuantERA II project MAESTRO.

## References

  * (1) L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, “Magnetometry with nitrogen-vacancy defects in diamond,” _Reports on Progress in Physics_ , vol. 77, no. 5, p. 056503, may 2014. [Online]. Available: https://dx.doi.org/10.1088/0034-4885/77/5/056503
  * (2) F. Dolde, H. Fedder, M. W. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. Hollenberg, F. Jelezko _et al._ , “Electric-field sensing using single diamond spins,” _Nature Physics_ , vol. 7, no. 6, pp. 459–463, 2011. [Online]. Available: https://rdcu.be/c3wnW
  * (3) P. Neumann, I. Jakobi, F. Dolde, C. Burk, R. Reuter, G. Waldherr, J. Honert, T. Wolf, A. Brunner, J. H. Shim, D. Suter, H. Sumiya, J. Isoya, and J. Wrachtrup, “High-precision nanoscale temperature sensing using single defects in diamond,” _Nano Letters_ , vol. 13, no. 6, pp. 2738–2742, 2013, pMID: 23721106. [Online]. Available: https://doi.org/10.1021/nl401216y
  * (4) S. Choe, J. Yoon, M. Lee, J. Oh, D. Lee, H. Kang, C.-H. Lee, and D. Lee, “Precise temperature sensing with nanoscale thermal sensors based on diamond nv centers,” _Current Applied Physics_ , vol. 18, no. 9, pp. 1066–1070, 2018\. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1567173918301573
  * (5) M. W. Doherty, V. V. Struzhkin, D. A. Simpson, L. P. McGuinness, Y. Meng, A. Stacey, T. J. Karle, R. J. Hemley, N. B. Manson, L. C. L. Hollenberg, and S. Prawer, “Electronic properties and metrology applications of the diamond ${\mathrm{nv}}^{-}$ center under pressure,” _Phys. Rev. Lett._ , vol. 112, p. 047601, Jan 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.112.047601
  * (6) E. van Oort, P. Stroomer, and M. Glasbeek, “Low-field optically detected magnetic resonance of a coupled triplet-doublet defect pair in diamond,” _Phys. Rev. B_ , vol. 42, pp. 8605–8608, Nov 1990. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.42.8605
  * (7) A. Wickenbrock, H. Zheng, L. Bougas, N. Leefer, S. Afach, A. Jarmola, V. M. Acosta, and D. Budker, “Microwave-free magnetometry with nitrogen-vacancy centers in diamond,” _Applied Physics Letters_ , vol. 109, no. 5, p. 053505, 2016. [Online]. Available: https://doi.org/10.1063/1.4960171
  * (8) H. Zheng, Z. Sun, G. Chatzidrosos, C. Zhang, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, J. Wrachtrup, A. Wickenbrock, and D. Budker, “Microwave-free vector magnetometry with nitrogen-vacancy centers along a single axis in diamond,” _Phys. Rev. Appl._ , vol. 13, p. 044023, Apr 2020\. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevApplied.13.044023
  * (9) H. Zheng, J. Xu, G. Z. Iwata, T. Lenz, J. Michl, B. Yavkin, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, J. Wrachtrup, A. Wickenbrock, and D. Budker, “Zero-field magnetometry based on nitrogen-vacancy ensembles in diamond,” _Phys. Rev. Appl._ , vol. 11, p. 064068, Jun 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevApplied.11.064068
  * (10) T. Lenz, A. Wickenbrock, F. Jelezko, G. Balasubramanian, and D. Budker, “Magnetic sensing at zero field with a single nitrogen-vacancy center,” _Quantum Science and Technology_ , vol. 6, no. 3, p. 034006, jun 2021. [Online]. Available: https://dx.doi.org/10.1088/2058-9565/abffbd
  * (11) Y. Schlussel, T. Lenz, D. Rohner, Y. Bar-Haim, L. Bougas, D. Groswasser, M. Kieschnick, E. Rozenberg, L. Thiel, A. Waxman, J. Meijer, P. Maletinsky, D. Budker, and R. Folman, “Wide-field imaging of superconductor vortices with electron spins in diamond,” _Phys. Rev. Appl._ , vol. 10, p. 034032, Sep 2018. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevApplied.10.034032
  * (12) J. W. Blanchard, D. Budker, and A. Trabesinger, “Lower than low: Perspectives on zero- to ultralow-field nuclear magnetic resonance,” _Journal of Magnetic Resonance_ , vol. 323, p. 106886, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1090780720302044
  * (13) D. A. Simpson, J.-P. Tetienne, J. M. McCoey, K. Ganesan, L. T. Hall, S. Petrou, R. E. Scholten, and L. C. Hollenberg, “Magneto-optical imaging of thin magnetic films using spins in diamond,” _Scientific reports_ , vol. 6, no. 1, pp. 1–8, 2016. [Online]. Available: https://rdcu.be/c3ws5
  * (14) K. Holliday, N. B. Manson, M. Glasbeek, and E. van Oort, “Optical hole-bleaching by level anti-crossing and cross relaxation in the n-v centre in diamond,” _Journal of Physics: Condensed Matter_ , vol. 1, no. 39, p. 7093, oct 1989. [Online]. Available: https://dx.doi.org/10.1088/0953-8984/1/39/021
  * (15) A. Dmitriev and A. Vershovskii, “High-contrast two-quantum optically detected resonances in nv centers in diamond in zero magnetic field,” _arXiv preprint arXiv:1902.10481_ , 2019.
  * (16) A. K. Dmitriev and A. K. Vershovskii, “Radio-frequency response of the optically detected level anticrossing signal in nitrogen-vacancy color centers in diamond in zero and weak magnetic fields,” _Phys. Rev. A_ , vol. 105, p. 043509, Apr 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.105.043509
  * (17) H. Clevenson, E. H. Chen, F. Dolde, C. Teale, D. Englund, and D. Braje, “Diamond-nitrogen-vacancy electronic and nuclear spin-state anticrossings under weak transverse magnetic fields,” _Phys. Rev. A_ , vol. 94, p. 021401, Aug 2016. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.94.021401
  * (18) S. V. Anishchik, V. G. Vins, A. P. Yelisseyev, N. N. Lukzen, N. L. Lavrik, and V. A. Bagryansky, “Low-field feature in the magnetic spectra of nv-centers in diamond,” _New Journal of Physics_ , vol. 17, no. 2, p. 023040, feb 2015\. [Online]. Available: https://dx.doi.org/10.1088/1367-2630/17/2/023040
  * (19) E. van Oort and M. Glasbeek, “Cross-relaxation dynamics of optically excited n-v centers in diamond,” _Phys. Rev. B_ , vol. 40, pp. 6509–6517, Oct 1989\. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.40.6509
  * (20) R. Akhmedzhanov, L. Gushchin, N. Nizov, V. Nizov, D. Sobgayda, I. Zelensky, and P. Hemmer, “Microwave-free magnetometry based on cross-relaxation resonances in diamond nitrogen-vacancy centers,” _Phys. Rev. A_ , vol. 96, p. 013806, Jul 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.96.013806
  * (21) R. Akhmedzhanov, L. Gushchin, N. Nizov, V. Nizov, D. Sobgayda, l. Zelensky, and P. Hemmer, “Magnetometry by cross-relaxation-resonance detection in ensembles of nitrogen-vacancy centers,” _Phys. Rev. A_ , vol. 100, p. 043844, Oct 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.100.043844
  * (22) S. Felton, A. M. Edmonds, M. E. Newton, P. M. Martineau, D. Fisher, D. J. Twitchen, and J. M. Baker, “Hyperfine interaction in the ground state of the negatively charged nitrogen vacancy center in diamond,” _Phys. Rev. B_ , vol. 79, p. 075203, Feb 2009. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.79.075203
  * (23) J. Choi, S. Choi, G. Kucsko, P. C. Maurer, B. J. Shields, H. Sumiya, S. Onoda, J. Isoya, E. Demler, F. Jelezko, N. Y. Yao, and M. D. Lukin, “Depolarization dynamics in a strongly interacting solid-state spin ensemble,” _Phys. Rev. Lett._ , vol. 118, p. 093601, Mar 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.118.093601
  * (24) C. Pellet-Mary, M. Perdriat, P. Huillery, and G. Hétet, “Spin-relaxation of dipolar-coupled nitrogen-vacancy centers: The role of double-flip processes,” _arXiv preprint arXiv:2207.13899_ , 2022.
  * (25) T. Lenz, G. Chatzidrosos, Z. Wang, L. Bougas, Y. Dumeige, A. Wickenbrock, N. Kerber, J. Zázvorka, F. Kammerbauer, M. Kläui, Z. Kazi, K.-M. C. Fu, K. M. Itoh, H. Watanabe, and D. Budker, “Imaging topological spin structures using light-polarization and magnetic microscopy,” _Phys. Rev. Appl._ , vol. 15, p. 024040, Feb 2021. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevApplied.15.024040
  * (26) J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, “Sensitivity optimization for nv-diamond magnetometry,” _Rev. Mod. Phys._ , vol. 92, p. 015004, Mar 2020. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.92.015004
  * (27) G. m. H. Thiering and A. Gali, “Theory of the optical spin-polarization loop of the nitrogen-vacancy center in diamond,” _Phys. Rev. B_ , vol. 98, p. 085207, Aug 2018. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.98.085207
  * (28) V. M. Acosta, A. Jarmola, E. Bauch, and D. Budker, “Optical properties of the nitrogen-vacancy singlet levels in diamond,” _Phys. Rev. B_ , vol. 82, p. 201202, Nov 2010. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.82.201202
  * (29) M. Bockstedte, F. Schütz, T. Garratt, V. Ivády, and A. Gali, “Ab initio description of highly correlated states in defects for realizing quantum bits,” _npj Quantum Materials_ , vol. 3, no. 1, p. 31, Jun. 2018. [Online]. Available: https://www.nature.com/articles/s41535-018-0103-6
  * (30) A. Csóré, I. G. Ivanov, N. T. Son, and A. Gali, “Fluorescence spectrum and charge state control of divacancy qubits via illumination at elevated temperatures in $4H$ silicon carbide,” _Physical Review B_ , vol. 105, no. 16, p. 165108, Apr. 2022, publisher: American Physical Society. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.105.165108


# On the Symmetric Difference Property in Difference Sets under Product
Construction

Andrew Clickard  Department of Mathematical and Digital Sciences, Bloomsburg
University, Bloomsburg, Pennsylvania 17815<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

A $(v,k,\lambda)$ symmetric design is said to have the symmetric difference
property (SDP) if the symmetric difference of any three blocks is either a
block or the complement of a block. Symmetric designs fulfilling this property
have the nice property of having minimal rank, which makes them interesting to
study. Thus, SDP designs become useful in coding theory applications. We show
in this paper that difference sets formed by direct product construction of
difference sets whose developments have the SDP also have the SDP. We also
establish a few results regarding isomorphisms in product constructed SDP
designs.

###### Key words and phrases:

symmetric design, design, difference sets, product construction, isomorphic
designs, semi-direct product, symmetric difference property, block design,
coding theory, algebraic coding theory

Thank you to Drs. James Davis, John Polhill, and Ken Smith for their counsel
and teaching.

##  1\. Introduction

Extremal error correcting codes, (codes whose parameters meet a bound) have
long been studied in the coding theory community, as codes that optimize the
minimal distance between codes allows for more errors to be corrected. One way
to generate these codes is to consider symmetric designs and their incidence
matrices. From this effort, the concept of the symmetric difference property
(SDP) was established; a property that minimizes the rank of the incidence
matrix. The properties and interactions that SDP designs hold are often
obfuscated by having to interact with extremely large matrices, and so there
is a great deal that is simply not known. This paper seeks to extend the
knowledge of designs with this property, particularly SDP designs coming from
difference sets in groups of order $2^{2n}$. The main results of this paper
are as follows:

  1. (1)

If $D_{1},D_{2}$ are difference sets in groups $G_{1},G_{2}$, then
$D=(D_{1}\times(G_{2}-D_{2}))\cup((G_{1}-D_{1})\times D_{2})$ has the SDP if
and only if $D_{1}$ and $D_{2}$ have the SDP. (Section 3, Theorem 1)

  2. (2)

Given symplectic difference sets $D_{1}$ and $D_{2}$ in groups $G_{1}$ and
$G_{2}$ and a homomorphism $\varphi:G_{2}\to Aut(G_{2})$, the product
construction $D$ of $D_{1}$ and $D_{2}$ is a symplectic difference set in
$G_{1}\rtimes_{\varphi}G_{2}$ if the automorphism induced by each generator of
$G_{2}$ under $\varphi$ fixes $D_{1}$. (Section 4, Theorem 3)

  3. (3)

If $D,D^{\prime}$ are the product construction of SDP difference sets
$D_{1},D_{2}$ and $D_{1}^{\prime},D_{2}^{\prime}$ in $G_{1}\times G_{2}$ and
$G_{1}^{\prime}\times G_{2}^{\prime}$ with $|G_{1}|=|G_{1}^{\prime}|$,
$|G_{2}|=|G_{2}^{\prime}|$, then the developments of $D$ and $D^{\prime}$ are
isomorphic if the developments of $D_{1}$ and $D_{1}^{\prime}$ are isomorphic
and the developments of $D_{2}$ and $D_{2}^{\prime}$ are isomorphic. (Section
5, Theorem 5)

The symmetric difference property being closed under direct product
construction allows for a whole new line of questioning: In what groups is it
possible to produce SDP difference sets from product construction? And more
generally, we can ask, given two difference sets with the SDP, what effects do
different semi-direct products have on the product construction of the sets?

##  2\. Preliminaries

Let us first lay out some notation and definitions to be used going forward.

######  Definition 1.

Let $A,B$ be sets. Then the operation $\Delta$ is defined by $A\Delta
B=(A-B)\cup(B-A)$ and is called the symmetric difference of $A$ and $B$.

The sets we want to take the symmetric differences of are the blocks of a
symmetric design, which is defined thusly:

######  Definition 2.

A $(v,k,\lambda)$ symmetric design is a set of points $P$ together with a set
of blocks $B$ such that there are $v$ points and $v$ blocks such that any
point is incident on $k$ blocks, any block is incident on $k$ points, any two
points share incidence on $\lambda$ blocks, and any two blocks share incidence
on $\lambda$ points.

Importantly for this paper, we may organize a symmetric design into what is
know as an incidence matrix, which is defined here:

######  Definition 3.

The incidence matrix of a $(v,k,\lambda)$ symmetric design with point set $P$
and block set $B$ is the $v\times v$ matrix with columns labelled by the
elements of $P$ and rows labelled by the elements of $B$, with the $ij$-th
entry being 1 if the $j$-th point is incident on the $i$-th block, and 0
otherwise.

Now, we are most interested in symmetric designs that come from groups, and in
particular, the subsets of groups known as difference sets:

######  Definition 4.

A $(v,k,\lambda)$ difference set is a set $D$ of order $k$ in a group $G$ of
order $v$ such that the multiset $\\{d_{1}d_{2}^{-1}\ |\ d_{1},d_{2}\in D\\}$
contains every non-identity element of the group $\lambda$ times. The
development of a difference set $D$ is the symmetric design with the elements
of $G$ as points, the left translates of $D$ by the elements of $G$ as blocks,
and the incidence relation defined by set inclusion.

And now, we may finally define the symmetric difference property in symmetric
difference sets.

######  Definition 5.

A $(v,k,\lambda)$ symmetric design has the symmetric difference property (SDP)
if the symmetric difference of any three blocks is either a block or the
complement of a block. For further motivation and reading regarding this
property, see [Kan75].

Note that if we consider the incidence matrix of a symmetric design, having
the SDP is equivalent to saying that the addition of three rows (addition here
defined as element-wise addition modulo 2) of the matrix is either a row or
the complement of a row. This is the definition to be used throughout the
remainder of the paper. The complement of the incidence matrix $A$ is defined
to be $A^{c}=A+J$, where $J$ is the all-ones matrix of the same size as $A$.

######  Definition 6.

If $D_{1}=(P_{1},B_{1}),\ D_{2}=(P_{2},B_{2})$ are two $(v,k,\lambda)$
symmetric designs, they are isomorphic if and only if there exists a bijection
$\varphi:P_{1}\to P_{2}$ that preserves incidence. Equivalently, let
$A_{1},A_{2}$ be the incidence matrices for $D_{1},D_{2}$, respectively. Then
$D_{1}$ and $D_{2}$ are isomorphic if and only if there exist permutation
matrices $P,Q$ such that $A_{1}=PA_{2}Q$.

One of the earliest examples of an error-correcting code is the Reed-Muller
code, which was first introduced in 1954 by the titular mathematicians Reed
and Muller in [Ree54] and [Mul54]. Our naming scheme for the basis elements of
the first-order Reed-Muller code $RM(1,m)$ is as follows. The first $m$ basis
elements have $2^{m-1}$ zeros and $2^{m-1}$ ones. The first basis element
$c_{1}$ has the full string of zeros followed by the full string of ones.
$c_{2}$ has a string of $2^{m-2}$ zeros, $2^{m-2}$ ones, and repeats that
pattern again. In general, the $i$-th basis element has $2^{i}$ alternating
strings of $2^{m-i}$ zeros and ones. The $(m+1)$-th basis word, $\mathds{1}$
is the all-ones word, and is denoted as such. As an illustrative example, the
basis words for $RM(1,4)$ are:

$\begin{matrix}c_{1}:&0000&0000&1111&1111\\\ c_{2}:&0000&1111&0000&1111\\\
c_{3}:&0011&0011&0011&0011\\\ c_{4}:&0101&0101&0101&0101\\\
\mathds{1}:&1111&1111&1111&1111\end{matrix}$

The Reed-Muller codes may also be defined recursively by $RM(1,m)=\\{\langle
u\ \ |\ v\rangle\ |\ u,v\in RM(1,m)\\}$, where $\langle u\ |\ v\rangle$
denotes the concatenation of strings, but the basis element definition becomes
a useful construction in the proof of Theorem 1.

Section 4 deals with semi-direct products of difference sets with developments
that are isomorphic to the symplectic design on $2^{2n}$ points. The
symplectic design on $2^{2n}$ points, first developed by Kantor in [Kan75], is
the design formed by iterative product construction of the trivial SDP in
$C_{2}^{2}$, and thus has incidence matrix $A$ defined by

$A=\frac{-1}{2}\bigg{(}\big{(}\underbrace{(J_{4}-2I_{4})\otimes(J_{4}-2I_{4})\otimes\cdots\otimes(J_{4}-2I_{4})}_{n\text{
times}}\big{)}-J_{4}\bigg{)},$

where $\otimes$ denotes the Kronecker product, $I_{4}$ is the $4\times 4$
identity matrix and $J_{4}$ is the $4\times 4$ all-ones matrix. The definition
for the semi-direct product of groups is as follows:

######  Definition 7.

Let $N$, $H$ be groups, $Aut(N)$ be the automorphism group of $N$, and
$\varphi:H\to Aut(N)$ a homomorphism. Then the semi-direct product
$N\rtimes_{\varphi}H$ of $N$ and $H$ by $\varphi$ is the group which has
underlying set $N\times H$ and operation defined by
$(n_{1},h_{1})(n_{2},h_{2})=(n_{1}\varphi(h_{1})(n_{2}),h_{1}h_{2})$.

The cyclic group of order $n$ is denoted by $C_{n}$, and is written
multiplicatively. The notation $C_{n}^{m}$ denotes the direct product of $m$
copies of $C_{n}$.

The last piece of background we need to address upon which this paper is based
is the product construction of difference sets. First, we must discuss doing
calculations in the group ring $\mathbb{Z}[G]$. There are two main ways of
representing difference sets in this group ring, each of which has its own set
of strengths and weaknesses. The one we use for illustration purposes with a
difference set $D$ in a group $G$ is defined by $D=\sum_{g\in G}(-1)^{j}g$,
where $j=0$ if $g\not\in D$ and $j=1$ otherwise. We also define
$D^{(-1)}=\sum_{g\in G}(-1)^{j}g^{-1}$, where $j=0$ if $g\not\in D$ and $j=1$
otherwise (hence, the element-wise inverses). From [Dil74], we have that if
$D$ is a difference set in $G$, where $G$ is a 2-group, then $DD^{(-1)}=|G|$.
Now, if $D_{1},D_{2}$ are difference sets in the groups $G_{1},G_{2}$, then
the set
$D=\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$ is a difference set in the group $G_{1}\times G_{2}$. In the
group ring $\mathbb{Z}[G_{1}\times G_{2}]$, we let $D=D_{1}D_{2}$, where
$D_{1}$ and $D_{2}$ are the sums as above over $G_{1}$ and $G_{2}$,
respectively. We then see that
$DD^{(-1)}=D_{1}D_{2}D_{1}^{(-1)}D_{2}^{(-1)}=|G_{1}||G_{2}|=|G_{1}\times
G_{2}|$, and thus $D$ is a difference set in $G_{1}\times G_{2}$ which,
written in standard set theory notation, is
$\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$. Since the underlying set is $G_{1}\times G_{2}$ for any semi-
direct product, this construction works for any homomorphism $\varphi:G_{2}\to
Aut(G_{1})$.

##  3\. Closure of the SDP under Direct Product Construction

We first begin with the closure of the SDP under direct product, followed by a
discussion of the effects of group “factoring” on the resulting product
construction. A very specific case of the following theorem is found in
[DHK21], which showed that the SDP is closed when product constructed with the
trivial difference set in $C_{4}$.

######  Theorem 1.

Let $G=G_{1}\times G_{2}$ be a group of order $2^{2n}$ with
$|G_{1}|=2^{v_{1}}$, $|G_{2}|=2^{v_{2}}$, with $v_{1}$, $v_{2}$ even. Let
$D_{1}$ and $D_{2}$ be difference sets in $G_{1},G_{2}$, respectively. Then
$D=\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$ has the SDP if and only if $D_{1}$ and $D_{2}$ have the SDP.

###### Proof.

That $D$ is a difference set follows immediately from the product construction
of difference sets, so all we must just verify is that $D$ has the SDP. Let
$A_{1}$ and $A_{2}$ be the incidence matrices of the developments of $D_{1}$
and $D_{2}$, and keep the ordering of $G_{1}$ as used in $A_{1}$ and likewise
for $G_{2}$ consistent throughout this proof. Let $G$ have the ordering given
by
$G_{1}\times\\{g_{1}\\},G_{1}\times\\{g_{2}\\},\cdots,G_{1}\times\\{g_{|G_{2}|}\\}$,
where $g_{i}$ is the $i$-th element of $G_{2}$ in its ordering and $G_{1}$ is
internally ordered as in $A_{1}$, and let $A$ be the development of $D$ with
this ordering of $G$.

From here, consider the sub-matrices formed by the rows labelled by
$G_{1}\times\\{g_{i}\\}$ and the columns labelled by $G_{1}\times\\{g_{j}\\}$.
Firstly, note that we are now considering $A$ as a $2^{v_{2}}\times 2^{v_{2}}$
block matrix with $2^{v_{1}}\times 2^{v_{1}}$ matrices as entries. Note that
if $g_{j}\in g_{i}D_{2}$, then the sub-matrix will be equal to $A_{1}^{c}$,
since we are forced to consider incidence in $(G_{1}-D_{1})\times D_{2}$. But
if $g_{j}\not\in g_{i}D_{2}$, then the sub-matrix is identical to $A_{1}$ by
the same reasoning. Let $A^{\prime}$ denote the $2^{v_{2}}\times 2^{v_{2}}$
block matrix with $A_{1}$ in every entry, and $A_{2}^{\prime}$ denote the
$2^{v_{2}}\times 2^{v_{2}}$ block matrix whose $ij$-th matrix is the all-ones
matrix $J$ if the $ij$-th entry of $A_{2}$ is 1, and the zero matrix
otherwise. Then we see that $A=A^{\prime}+A_{2}^{\prime}$, and thus $A$ has a
similar structure to $A_{2}$.

Since $D_{2}$ has the SDP, this implies that the addition of any three of
these block rows is either a block row or the complement of a block row. Let
$R_{i},R_{j},R_{k}$ be three arbitrary block rows of $A$. Then
$R_{i}+R_{j}+R_{k}\in\\{R_{u},R_{u}^{c}\\}$ for some $u$. Let
$r_{i},r_{j},r_{k}$ be rows contained in $R_{i},R_{j},R_{k}$, respectively. If
$R_{i}=R_{j}=R_{k}$, then since $D_{1}$ has the SDP, and $r_{i},r_{j},r_{k}$
are the concatenation of $2^{v_{2}}$ copies of rows in $A_{1}$, then
$r_{i}+r_{j}+r_{k}\in\\{r_{v},r_{v}^{c}\\}$ for some $r_{v}\in R_{i}$. If,
without loss of generality, $R_{i}=R_{j}$, then $R_{i}+R_{j}+R_{k}=R_{k}$.
Since, as was discussed in Section 2, each row of $A_{1}$ defines a bent
function in $RM(1,v_{1})$, the addition $r_{i}+r_{j}$ is in $RM(1,2n)$.
Therefore $r_{i}+r_{j}+r_{k}\in\\{r_{v},r_{v}^{c}\\}$ for some $r_{v}\in
R_{k}$. The other equalities follow by symmetry. If all three block rows are
pairwise unequal, we consider the following. Let $r_{i},r_{j},r_{k}$ be
arbitrary rows of the block rows $R_{i},R_{j},R_{k}$, respectively. Note that
$r_{i},r_{j},r_{k}$ are each concatenations of a combination of rows
$\rho_{i},\rho_{j},\rho_{k}$ of $A_{1}$ and their complements based on the
$i$-th, $j$-th, and $k$-th row of $A_{2}$ (denote these rows
$\alpha_{i},\alpha_{j},\alpha_{k}$, and the large versions of these rows
$a_{i},a_{j},a_{k}$), respectively. Since $D_{1}$ has the SDP, this implies
that the addition of each of these concatenated rows will result in the
concatenation of some combination of a row $\rho_{v}$ and its complement. Then

$r_{i}+r_{j}+r_{k}=\begin{matrix}&\langle&\rho_{i}&|&\rho_{i}&|&\cdots&|&\rho_{i}&\rangle+a_{i}\\\
&\langle&\rho_{j}&|&\rho_{j}&|&\cdots&|&\rho_{j}&\rangle+a_{j}\\\
+&\langle&\rho_{k}&|&\rho_{k}&|&\cdots&|&\rho_{k}&\rangle+a_{k}\\\
\end{matrix}$

Which is one of

$\left\\{\langle\rho_{v}\ |\ \rho_{v}\ |\ \cdots\ |\
\rho_{v}\rangle,\langle\rho_{v}^{c}\ |\ \rho_{v}^{c}\ |\ \cdots\ |\
\rho_{v}^{c}\rangle\right\\}+\left\\{a_{v},a_{v}^{c}\right\\},$

all of which are a row of $A$ or the complement thereof. Thus, the addition of
any three rows of $A$ is either a row of $A$ or the complement of a row, and
so $D$ has the SDP.

Conversely, without loss of generality suppose that $D_{1}$ does not have the
SDP. By way of contradiction, suppose $D$ has the SDP. Consider the sub-matrix
$M$ formed by the rows of the form $r_{1}+\\{RM(1,2n)-\langle
c_{1},\cdots,c_{v_{2}}\rangle\\}$ and the columns labelled by
$G_{1}\times\\{1\\}$. Firstly, note that $M$ is a $2^{v_{1}}\times 2^{v_{1}}$
square matrix. By construction, the first row $\rho_{1}$ of $M$ must be the
incidence of either $D_{1}$ or $G_{1}-D_{1}$. Also, since $D$ has the SDP, and
$M$ is formed by a subgroup of the code, the addition of any three rows of $M$
must also be a row or the complement of a row, which implies that the
incidence of $D_{1}$ or $G_{1}-D_{1}$ is a bent function on $RM(1,v_{1})$.
Thus, the development of $D_{1}$ must have the SDP contrary to supposition,
and therefore $D$ cannot have the SDP. ∎

Note that this implies that the possible product-constructed SDP difference
sets in $G$ depends on the “factoring” of $G$. For example, in the Abelian
group $C_{8}\times C_{8}\times C_{2}\times C_{2}$, there are no SDP difference
sets coming from the grouping $(C_{8}^{2})\times(C_{2}^{2})$, but there is
coming from $(C_{8}\times C_{2})^{2}$. From this, we can state the following
corollaries:

######  Remark.

The Abelian group $C_{8}\times C_{8}\times C_{4}$ has no SDP difference sets
from product construction.

###### Proof.

The only eligible grouping of $C_{8}\times C_{8}\times C_{4}$ into difference
set-containing groups is $(C_{8})^{2}\times C_{4}$, so since $C_{8}^{2}$ has
no SDP difference sets, $C_{8}\times C_{8}\times C_{4}$ cannot have any SDP
difference sets by product construction by Theorem 1. ∎

######  Remark.

The abelian groups $C_{16}\times(C_{8}\times C_{2})$, $C_{16}\times
C_{4}^{2}$, $C_{16}\times(C_{8}\times C_{2})\times(C_{4})\times(C_{2}^{2})$,
when grouped as such, have no SDP difference sets from product construction
for $i,j,k\geq 0$.

###### Proof.

None of $C_{16}\times(C_{8}\times C_{2})$, $C_{16}\times C_{4}$, nor
$C_{16}\times(C_{2}^{2})$ contain SDP difference sets from product
construction by Theorem 1, from which it immediately follows that any grouping
of these groups has no SDP difference set from product construction. ∎

######  Remark.

This grouping as in the last remark is some sense the “best” factoring of the
group. That is, it is grouped so as to maximize the number of SDP-containing
groups and placing priority on larger “primitive” (not product constructed)
SDP difference set-containing groups. Another example of this factoring is the
factoring of the group $C_{8}\times C_{8}\times C_{2}\times C_{2}$ as we
discussed earlier.

With this result under our belts, we may now expand to the more general semi-
direct product.

##  4\. Closure of the SDP under Semi-Direct Product Construction

In general, the SDP will not be closed under product construction. Section 3
is dedicated to showing a very specific case of the general product
construction. The question then becomes: under what semi-direct products does
the product construction of two SDP difference sets yield the SDP? Note that
the SDP is not always preserved under the general product, as, for example, in
the groups of the form $(C_{8}\times C_{2})\rtimes(C_{8}\times C_{2})$, there
are seven distinct (nonisomorphic) designs. Of these, only two have the SDP.
For the purpose of illustration, the following table shows the mappings of $x$
and $y$ that produces each design (and so the homomorphism will be defined by
a choice of mapping for $x$ and a choice of mapping for $y$). Note that for
designs 3 and 4, the combinations of the mappings of $x$ and $y$ are
restricted. These restrictions for Design 3 will be discussed following the
table.

Design 1 | Rank: 10 | Total Designs: 16 | Design 2 | Rank: 10 | Total Designs: 8
---|---|---|---|---|---
$x\mapsto$ | $\varphi_{x}(z)=z,$ | $\varphi_{x}(w)=w$ | $x\mapsto$ | $\varphi_{x}(z)=zw,$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{3},$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{x}(z)=z^{3}w,$ | $\varphi_{x}(w)=z^{4}w$
| $\varphi_{x}(z)=z^{5},$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{5}w,$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{7},$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{x}(z)=z^{7}w,$ | $\varphi_{x}(w)=z^{4}w$
$y\mapsto$ | $\varphi_{y}(z)=z,$ | $\varphi_{y}(w)=w$ | $y\mapsto$ | $\varphi_{y}(z)=z,$ | $\varphi_{y}(w)=w$
| $\varphi_{y}(z)=z^{3},$ | $\varphi_{y}(w)=z^{4}w$ |  |  |
| $\varphi_{y}(z)=z^{5},$ | $\varphi_{y}(w)=w$ |  | $\varphi_{y}(z)=z^{5},$ | $\varphi_{y}(w)=w$
| $\varphi_{y}(z)=z^{7},$ | $\varphi_{y}(w)=z^{4}w$ |  |  |
Design 3 | Rank: 11 | Total Designs: 24 | Design 4 | Rank: 12 | Total Designs: 40
$x\mapsto$ | $\varphi_{x}(z)=z$ | $\varphi_{x}(w)=z^{4}w$ | $x\mapsto$ | $\varphi_{x}(z)=z$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{3}$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z$ | $\varphi_{x}(w)=z^{4}w$
| $\varphi_{x}(z)=z^{5}$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{x}(z)=z^{3}$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{7}$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{3}$ | $\varphi_{x}(w)=z^{4}w$
| $\varphi_{x}(z)=zw$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{x}(z)=z^{5}$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{3}w$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{5}$ | $\varphi_{x}(w)=z^{4}w$
| $\varphi_{x}(z)=z^{5}w$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{x}(z)=z^{7}$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{7}w$ | $\varphi_{x}(w)=z$ |  | $\varphi_{x}(z)=z^{7}$ | $\varphi_{x}(w)=z^{4}w$
|  |  |  | $\varphi_{x}(z)=z^{3}w$ | $\varphi_{x}(w)=w$
|  |  |  | $\varphi_{x}(z)=z^{7}w$ | $\varphi_{x}(w)=w$
$y\mapsto$ | $\varphi_{y}(z)=z$ | $\varphi_{y}(w)=w$ | $y\mapsto$ | $\varphi_{y}(z)=z$ | $\varphi_{y}(w)=z^{4}w$
| $\varphi_{y}(z)=z^{3}$ | $\varphi_{y}(w)=z^{4}w$ |  | $\varphi_{y}(z)=z^{3}$ | $\varphi_{y}(w)=w$
| $\varphi_{y}(z)=z^{5}$ | $\varphi_{y}(w)=w$ |  | $\varphi_{y}(z)=z^{5}$ | $\varphi_{y}(w)=z^{4}w$
| $\varphi_{y}(z)=z^{7}$ | $\varphi_{y}(w)=z^{4}w$ |  | $\varphi_{y}(z)=z^{7}$ | $\varphi_{y}(w)=w$
|  |  |  | $\varphi_{y}(z)=z^{3}w$ | $\varphi_{y}(w)=w$
|  |  |  | $\varphi_{y}(z)=z^{7}w$ | $\varphi_{y}(w)=w$
Design 5 | Rank: 12 | Total Designs: 24 |  |  |
---|---|---|---|---|---
$x\mapsto$ | $\varphi_{x}(z)=z^{3}$ | $\varphi_{x}(w)=w$ | $y\mapsto$ | $\varphi_{y}(z)=z^{3}$ | $\varphi_{y}(w)=w$
| $\varphi_{x}(z)=z^{7}$ | $\varphi_{x}(w)=w$ |  | $\varphi_{y}(z)=z^{7}$ | $\varphi_{y}(w)=w$
| $\varphi_{x}(z)=zw$ | $\varphi_{x}(w)=w$ |  | $\varphi_{y}(z)=z^{3}w$ | $\varphi_{y}(w)=w$
| $\varphi_{x}(z)=z^{3}w$ | $\varphi_{x}(w)=z^{4}w$ |  | $\varphi_{y}(z)=z^{7}w$ | $\varphi_{y}(w)=w$
| $\varphi_{x}(z)=z^{5}w$ | $\varphi_{x}(w)=z^{4}w$ |  |  |
| $\varphi_{x}(z)=z^{7}w$ | $\varphi_{x}(w)=w$ |  |  |
Design 6 | Rank: 11 | Total Designs: 8 | Design 7 | Rank: 11 | Total Designs: 8
$x\mapsto$ | $\varphi_{x}(z)=z,$ | $\varphi_{x}(w)=w$ | $x\mapsto$ | $\varphi_{x}(z)=z^{3},$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{5},$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{7},$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=zw,$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{3}w,$ | $\varphi_{x}(w)=w$
| $\varphi_{x}(z)=z^{5}w,$ | $\varphi_{x}(w)=w$ |  | $\varphi_{x}(z)=z^{7}w,$ | $\varphi_{x}(w)=w$
$y\mapsto$ | $\varphi_{y}(z)=zw,$ | $\varphi_{y}(w)=w$ | $y\mapsto$ | $\varphi_{y}(z)=zw,$ | $\varphi_{y}(w)=w$
| $\varphi_{y}(z)=z^{5}w,$ | $\varphi_{y}(w)=w$ |  | $\varphi_{y}(z)=z^{5}w,$ | $\varphi_{y}(w)=w$

When discussing the maps of $x$ and $y$, for brevity’s and notation’s sake we
refer only to $\varphi_{x}(z)$ and $\varphi_{y}(z)$. In Design 3, any of the
maps of the form $\varphi_{x}(z)=z^{i}$ may be freely combined with any map of
$y$, but the maps of the form $\varphi_{x}(z)=z^{i}w$ can only be combined
with $\varphi_{y}(z)=z$ and $\varphi_{y}(z)=z^{5}$.

Let us narrow the purview of the discussion to semi-direct products of
difference sets whose developments are isomorphic to the symplectic design
(henceforth referred to as “symplectic difference sets”). Toward this, we
first note that for groups $G_{1}$ and $G_{2}$ with SDP difference sets
$D_{1}$ and $D_{2}$, respectively, the product construction $D$ of $D_{1}$ and
$D_{2}$ in $G_{1}\rtimes G_{2}$ is the same (set-equivalent) regardless of
choice of homomorphism from $G_{2}$ into $Aut(G_{1})$, as the homomorphism
only affects the multiplication of elements, and no such operation is
occurring in the product construction. Thus, since the difference sets are
element-wise equal, the question must come down to the blocks of the design,
since this is the only part of the design that relies upon multiplication
within each group.

Recall that the incidence matrix for the symplectic design on $2^{2n}$ points
is

$A=\frac{-1}{2}\bigg{(}\big{(}\underbrace{(J_{4}-2I_{4})\otimes(J_{4}-2I_{4})\otimes\cdots\otimes(J_{4}-2I_{4})}_{n\text{
times}}\big{)}-J_{4}\bigg{)},$

where $\otimes$ denotes the Kronecker product, $I_{4}$ is the $4\times 4$
identity matrix, and $J_{4}$ is the $4\times 4$ all-ones matrix. Let $D_{1}$
and $D_{2}$ be symplectic difference sets in $G_{1}$ and $G_{2}$,
respectively, and let $\varphi:G_{2}\to Aut(G_{1})$ be a homomorphism. Note
that if $ker(\varphi)=G_{2}$, then $G_{1}\rtimes_{\varphi}G_{2}=G_{1}\times
G_{2}$, and $D=(D_{1}\times(G_{2}-D_{2}))\cup((G_{1}-D_{1})\times D_{2})$ has
the SDP by Theorem 1, and in fact is the symplectic design, since the
construction in Theorem 1 is homologous to the Kronecker product construction
of the incidence matrix.

######  Theorem 2.

Let $D_{1}$ and $D_{2}$ be symplectic SDP difference sets in $G_{1}$ and
$G_{2}$, respectively. Let $\varphi:G_{2}\to Aut(G_{1})$ be a homomorphism,
and let $G=G_{1}\rtimes_{\varphi}G_{2}$ be the semi-direct product of $G_{1}$
by $G_{2}$ under $\varphi$. Then
$D=\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$ is a symplectic difference set if
$(g_{i},g_{j})*D=(g_{i},g_{j})\cdot D$ for all $(g_{i},g_{j})\in G_{1}\times
G_{2}$, where $*$ is the operation defined by $\varphi$, and $\cdot$ is the
operation of the direct product.

###### Proof.

Let $\varphi$ be such that $(g_{i},g_{j})*D=(g_{i},g_{j})\cdot D$ for all
$(g_{i},g_{j})\in G_{1}\times G_{2}$. Since the underlying sets of
$G_{1}\rtimes_{\varphi}G_{2}$ and $G_{1}\times G_{2}$ are equal, form the
incidence matrix $A$ for the development of $D$ in
$G_{1}\rtimes_{\varphi}G_{2}$ and the incidence matrix $A^{\prime}$ for the
development $D$ in $G_{1}\times G_{2}$ with the same ordering of the
underlying set. Consider the incidence of some arbitrary point $y$ on some
arbitrary block $rD$ in both designs. Then since $r,y\in G_{1}\times G_{2}$,
we have that $r=(g_{i},g_{j})$ and $y=(g_{k},g_{l})$ for some
$(g_{i},g_{j}),(g_{k},g_{l})\in G_{1}\times G_{2}$. Thus since $r*D=r\cdot D$
by supposition, the incidence of $y$ in $r*D$ is the same as the incidence of
$y$ in $r\cdot D$. Thus, since $r,y$ were arbitrary, $A$ must equal
$A^{\prime}$, and so the designs are isomorphic. Also, $D$ is a symplectic
difference set in $G_{1}\times G_{2}$ by the definition of the symplectic
design, so $D$ is also a symplectic difference set in
$D_{1}\rtimes_{\varphi}D_{2}$. ∎

This condition by itself is somewhat nebulous: for what (nontrivial) choices
of $\varphi$ is this the case? Firstly, note that we need only work with
$(1,g_{j})*D$, since $G_{1}$ applies none of the “twisting” from the semi-
direct product. Further, $\varphi_{g_{j}}$ must induce a permutation on
$D_{1}$ for all $g_{j}$. This leads us to the following theorem:

######  Theorem 3.

Let $G_{1}$ and $G_{2}$ be groups of even power of 2 order with $G_{2}$ having
generators $\\{x_{1},x_{2},\cdots,x_{n}\\}$, and let $D_{1}$ and $D_{2}$ be
symplectic difference sets in $G_{1}$ and $G_{2}$, respectively. Let
$\varphi:G_{2}\to Aut(G_{1})$ be a homomorphism. Then
$D=\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$ is a symplectic difference set in $G_{1}\rtimes_{\varphi}G_{2}$
if $\varphi_{x_{i}}(D_{1})=D_{1}$ for all $x_{i}$.

###### Proof.

Suppose that $\varphi_{x_{i}}(D_{1})=D_{1}$ for all $x_{i}$. Let
$g=x_{1}^{e_{1}}x_{2}^{e_{2}}\cdots x_{n}^{e_{n}}\in G_{2}$ be arbitrary. Then
since $\varphi$ is a homomorphism,
$\varphi_{g}(D_{1})=\varphi_{x_{1}^{e_{1}}}\varphi_{x_{2}^{e_{2}}}\cdots\varphi_{x_{n}^{e_{n}}}(D_{1})=D_{1}$
by supposition. ∎

Thus, we can determine whether the condition in Theorem 3 is met using only
the generators of $G_{2}$. This significantly decreases the search space for
such homomorphisms, as if there are $m_{1}$ mappings of $x_{1}$, $m_{2}$
mappings of $x_{2}$, and so on up to $m_{n}$ mappings of $x_{n}$ that keep $D$
fixed, there are then $\prod_{i=1}^{n}m_{i}$ homomorphisms that satisfy the
property in Theorem 3. We present the following as a non-trivial example of
this property:

######  Example 4.

Let $G=(C_{8}\times C_{2})\rtimes_{\varphi}C_{4}$, ($x^{8}=y^{2}=z^{4}=1$),
where $\varphi$ is defined by $\varphi_{z}(x)=x^{5}$, $\varphi_{z}(y)=y$.
Recall that $C_{8}\times C_{2}$ has a symplectic SDP difference set
$D=\\{1,x,x^{2},x^{5},y,x^{6}y\\}$, and $C_{4}$ has the trivial symplectic SDP
difference set $\\{1\\}$. Note that
$\varphi_{z}(D)=\\{1,x^{5},(x^{5})^{2},(x^{5})^{5},y,(x^{5})^{6}y\\}=\\{1,x^{5},x^{2},x,y,x^{6}y\\}$,
and thus since the automorphism induced by the generator of $C_{4}$ fixes $D$,
we must have that the incidence matrix of
$(D\times\\{z,z^{2},z^{3}\\})\cup((C_{8}\times C_{2}-D)\times\\{1\\})$ under
this semi-direct product is equal to the incidence matrix under the direct
product, and so the design is symplectic.

##  5\. Isomorphic Designs under Product Construction

The closure of the symmetric difference property under direct product
construction also opens a new line of questioning: how does product
construction interact with isomorphic designs? In particular, given two SDP
difference sets $D,D^{\prime}$ produced from the product construction of
isomorphic designs, how do different semi-direct products effect the
equivalence of $D$ and $D^{\prime}$? If the developments of two difference
sets $D_{1}$ and $D_{2}$ are isomorphic, by an abuse of notation we denote
this relation by $D_{1}\simeq D_{2}$. For the direct product, we present the
following theorem:

######  Theorem 5.

Let $G=G_{1}\times G_{2}$ and $G^{\prime}=G_{1}^{\prime}\times G_{2}^{\prime}$
be two distinct groups of order $2^{2n}$ with respective SDP difference sets
$D=\big{(}D_{1}\times(G_{2}-D_{2})\big{)}\cup\big{(}(G_{1}-D_{1})\times
D_{2}\big{)}$ and
$D^{\prime}=\big{(}D_{1}^{\prime}\times(G_{2}^{\prime}-D_{2}^{\prime})\big{)}\cup\big{(}(G_{1}^{\prime}-D_{1}^{\prime})\times
D_{2}^{\prime}\big{)}$ for SDP difference sets $D_{i}\in G_{i}$ and
$D_{i}^{\prime}\in G_{i}^{\prime}$. Then if $D_{1}\simeq D_{1}^{\prime}$ and
$D_{2}\simeq D_{2}^{\prime}$, then $D\simeq D^{\prime}$.

###### Proof.

Suppose $D_{1}\simeq D_{1}^{\prime}$ and $D_{2}\simeq D_{2}^{\prime}$. Then
$|G_{1}|=|G_{1}^{\prime}|=v_{1}$ and $|G_{2}|=|G_{2}^{\prime}|=v_{2}$. Let
$A_{1},A_{1}^{\prime},A_{2},A_{2}^{\prime}$ be the incidence matrices for the
developments of $D_{1},D_{1}^{\prime},D_{2},D_{2}^{\prime}$, respectively, and
maintain the ordering of $G_{1},G_{1}^{\prime},G_{2},G_{2}^{\prime}$ as in
these incidence matrices for the remainder of the proof. Then using the same
ordering of $G$ as in Theorem 1, we have $A$ and $A^{\prime}$ being block
matrices made of copies of $A_{1},A_{1}^{c}$ and $A_{1}^{\prime},A_{1}^{\prime
c}$, respectively. Since the designs of $D_{1}$ and $D_{1}^{\prime}$ are
isomorphic, there exist permutation matrices $P_{1},Q_{1}$ such that
$A_{1}^{\prime}=P_{1}A_{1}Q_{1}$, and likewise there exist permutation
matrices $P_{2},Q_{2}$ for $A_{2}$ into $A_{2}^{\prime}$. Then consider the
diagonal block matrices $P_{1}^{\prime},Q_{1}^{\prime}$ with $P_{1}$ on the
diagonal of one and $Q_{1}$ on the diagonal of the other. Then
$P_{1}^{\prime}AQ_{1}^{\prime}$ permutes each $A_{1}$ or $A_{1}^{c}$ in $A$
into $A_{1}^{\prime}$ or $A_{1}^{\prime c}$, respectively. Now, recalling that
$A$ is $A_{2}$ as a block matrix, let $P_{2}^{\prime}$ and $Q_{2}^{\prime}$ be
the block matrix equivalents (zero $v_{1}\times v_{1}$ matrix in place of 0,
and $I_{v_{1}}$ in place of 1) of $P_{2}$ and $Q_{2}$, respectively. Then
$(P_{2}^{\prime}P_{1}^{\prime})A(Q_{1}^{\prime}Q_{2}^{\prime})=A^{\prime}$,
and we thus have that $D\simeq D^{\prime}$. ∎

From this theorem, we have the following corollaries:

######  Corollary 5.1.

The developments of all direct product-constructed SDP difference sets of
order 64 are isomorphic to the symplectic design on 64 points.

###### Proof.

We first note the SDP on $C_{2}^{6}$ by iterative product construction is
isomorphic to the symplectic design on 64 points. Let $T$ be the trivial SDP
difference set on $C_{2}^{2}$, and let $S$ be the product constructed SDP
difference set on $C_{2}^{4}$. Note as well that there is only one SDP design
each on 16 and 4 points. Let $G_{1}$ and $G_{2}$ be arbitrary groups of order
16 and 4 that contain SDP difference sets $D_{1}$ and $D_{2}$, respectively.
Then since there is only one SDP design on 16 points and 4 points,
$D_{1}\simeq S$ and $D_{2}\simeq T$. Thus, by Theorem 5, the product
construction of $D_{1}$ and $D_{2}$ is isomorphic to the symplectic design. ∎

######  Corollary 5.2.

There are four nonisomorphic SDP designs on 256 points coming from product
construction.

###### Proof.

There are four nonisomorphic SDP designs on 64 points and one SDP design on 4
points, and so there is exactly one nonisomorphic design on 256 for each
nonisomorphic design on 64 points by Theorem 5. ∎

## References

  * [DHK21] J. A. Davis, J.J. Hoo, C. Kissane et al. Abelian difference sets with the symmetric difference property. Des. Codes Cryptogr. 89, (2021), 517–523.
  * [Dil74] J. F. Dillon. Elementary Hadamard Difference Sets. Dissertation. (1974). 10-23.
  * [Kan75] W. M. Kantor. Symplectic Groups, Symmetric Designs, and Line Ovals. J. Algebra. 33, (1975), 43-58.
  * [Lan83] E. Lander, Symmetric Designs: An Algebraic Approach, London Math. Soc. Lecture Note Series, Cambridge University Press, (1983), pgs.
  * [Mul54] D. E. Muller, Application of Boolean algebra to switching circuit design and to error detection, Transactions of the I.R.E. Professional Group on Information Theory, (1954), 6-12.
  * [Ree54] I. S. Reed, A class of multiple-error-correcting codes and the decoding scheme. Transactions of the I.R.E. Professional Group on Information Theory. 4 (1954), 38-49.