RBI-ThPhys-2022-9, CERN-TH-2022-041

# New Physics with missing energy at future lepton colliders - Snowmass White
Paper

Jan Kalinowski<EMAIL_ADDRESS>Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02–093 Warsaw, Poland  Tania Robens<EMAIL_ADDRESS>Ruder Boskovic Institute, Bijenicka cesta 54, 10000 Zagreb, Croatia
Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland  Aleksander
Filip Żarnecki<EMAIL_ADDRESS>Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02–093 Warsaw, Poland

######  Abstract

Two models that extend the particle content of the SM and provide dark matter
candidates, namely the Inert Doublet Model and the Two-Higgs Doublet model
with additional pseudoscalar, are confronted with current experimental and
theoretical constraints and predictions for production cross sections for
various standard pair-production modes within these models at future lepton
colliders are presented.

##  I Introduction

We discuss two new physics models that extend the Standard Model (SM) particle
sector by additional scalars and provide a dark matter candidate, namely the
Inert Doublet Model (IDM) and the Two-Higgs Doublet Model with additional
pseudoscalar (THDMa). Both models are confronted with current theoretical and
experimental constraints. From the theoretical side, these include the
minimization of the vacuum as well as the requirement of vacuum stability and
positivity. We also require perturbative unitarity to hold, and perturbativity
of the couplings at the electroweak scale.

Experimental bounds include the agreement with current measurements of the
properties of the 125 GeV resonance discovered by the LHC experiments, as well
as agreement with the null-results from searches for additional particles at
current or past colliders. We also confront the models with bounds from
electroweak precision observables (via $S,\,T,\,U$ parameters), B-physics
observables
$\left(B\,\rightarrow\,X_{s}\,\gamma,\,B_{s}\,\rightarrow\,\mu^{+}\,\mu^{-},\,\Delta
M_{s}\right)$, as well as agreement with astrophysical observables (relic
density and direct detection bounds). We use a combination of private and
public tools in these analyses, where the latter include HiggsBounds
Bechtle:2020pkv, HiggsSignals Bechtle:2020uwn, 2HDMC Eriksson:2009ws, SPheno
Porod:2011nf, Sarah Staub:2013tta, micrOMEGAs Belanger:2018ccd;
Belanger:2020gnr, and MadDM Ambrogi:2018jqj. Experimental numbers are taken
from Baak:2014ora; Haller:2018nnx for electroweak precision observables, combi
for $B_{s}\,\rightarrow\,\mu^{+}\,\mu^{-}$, Amhis:2019ckw for $\Delta M_{s}$
and Planck:2018vyg and Aprile:2018dbl for relic density and direct detection,
respectively. Bounds from $B\,\rightarrow\,X_{s}\gamma$ are implemented using
a fit function from Misiak:2020vlo; mm. Predictions for production cross
sections shown here have been obtained using Madgraph5 Alwall:2011uj.

##  II The Inert Doublet Model

###  II.1 The model

The Inert Doublet Model (IDM) Deshpande:1977rw; Cao:2007rm; Barbieri:2006dq is
an intriguing new physics model. In this model, the SM scalar sector is
enhanced by an additional $SU(2)\,\times\,U(1)$ gauge doublet $\phi_{D}$. As
in the SM, the first doublet $\phi_{S}$ contains the SM-like Higgs boson $h$,
while the inert one $\phi_{D}$ contains four scalar states $H$, $A$ and
$H^{\pm}$. A discrete exact $\mathbb{Z}_{2}$ symmetry is introduced with the
following transformation properties

$\phi_{S}\to\phi_{S},\,\,\phi_{D}\to-\phi_{D},\,\,\text{SM}\to\text{SM}.$ (1)

The additional doublet does not acquire a vacuum expectation value (vev) and
does not couple to fermions. Electroweak symmetry breaking is as in the SM.
The symmetry also insures that the lightest particle of $\phi_{D}$ is stable,
making this a good dark matter candidate.

The scalar potential of the model is given by

$\begin{array}[]{c}V=-\frac{1}{2}\left[m_{11}^{2}(\phi_{S}^{\dagger}\phi_{S})\\!+\\!m_{22}^{2}(\phi_{D}^{\dagger}\phi_{D})\right]+\frac{\lambda_{1}}{2}(\phi_{S}^{\dagger}\phi_{S})^{2}\\!+\\!\frac{\lambda_{2}}{2}(\phi_{D}^{\dagger}\phi_{D})^{2}\\\\[5.69054pt]
+\\!\lambda_{3}(\phi_{S}^{\dagger}\phi_{S})(\phi_{D}^{\dagger}\phi_{D})\\!\\!+\\!\lambda_{4}(\phi_{S}^{\dagger}\phi_{D})(\phi_{D}^{\dagger}\phi_{S})+\frac{\lambda_{5}}{2}\left[(\phi_{S}^{\dagger}\phi_{D})^{2}\\!+\\!(\phi_{D}^{\dagger}\phi_{S})^{2}\right].\end{array}$
(2)

The model features 7 free parameters, which we chose in the so-called physical
basis Ilnicka:2015jba

$v,M_{h},M_{H},M_{A},M_{H^{\pm}},\lambda_{2},\lambda_{345},$ (3)

with $\lambda_{345}\,\equiv\,\lambda_{3}+\lambda_{4}+\lambda_{5}$. As two
parameters (vev $v$ and $M_{h}\,\sim\,125\,{\rm GeV}$) are fixed by
experimental measurements, we end up with a total number of 5 free parameters.
Here, we consider the case where $H$ is the dark matter candidate, which
implies $M_{A,\,H^{\pm}}\,\geq\,M_{H}$. 111Note that the new scalars in the
IDM do not have CP quantum numbers, as they do not couple to fermions. In the
subsequent discussion, we can replace $H\,\longleftrightarrow\,A$ if we
simultaneously use $\lambda_{5}\,\longleftrightarrow\,-\lambda_{5}$. All
phenomenological considerations are identical for these cases.

The model is subject to a large number of theoretical and experimental
constraints, discussed at length e.g. in Ilnicka:2015jba; Ilnicka:2018def;
Dercks:2018wch; Kalinowski:2018ylg; Kalinowski:2020rmb. In the scan, we make
use of the publicly available tools 2HDMC Eriksson:2009ws, HiggsBounds-5.10.1
Bechtle:2008jh; Bechtle:2011sb; Bechtle:2013wla; Bechtle:2015pma;
Bechtle:2020pkv, HiggsSignals-2.6.2 Bechtle:2013xfa; Bechtle:2020uwn, as well
as micrOMEGAs$\\_$5.2.4 Belanger:2020gnr. Cross sections are calculated using
Madgraph5 Alwall:2011uj with a UFO input file from Goudelis:2013uca222 Note
the official version available at ufo_idm exhibits a wrong CKM structure,
leading to false results for processes involving electroweak gauge bosons
radiated off quark lines. In our implementation, we corrected for this. Our
implementation corresponds to the expressions available from Zyla:2020zbs..
Experimental values are taken from GFitter gfitter; Haller:2018nnx, and the
Planck Aghanim:2018eyx and XENON1T Aprile:2018dbl experiments. Direct collider
searches as well as agreement with the 125 GeVcoupling strength measurements
are implemented via HiggsBounds and HiggsSignals, where we also compare to the
total width upper limit Sirunyan:2019twz and invisible branching ratio ATLAS-
CONF-2020-052 of $h$. Finally, recast results from a LEP-SUSY search
Lundstrom:2008ai were included.

###  II.2 Current Status

The experimental and theoretical constraints lead to a large reduction of the
allowed parameter space. As an example, the masses are usually quite
degenerate, as can be seen from figure 1. This is caused by an interplay of
electroweak constraints and theoretical requirements on the potential.

Figure 1: Masses are requested to be quite degenerate after all constraints
have been taken into account. Left: In the $\left(M_{A},\,M_{H^{\pm}}\right)$
plane (taken from Ilnicka:2015jba). Right: In the
$\left(M_{H^{\pm}}-M_{H},\,M_{A}-M_{H}\right)$ plane (taken from
Kalinowski:2018ylg).

We also consider the case when $M_{H}\,\leq\,M_{h}/2$, where constraints from
$h\,\rightarrow\,\text{inv{isible}}$ start to play an important role and an
interesting interplay arises, between bounds from signal strength
measurements, that require $|\lambda_{345}|$ to be rather small
$\lesssim\,0.3$, and bounds from dark matter relic density, where too low
values of that parameter lead to small annihilation cross sections and
therefore too large relic density values. The resulting parameter space is
shown in figure 2. In Ilnicka:2015jba, it was found that this in general leads
to a lower bound of $M_{H}\,\sim\,50\,{\rm GeV}$, with exceptions presented in
Kalinowski:2020rmb.

Figure 2: Interplay of signal strength and relic density constraints in the
$\left(M_{H},\,\lambda_{345}\right)$ plane. Left: Using LUX constraints
Akerib:2013tjd, bounds labelled ”excluded from collider data” have been tested
using HiggsBounds and HiggsSignals (taken from Ilnicka:2015jba). Right: Using
XENON1T results, with golden points labelling those points that produce exact
relic density (taken from Ilnicka:2018def).

###  II.3 Discovery prospects at ILC and CLIC

So far, no publicly available search exists that investigates the IDM
parameter space with actual collider data. The discovery potential of ILC and
CLIC was investigated in Kalinowski:2018kdn; deBlas:2018mhx; Zarnecki:2019poj;
Zarnecki:2020swm; Sokolowska:2019xhe; Klamka:2022ukx for several benchmark
points proposed in Kalinowski:2018ylg, for varying center-of-mass energies
from $250\,{\rm GeV}$ up to $3\,{\rm TeV}$. We concentrated on $AH$ and
$H^{+}H^{-}$ production with $A\,\rightarrow\,Z\,H$ and
$H^{\pm}\,\rightarrow\,W^{\pm}H$, where the electroweak gauge bosons
subsequently decay leptonically. For event generation, we used WHizard 2.2.8
Moretti:2001zz; Kilian:2007gr, with an interface via SARAH Staub:2015kfa and
SPheno 4.0.3 Porod:2003um; Porod:2011nf for model implementation. For CLIC
results energy spectra Linssen:2012hp were also taken into account.

For the production modes above, we considered leptonic decays of the
electroweak gauge bosons. The investigated final states were

$e^{+}\,e^{-}\,\rightarrow\,\mu^{+}\mu^{-}+\not{E},\;\;\;e^{+}\,e^{-}\,\rightarrow\,\mu^{\pm}\,e^{\mp}+\not{E}$

for $HA$ and $H^{+}\,H^{-}$ production, respectively. For a more accurate
description, we did not specify the intermediate states in the event
generation, which in turn means all processes leading to the above signatures
were taken into account, including interference between the contributing
diagrams. This includes final states, where the missing energy stems from
additional neutrinos, $e.g.$ from $\tau^{\pm}$ decays. Event selection was
performed using a set of preselection cuts as well as boosted decision trees,
as implemented in the TMVA toolkit Hocker:2007ht. Results for the ILC running
at 250 GeV and 500 GeV, and the first CLIC stage at 380 GeV are shown in
figure 3. The expected discovery reach of 500 GeV ILC extends up to neutral
scalar mass sum of 330 GeV up to charged scalar masses of 200 GeV.

Figure 3: Significance of the deviations from the Standard Model predictions
expected for 1 ab-1 of data collected at centre-of-mass energy of 250 GeV, 380
GeV and 500 GeV, for: (left) events with two muons in the final state
($\mu^{+}\mu^{-}$) as a function of the sum of neutral inert scalar masses and
(right) events with an electron and a muon in the final state ($e^{+}\mu^{-}$
or $e^{-}\mu^{+}$) as a function of twice the charged scalar mass.

Results for the discovery reach of CLIC, including center-of-mass energies of
1.5 TeV and 3 TeV, are shown in figure 4. In general, production cross
sections $\gtrsim\,0.5\,{\rm fb}$ seem to be accessible, where best prospects
for the considered benchmark points are given for 380 GeVor 1.5 TeVcenter-of-
mass energies. Along similar lines, mass sums up to 1 TeVseem accessible,
where the $\mu^{\pm}\,e^{\mp}$ channel seems to provide a larger discovery
range in general.

Figure 4: Discovery prospects at CLIC for the IDM in $\mu^{+}\mu^{-}+\not{E}$
(left) and $\mu^{\pm}\,e^{\mp}+\not{E}$ (right) final states, as a function of
the respective production cross-sections (top) and mass sum of the produced
particles (bottom). Taken from Kalinowski:2018kdn.

For leptonic signatures, the sensitivity of high energy $e^{+}e^{-}$ colliders
to pair-production of IDM scalars is limited by the production cross section
in the considered channel. The signal cross section for the charged scalar
pair-production increases by about an order of magnitude when the semi-
leptonic final state is considered, i.e. when the hadronic decay of one of the
$W$ bosons is considered. The sensitivity of high energy CLIC to
$H^{+}\,H^{-}$ production with the semi-leptonic final state is shown in Fig.
5. The accessible scalar mass range increases by about a factor of two, to
about 2 TeV Sokolowska:2019xhe; Zarnecki:2020swm; Klamka:2022ukx. It is also
interesting to note that highest signal observation significance is obtained
for scenarios with scalar mass difference of 20 to 60 GeV.

Figure 5: Expected statistical significance of IDM charged scalar pair-
production observation as a function of the IDM scalar mass difference (left)
and of the total mass of the produced IDM scalars (right). Results are
presented for CLIC running at 1.5 TeV (orange circles) and 3 TeV (blue
points). The red horizontal lines indicate the 5$\sigma$ threshold. Figure
taken from Klamka:2022ukx.

###  II.4 Sensitivity comparison at future colliders

After a dedicated analysis of the IDM benchmarks in the high energy CLIC
environment, an important question is whether other current or future collider
options provide similar or better discovery prospects. For the same set of
benchmarksKalinowski:2018ylg; Kalinowski:2018kdn, production cross sections
for a variety of processes have been presented in Kalinowski:2020rmb,
including VBF-type topologies. Cross sections were calculated using Madgraph5.
We concentrate on production modes at muon colliders and refer to
Kalinowski:2020rmb regarding processes at proton-proton machines. We list the
considered collider types and nominal center-of-mass energies as well as
integrated luminosities in table 1.

collider | cm energy [TeV] | $\int\mathcal{L}$ |  $\sigma_{{}_{1000}}$ [fb]
---|---|---|---
ee | 3 | $5\,{\rm ab}^{-1}$ | 0.2
$\mu\mu$ | 10 | $10\,{\rm ab}^{-1}$ | 0.1
$\mu\mu$ | 30 | $90\,{\rm ab}^{-1}$ | 0.01
Table 1: Collider parameters used in the discovery reach study performed in
Kalinowski:2020rmb. Collider specifications have been taken from
Delahaye:2019omf for the muon collider. The last column denotes the minimal
cross section required to produce 1000 events using full target luminosity.

A scenario is called ”realistic” when 1000 events can be produced using target
luminosity and center-of-mass energies as specified above. Obviously, more
detailed studies, including both background contribution and detector response
simulation, are necessary to assess the actual collider reach.

We here concentrate on production at future muon colliders:

$\mu^{+}\,\mu^{-}\,\rightarrow\,\nu_{\mu}\,\bar{\nu}_{\mu}AA,\;\;\;\mu^{+}\,\mu^{-}\,\rightarrow\,\nu_{\mu}\,\bar{\nu}_{\mu}H^{+}H^{-}.$

which corresponds to VBF-like production modes. Here again intermediate states
are not specified, so in fact several diagrams contribute which not all have a
typical VBF topology. See appendix B and C of Kalinowski:2020rmb for details.

Figure 6 shows the production cross sections as a function of the mass sum of
produced particles for various center-of-mass energies. Understanding the
behaviour of the VBF-induced channels is non-trivial; this can be attributed
to the fact that more diagrams, apart from the naive pair-production process,
contribute. Large jumps between cross-section predictions for scenarios with
similar mass scales can be traced back mainly to a fine-tuned cancellation of
various contributing diagrams, as discussed in greater detail in
Kalinowski:2020rmb.

Figure 6: Cross sections as a function of the mass sum at muon colliders in the VBF-type production mode. Taken from Kalinowski:2020rmb. collider | all others | $AA$ |  $AA$ +VBF
---|---|---|---
HL-LHC | 1 TeV | 200-600 GeV | 500-600 GeV
HE-LHC | 2 TeV | 400-1400 GeV | 800-1400 GeV
FCC-hh | 2 TeV | 600-2000 GeV | 1600-2000 GeV
CLIC, 3 TeV | 2 TeV | - | 300-600 GeV
$\mu\mu$, 10 TeV | 2 TeV | - | 400-1400 GeV
$\mu\mu$, 30 TeV | 2 TeV | - | 1800-2000 GeV
Table 2: Sensitivity of different collider options specified in table 1, using
the ”realistic” criterium of 1000 generated events in the specific channel.
Shown are minimal and maximal mass scales that are reachable. Numbers for CLIC
correspond to results from detailed investigations Kalinowski:2018kdn;
deBlas:2018mhx. Table taken from Robens:2021zvr.

The summary of sensitivities in terms of mass scales is given in table 2,
where for completeness the reach for proton colliders as estimated in the
above work has been added. It is seen that especially for $AA$ production the
VBF mode at both proton and muon colliders serves to significantly increase
the discovery reach of the respective machine. Using the simple counting
criterium above, we can furthermore state that a 27 TeV proton-proton machine
has a similar reach as a 10 TeV muon collider, while 100 TeV FCC-hh would
correspond to a 30 TeV muon-muon machine. Obviously, detailed investigations
including SM background are needed to give a more realistic estimate of the
respective collider reach.

##  III THDMa

The THDMa is a type II two-Higgs-doublet model that is extended by an
additional pseudoscalar $a$ mixing with the ”standard” pseudoscalar $A$ of the
THDM. In the gauge-eigenbasis, the additional scalar serves as a portal to the
dark sector, with a fermionic dark matter candidate, denoted by $\chi$. More
details can e.g. be found in Ipek:2014gua; No:2015xqa; Goncalves:2016iyg;
Bauer:2017ota; Tunney:2017yfp; LHCDarkMatterWorkingGroup:2018ufk;
Robens:2021lov.

The model contains the following particles in the scalar and dark matter
sector: ${h,\,H,\,H^{\pm}}$, ${a,}\,{A,}\,{{\chi}}$. It depends on 12
additional new physics parameters

$\displaystyle{v,\,m_{h},\,m_{H},}\,{m_{a},}\,{m_{A},\,m_{H^{\pm}},}\,{m_{\chi}};\;{\cos\left(\beta-\alpha\right),\,\tan\beta,}\,{\sin\theta;\;y_{\chi},}\,{\lambda_{3},}\,{\lambda_{P_{1}},\,\lambda_{P_{2}}},$

where $v$ and either $m_{h}$ or $m_{H}$ are fixed by current measurements in
the electroweak sector.

We here report on results of a scan that allows all of the above novel
parameters float in specific predefined ranges Robens:2021lov. In such a
scenario, it is not always straightforward to display bounds from specific
constraints in 2-dimensional planes. Two examples for scenarios where this is
possible are shown in figure 8. The first plot shows bounds in the
$\left(m_{H^{\pm}},\,\tan\beta\right)$ plane from B-physics observables. The
result is similar to a simple THDM, and shows that in general low masses
$m_{H^{\pm}}\lesssim\,800\,{\rm GeV}$ as well as values $\tan\beta\lesssim\,1$
are excluded. The second plot displays the relic density as a function of the
mass difference $m_{a}-2\,m_{\chi}$. Here, a behaviour can be observed that is
typical in many models with dark matter candidates: in the region where this
mass difference remains small, relic density annihilates sufficiently to stay
below the observed relic density bound. On the other hand, too large
differences lead to values $\Omega\,h_{c}\,\gtrsim\,0.12$ and therefore are
forbidden from dark matter considerations.

Finally, it was investigated which cross-section values would still be
feasible for points that fulfill all constraints Robens:2021lov at
$e^{+}e^{-}$ colliders. We here concentrate on signatures that include missing
energy and therefore do not exist in a THDM without a portal to the dark
sector. Processes like $e^{+}e^{-}\,\rightarrow\,hA,ha$ are suppressed due to
alignment, which makes $e^{+}e^{-}\,\rightarrow\,HA,Ha$ the most interesting
channel that contains novel signatures. Due to the interplay of B-physics and
electroweak constraints, such points typically have mass scales
$\gtrsim\,1\,{\rm TeV}$. Therefore production cross sections for an
$e^{+}e^{-}$ collider with a center-of-mass energy of 3 TeV are of interest.
The corresponding production cross sections are shown in figure 9, which
displays predictions for $t\,\bar{t}\,t\,\bar{t}$ and $t\,\bar{t}+\not{E}$
final states using a factorized approach. There is a non-negligible number of
points where the second channel is dominant. A ”best” point with a large rate
for $t\,\bar{t}+\not{E}_{\perp}$ has been presented in Robens:2021lov and is
repeated here for completeness

$\displaystyle\sin\theta\,=\,-0.626,\;\cos\left(\beta-\alpha\right)\,=\,0.0027,\,\tan\beta\,=\,3.55$
$\displaystyle m_{H}\,=\,643\,{\rm GeV},\,m_{A}\,=\,907\,{\rm
GeV},\,m_{H^{\pm}}\,=\,814\,{\rm GeV},$ $\displaystyle m_{a}\,\,=\,653\,{\rm
GeV},\,m_{\chi}\,=\,277\,{\rm GeV},$ $\displaystyle
y_{\chi}\,\,=\,-1.73,\,\lambda_{P_{1}}\,=\,0.18,\,\lambda_{P_{2}}\,=\,2.98,\,\lambda_{3}\,=\,8.63.$
(4)

For this point, all width/ mass ratios are $\lesssim\,6\,\%$. In addition,
branching ratios for various final states as a function of the mass sum for
the $HA$ channel are given in figure 7.

Figure 7:  Branching ratios into various final states for $A\,H$ production,
as a function of the mass sum.

Figure 8:  Left: Bounds on the $\left(m_{H^{\pm}},\,\tan\beta\right)$ plane
from B-physics observables, implemented via the SPheno Porod:2011nf/ Sarah
Staub:2013tta interface, and compared to experimental bounds combi;
Amhis:2019ckw. The contour for low $\left(m_{H^{\pm},\,\tan\beta}\right)$
values stems from Misiak:2020vlo; mm. Right: Dark matter constraints in the
THDMa model. Right: Dark matter relic density as a function of
$m_{a}-2\,m_{\chi}$, with $m_{\chi}$ defining the color coding. The typical
resonance-enhanced relic density annihilation is clearly visible. Figures
taken from Robens:2021lov.

Figure 9:  Production cross sections for $t\bar{t}t\bar{t}$ (x-axis) and
$t\bar{t}+\not{E}$ (y-axis) final state in a factorized approach, for an
$e^{+}e^{-}$ collider with a 3 TeVcenter-of-mass energy. Left: mediated via
$HA$, right: mediated via $HA$ and $Ha$ intermediate states. Color coding
refers to $m_{H}+m_{A}$ (left) and $M_{H}+0.5\times\,\left(m_{A}+m_{a}\right)$
(right). Figures taken from Robens:2021lov.

##  IV Conclusion and Outlook

In this work, we presented two models that extend the particle content of the
SM and also provide at least one dark matter candidate. We have presented
production cross sections for various standard pair-production modes within
these models; for the IDM, we have given an estimate of mass range that can be
reached based on a simple counting criterium. A more dedicated investigation
of the corresponding signatures, including background simulation and cut
optimization, is in the line of future work.

## Acknowledgements

This research was supported in parts by the National Science Centre, Poland,
the HARMONIA project under contract UMO-2015/18/M/ST2/00518 (2016-2021), OPUS
project under contract UMO-2017/25/B/ST2/00496 (2018-2021), as well as the
COST action CA16201 - Particleface.

## References

  * (1)


# On a class of optimal constant weight ternary codes

Hadi Kharaghani Department of Mathematics and Computer Science, University of
Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada<EMAIL_ADDRESS>Sho Suda
Department of Mathematics, National Defense Academy of Japan, Yokosuka,
Kanagawa 239-8686, Japan<EMAIL_ADDRESS>Vlad Zaitsev Department of
Mathematics and Computer Science, University of Lethbridge, Lethbridge,
Alberta, T1K 3M4, Canada<EMAIL_ADDRESS>
###### Abstract

A weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight $p^{m}$ is
constructed and shown that the rows of $W$ and $-W$ together form optimal
constant weight ternary codes of length $n$, weight $p^{m}$ and minimum
distance $p^{m-1}(\frac{p+3}{2})$ for each odd prime power $p$ and integer
$m\geq 1$ and thus

$A_{3}\left(\frac{p^{m+1}-1}{p-1},p^{m-1}\big{(}\frac{p+3}{2}\big{)},p^{m}\right)=2\big{(}\frac{p^{m+1}-1}{p-1}\big{)}.$

##  1 Introduction

It is not hard to see that the rows of the incidence matrix of any symmetric
design form optimal binary codes. A _weighing matrix_ of order $n$ and weight
$p$, shown as $W(n,p)$, is a $(0,\pm 1)$-matrix $W$ of order $n$ such that
$WW^{t}=pI_{n}$. The case where $n=p+1$ is called a _conference_ matrix and
$n=p$ is a _Hadamard_ matrix. Optimal binary codes obtained from Hadamard
matrices constitute an important class of codes due to their error-correcting
capability. One expects that weighing matrices also provide sets of useful
codes. The rows of a weighing matrix $W(n,p)$, $n\neq p$, form a set of
constant weight _ternary_ codes. Theorem 16 of [5] relates, though in
disguise, to an optimal class of constant ternary codes from weighing
matrices. There seems to be no more optimal constant weight codes explicitly
related to the weighing matrices in the literature. A large class of optimal
constant weight ternary codes are shown to arise from weighing matrices in
Section 3 and Section 4 of this paper.

##  2 Preliminaries

Let $S_{3}=\\{0,1,-1\\}$. A _ternary_ code of length $n$ is any subset $C$ of
$S_{3}^{n}$. Elements of $C$ are called _codewords_. The _Hamming distance_
between two ternary codewords of length $n$ is the number of coordinates in
which they differ. The number of nonzero entries of a codeword is the _weight_
of the code. A ternary code of length $n$ containing $M$ codewords and having
minimum Hamming distance $d$ is denoted $(n,M,d)$-code. If all the codewords
have the same number of nonzero entires the code is said to be of _constant
weight_. The largest value of $M$ for which there is a ternary code of length
$n$, minimum distance $d$ and weight $w$ is denoted by $A_{3}(n,d,w)$ and the
code is said to be _optimal_. $\ddot{O}$sterg$\dot{a}$rd et al in [5] among
other interesting results have shown that if $p\geq 3$ is a prime power and
$m\geq 1$, then $A_{3}(p^{m}+1,(p^{m}+3)/2,p^{m})=2(p^{m}+1)$. To show an
extension of this result the restricted Johnson bound for $A_{3}(n,d,w)$ is
essential, see Theorem 2.3.4 of [2].

######  Theorem 1.

If $3w^{2}-4nw+2nd>0$, then

$\displaystyle A_{3}(n,d,w)$
$\displaystyle\leq\bigg{\lfloor}\frac{2nd}{3w^{2}-4nw+2nd}\bigg{\rfloor}.$ (1)

A weighing matrix $W(n,p)$ is said to be in _normal form_ if it has the block
configuration

$\begin{bmatrix}\mathbf{0}_{n-p}&R\\\ \mathbf{1}_{p}&D\end{bmatrix},$

for some $(0,\pm 1)$-matrices $R$ and $D$, where $\mathbf{0}_{n-p}$ is the
$(n-p)\times 1$ column vector with all entries $0$ and $\mathbf{1}_{p}$ is the
$p\times 1$ column vector of all entries $1$. We call $R$ the _residual_ and
$D$ the _derived_ parts of the weighing matrix. It follows that
$RR^{t}=pI_{n-p}$, and $DD^{t}=pI_{p}-J_{p}$, and $RD^{t}=DR^{t}=0$. By
permuting and negating some rows, if necessary, every weighing matrix can be
assumed to be in normal form. The Jacobsthal matrix, as described below, is
used extensively in this paper, see [6].

######  Theorem 2.

There is a $(0,\pm 1)$-matrix $Q$ of an odd prime power order $p$ having zero
on the diagonal and $\pm 1$ off-diagonal with row and column sum zero and
$QQ^{t}=pI_{p}-J_{p}$.

######  Theorem 3.

Let $C$ be a conference matrix $W(n+1,n)$ with the $n\times n$ matrix $D$
being the derived part of $C$. The rows of $D$ form an optimal constant weight
ternary code with minimum distance $\frac{n+3}{2}$ and
$A_{3}(n,\frac{n+3}{2},n-1)=n$.

###### Proof.

The inner product of two distinct rows of $D$ is $-1$. There are thus
$(n-1)/2$ minus ones and $(n-3)/2$ plus ones in the inner product. It follows
that the distance between any two rows is $\frac{n-1}{2}+2=\frac{n+3}{2}$.
Considering that $3w^{2}-4nw+2nd=3(n-1)^{2}-4n(n-1)+2n(\frac{n+3}{2})=n+3$, it
follows from Johnson bound (1) above that $A_{3}(n,\frac{n+3}{2},n-1)\leq n$.
This completes the proof. ∎

A second Johnson bound which will be used is as follows.

######  Theorem 4.

$\displaystyle A_{3}(n,d,w)$
$\displaystyle\leq\bigg{\lfloor}\frac{2n}{w}A_{3}(n-1,d,w-1).\bigg{\rfloor}$
(2)

As an application of Johnson bound (2) and Theorem 3 above a large set of
optimal ternary codes are obtained in the next theorem.

######  Theorem 5.

Let $C$ be a conference matrix of order $n+1$. Then the rows of $C$ and $-C$
together form an optimal constant weight ternary code and so

$A_{3}(n+1,\frac{n+3}{2},n)=2(n+1).$

###### Proof.

From Theorem 3 we know that $A_{3}(n,\frac{n+3}{2},n-1)=n$. By Johnson bound
(2):

$\displaystyle A_{3}(n+1,\frac{n+3}{2},n)$
$\displaystyle\leq\frac{2(n+1)}{n}A_{3}(n,\frac{n+3}{2},n-1)$
$\displaystyle=\frac{2(n+1)}{n}n=2(n+1).$

For convenience we may assume that

$\begin{bmatrix}C\\\ -C\end{bmatrix}=\begin{bmatrix}0&\mathbf{1}_{n}^{t}\\\
\mathbf{1}_{n}&D\\\ 0&-\mathbf{1}_{n}^{t}\\\ -\mathbf{1}_{n}&-D\end{bmatrix}.$

The codewords are of length $n+1$ and weight $n$. The minimum distance in $C$
and $-C$ stays as $\frac{n+3}{2}$. The only part requiring attention is taking
a row $i$ of $D$ and $-j$ in $-D$. If $i=|-j|$, then the distance between row
$i$ of $D$ and $-j$ of $-D$ is $n-1$ and thus the distance between the two
longer rows is $n$. For $i\neq|-j|$ the inner product of row $i$ of $D$ and
$-j$ of $-D$ is one and the distance between the two rows is
$\frac{n-3}{2}+2=\frac{n+1}{2}$. The entries in the first column now make the
difference and the minimum distance between the longer rows stays as
$\frac{n+3}{2}$. This completes the proof. ∎

Theorem 5 extends Theorem 16 of [5].

######  Corollary 6.

Let $p$ be an odd prime power, then
$A_{3}(p^{m}+1,\frac{p^{m}+3}{2},p^{m})=2(p^{m}+1)$ for every positive integer
$m$.

###### Proof.

There is a conference matrix $W(p^{m}+1,p^{m})$ for every odd prime power $p$
and positive integer $m$ and the result follows. ∎

######  Remark.

There are many orders not covered by the Theorem 16 of [5]. For example, there
is a conference matrix $W(16,15)$ and $15$ is not a prime number.

The main result of the paper is in part an application of orthogonal arrays.

######  Definition 7.

Let $S=\\{1,2,\dots,q\\}$ be some finite alphabet. An _orthogonal array_ of
strength $t$ and index $\lambda$ is an $N\times k$ matrix over $S$ such that
in every $N\times t$ subarray, each $t$-tuple in $S^{t}$ appears $\lambda$
times. We denote this property as $OA_{\lambda}(N,k,q,t)$.

######  Theorem 8.

For the prime power $p$ and the positive integer $m$ there is a

$p^{m+1}\times\left(\frac{p^{m+1}-1}{p-1}\right)$

array ${\bf O}$ in $p$ symbols such that any two distinct rows share a common
symbol in exactly

$\frac{p^{m}-1}{p-1}$

columns.

Theorem 8, see [4] for details, will be used extensively in the next section.
The main reference for orthogonal arrays is [1].

##  3 The main construction

There was no condition imposed on the weight of a conference matrix in Theorem
5. In this section the flat assumption is that all conference matrices have
odd prime power weights, unless otherwise specified. The following Lemma is
immediate.

######  Lemma 9.

Let $W$ be a weighing matrix $W(n,p)$ whose rows form constant ternary codes
of length $n$, weight $p$ and minimum distance $d$. Then the rows of
$W\otimes\begin{bmatrix}\mathbf{1}^{t}_{q}\end{bmatrix}$ are mutually
orthogonal and form constant weight ternary codes of length $qn$, weight $pq$
and minimum distance $qd$.

The main construction is recursive and consists of two parts. Before
proceeding to the main construction an example is helpful.

######  Example 1.

For the prime $p=3$

$\displaystyle A_{3}(13,9,9)=$ $\displaystyle\,26.$ (3)

In order to show equation (3) the construction is broken into seven steps.

  1. 1.

Starting with a normalized $W(4,3)$ ($-1$ is denoted by $-$):

$W=\begin{bmatrix}0&1&1&1\\\ 1&0&1&-\\\ 1&-&0&1\\\ 1&1&-&0\end{bmatrix}.$

  2. 2.

The derived part of $W$ is

$D=\begin{bmatrix}0&1&-\\\ -&0&1\\\ 1&-&0\end{bmatrix}.$

  3. 3.

A corresponding orthogonal array from Theorem 8 on symbols $\\{1,2,3\\}$ is

$\begin{array}[]{cccc}1&1&1&1\\\ 1&2&2&2\\\ 1&3&3&3\\\ 2&1&2&3\\\ 2&3&1&2\\\
2&2&3&1\\\ 3&1&3&2\\\ 3&2&1&3\\\ 3&3&2&1\end{array}.$

  4. 4.

Replacing the symbol $i$ with row $i$ of $D$ to find a matrix $\mathcal{D}$:

$\left[\begin{array}[]{cccccccccccc}0&1&-&0&1&-&0&1&-&0&1&-\\\
0&1&-&-&0&1&-&0&1&-&0&1\\\ 0&1&-&1&-&0&1&-&0&1&-&0\\\
-&0&1&0&1&-&-&0&1&1&-&0\\\ -&0&1&1&-&0&0&1&-&-&0&1\\\
-&0&1&-&0&1&1&-&0&0&1&-\\\ 1&-&0&0&1&-&1&-&0&-&0&1\\\
1&-&0&-&0&1&0&1&-&1&-&0\\\ 1&-&0&1&-&0&-&0&1&0&1&-\end{array}\right].$

Applying the restricted Johnson bound from Theorem 1, we can see that the rows
of this matrix form optimal constant weight ternary codes of length $n=12$,
weight $w=8$, minimum distance $d=9$ and $A_{3}(12,9,8)=9$. Applying the
second Johnson bound, Theorem 4, we note that $A_{3}(13,9,9)=26$.

  5. 5.

Forming a matrix $\mathcal{R}=W\otimes[111]$:

$\left[\begin{array}[]{cccccccccccc}0&0&0&1&1&1&1&1&1&1&1&1\\\
1&1&1&0&0&0&1&1&1&-&-&-\\\ 1&1&1&-&-&-&0&0&0&1&1&1\\\
1&1&1&1&1&1&-&-&-&0&0&0\end{array}\right].$

  6. 6.

Adding the two matrices $\mathcal{D}$ and $\mathcal{R}$ together with an
additional column of 1’s and 0’s the matrix

$C=\begin{bmatrix}\mathbf{0}&\mathcal{R}\\\
\mathbf{1}&\mathcal{D}\end{bmatrix}=\\\
\left[\begin{array}[]{ccccccccccccc}0&0&0&0&1&1&1&1&1&1&1&1&1\\\
0&1&1&1&0&0&0&1&1&1&-&-&-\\\ 0&1&1&1&-&-&-&0&0&0&1&1&1\\\
0&1&1&1&1&1&1&-&-&-&0&0&0\\\ 1&0&1&-&0&1&-&0&1&-&0&1&-\\\
1&0&1&-&-&0&1&-&0&1&-&0&1\\\ 1&0&1&-&1&-&0&1&-&0&1&-&0\\\
1&-&0&1&0&1&-&-&0&1&1&-&0\\\ 1&-&0&1&1&-&0&0&1&-&-&0&1\\\
1&-&0&1&-&0&1&1&-&0&0&1&-\\\ 1&1&-&0&0&1&-&1&-&0&-&0&1\\\
1&1&-&0&-&0&1&0&1&-&1&-&0\\\ 1&1&-&0&1&-&0&-&0&1&0&1&-\end{array}\right].$

is obtained. $C$ is a weighing matrix $W(13,9)$ and the rows of $C$ consist of
13 constant weight ternary codewords of length $n=13$, weight $w=9$ and
constant distance $d=9$.

  7. 7.

The final step is to form a $26\times 13$ matrix

$T_{13,9,9}=\begin{bmatrix}C\\\ -C\end{bmatrix}$

whose rows form the codewords of optimal constant weight ternary codes of
length $n=13$, $w=9$, and $d=9$, demonstrating that $A_{3}(13,9,9)=26$.

$T_{13,9,9}=\left[\begin{array}[]{ccccccccccccc}0&0&0&0&1&1&1&1&1&1&1&1&1\\\
0&1&1&1&0&0&0&1&1&1&-&-&-\\\ 0&1&1&1&-&-&-&0&0&0&1&1&1\\\
0&1&1&1&1&1&1&-&-&-&0&0&0\\\ 1&0&1&-&0&1&-&0&1&-&0&1&-\\\
1&0&1&-&-&0&1&-&0&1&-&0&1\\\ 1&0&1&-&1&-&0&1&-&0&1&-&0\\\
1&-&0&1&0&1&-&-&0&1&1&-&0\\\ 1&-&0&1&1&-&0&0&1&-&-&0&1\\\
1&-&0&1&-&0&1&1&-&0&0&1&-\\\ 1&1&-&0&0&1&-&1&-&0&-&0&1\\\
1&1&-&0&-&0&1&0&1&-&1&-&0\\\ 1&1&-&0&1&-&0&-&0&1&0&1&-\\\
0&0&0&0&-&-&-&-&-&-&-&-&-\\\ 0&-&-&-&0&0&0&-&-&-&1&1&1\\\
0&-&-&-&1&1&1&0&0&0&-&-&-\\\ 0&-&-&-&-&-&-&1&1&1&0&0&0\\\
-&0&-&1&0&-&1&0&-&1&0&-&1\\\ -&0&-&1&1&0&-&1&0&-&1&0&-\\\
-&0&-&1&-&1&0&-&1&0&-&1&0\\\ -&1&0&-&0&-&1&1&0&-&-&1&0\\\
-&1&0&-&-&1&0&0&-&1&1&0&-\\\ -&1&0&-&1&0&-&-&1&0&0&-&1\\\
-&-&1&0&0&-&1&-&1&0&1&0&-\\\ -&-&1&0&1&0&-&0&-&1&-&1&0\\\
-&-&1&0&-&1&0&1&0&-&0&-&1\end{array}\right].$

######  Remark.

The construction of ternary codes is recursive. The matrix $C$ obtained in the
preceding construction is used next to show that $A_{3}(40,27,27)=80$, etc.

The first class of optimal ternary codes which will be used in the proof of
the second class is introduced next.

######  Theorem 10.

For any odd prime power $p$ and positive integer $m$

$A_{3}\left(p\left(\frac{p^{m+1}-1}{p-1}\right),p^{m}\left(\frac{p+3}{2}\right),p^{m+1}-1\right)=p^{m+1}.$

###### Proof.

Let $W$ be the Jacobsthal matrix of order $p$ described in Theorem 2 and note
that the distance between any two rows is $\frac{p+3}{2}$. Consider the
orthogonal array $\bf O$ in $p$ symbols of Theorem 8 corresponding to the
positive integer $m$. By replacing the $p$ symbols with the rows of $D$ a
$p^{m+1}\times p(\frac{p^{m+1}-1}{p-1})$ array, say $T_{p^{m+1}}$, is obtained
in which any two distinct rows share exactly $\frac{p^{m}-1}{p-1}$ rows of $D$
in the same columns. Noting this, a careful calculation shows that the
distance between any two rows, considered as ternary codes, is

$\bigg{(}\frac{p^{m+1}-1}{p-1}-\frac{p^{m}-1}{p-1}\bigg{)}\bigg{(}\frac{p+3}{2}\bigg{)}=p^{m}\bigg{(}\frac{p+3}{2}\bigg{)}.$

The weight of each code is
$w=\big{(}\frac{p^{m+1}-1}{p-1}\big{)}(p-1)=p^{m+1}-1$. The rows of
$T_{p^{m+1}}$ form the codewords of constant weight ternary code of length
$n=p(\frac{p^{m+1}-1}{p-1})$, minimum distance $d=p^{m}(\frac{p+3}{2})$ and
weight $w=p^{m+1}-1$. The condition for the Johnson bound (1) is obtained to
be $3w^{2}-4nw+2nd=(\frac{p+3}{p-1})(p^{m+1}-1)$. This and the Johnson bound
shows that

$A_{3}\left(p\left(\frac{p^{m+1}-1}{p-1}\right),p^{m}\left(\frac{p+3}{2}\right),p^{m+1}-1\right)\leq
p^{m+1}.$

This completes the proof. ∎

The second class of optimal ternary codes is introduced next.

######  Theorem 11.

There is a weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight
$p^{m}$ for which the rows of $W$ and $-W$ together form the codewords of an
optimal constant weight ternary code of length $n$, weight $w=p^{m}$, and
minimum distance $d=p^{m-1}(\frac{p+3}{2})$ for each odd prime power $p$ and
integer $m\geq 1$ demonstrating that

$A_{3}\left(\frac{p^{m+1}-1}{p-1},p^{m-1}\left(\frac{p+3}{2}\right),p^{m}\right)=2\left(\frac{p^{m+1}-1}{p-1}\right).$

###### Proof.

The proof is by induction on $m$. For $m=1$ the statement is to show that
$A_{3}(p+1,\frac{p+3}{2},p)=2(p+1)$. By Theorems 2, 3 and Corollary 6, there
is a conference matrix $C=W(p+1,p)$ for which
$A_{3}(p+1,\frac{p+3}{2},p)=2(p+1)$.
Assuming the existence of a weighing matrix $C_{m}$ of order
$n=\frac{p^{m+1}-1}{p-1}$, weight $w=p^{m}$ and minimum distance
$d=p^{m-1}(\frac{p+3}{2})$ we proceed in three steps.

  1. 1.

Considering the $p^{m+1}\times p(\frac{p^{m+1}-1}{p-1})$ matrix $\mathcal{D}$
constructed in Theorem 10, the $(0,\pm 1)$ rows form $p^{m+1}$ codewords of
length $p(\frac{p^{m+1}-1}{p-1})$, weight $w=p^{m+1}-1$, and minimum distance
$p^{m}(\frac{p+3}{2})$.

  2. 2.

Let
$\mathcal{R}=C_{m}\otimes{\begin{bmatrix}\mathbf{1}_{p}^{t}\end{bmatrix}}$.
$\mathcal{R}$ is a $\frac{p^{m+1}-1}{p-1}\times
p\big{(}\frac{p^{m+1}-1}{p-1}\big{)}$ $(0,\pm 1)$-matrix. The weight of each
row is $w=p(p^{m})=p^{m+1}$ and the minimum distance is
$d=p^{m}(\frac{p+3}{2})$.

  3. 3.

The matrices

$W=\begin{bmatrix}\mathbf{0}_{\frac{p^{m+1}-1}{p-1}}&\mathcal{R}\\\
\mathbf{1}_{p^{m+1}}&\mathcal{D}\end{bmatrix},$

and $-W$ provides all the codewords.
There remains to show that the distance between one codeword from $W$ and one
from $-W$ is not smaller than $p^{m-1}(\frac{p+3}{2})$. This is easy to see.
The distance of a codeword $k$ in $W$ from a codeword $-j$ in $-W$ is either
$p^{m+1}$ or $p^{m}(\frac{p+3}{2})$ depending on whether $k=j$ or $k\neq j$,
respectively. $\frac{p+3}{2}\leq p$ for all $p\geq 3$. This completes the
proof. ∎

######  Remark.

The ternary codes from Theorem 11 corresponding to $p=3$ are in addition
_equidistant_. So, there is an optimal set of equidistant constant weight
ternary codes consisting of $n=3^{m+1}-1$ codewords of length
$n=\frac{3^{m+1}-1}{2}$, weight $w=3^{m}$ and minimum distance $d=3^{m}$ for
each positive integer $m$. All other ternary codes obtained from Theorem 11
are 2-distance codes.

##  4 Ternary codes from balanced weighing matrices

A weighing matrix $W(v,k)=[w_{ij}]$ is said to be _balanced_ if the matrix of
absolute values $[|w_{ij}|]$ is the incidence matrix of a symmetric balanced
incomplete design with parameters $(v,k,\lambda)$, [3] for details. To
emphasize that a weighing matrix is balanced it is denoted by
$BW(v,k,\lambda)$, where $\lambda=\frac{k(k-1)}{v-1}$. Examples of balanced
weighing matrices include conference matrices, those with classical parameters
$W\big{(}\frac{p^{m+1}-1}{p-1},p^{m}\big{)}$ and a few others. The balanced
structure of weighing matrices used in previous sections are instrumental with
the generation of optimal codes. A natural question is if all balanced
weighing matrices lead to optimal codes. The answer in general depends on the
parameters $(v,k,\lambda)$ and it seems to be difficult.

######  Lemma 12.

Let $C_{v}$ be the set of ternary codes consisting of the rows of a
$BW(v,k,\lambda)=[w_{ij}]$. Then the distance between the codewords is the
constant

$d_{3}=\frac{4k(v-1)-3k(k-1)}{2(v-1)},$

and the distance between the binary codewords consisting of the rows of
$|W|=[|w_{ij}|]$ is the constant

$d_{2}=2\left(k-\frac{k(k-1)}{v-1}\right).$

###### Proof.

The matrix $|W|=[|w_{ij}|]$ is the incidence matrix of a symmetric
$(v,k,\lambda)$ design, where $\lambda=\frac{k(k-1)}{v-1}$. Any two distinct
binary codewords have $\lambda$ ones in the same columns and the remaining
$k-\lambda$ ones have zero in the same columns. This shows that
$d_{2}=2\left(k-\frac{k(k-1)}{v-1}\right)$. The same arrangement happens for
the codewords in $C_{v}$. Since the two rows are orthogonal, there are
$\lambda/2$ (note that this forces $\lambda$ to be even) $-1$’s in the same
column contributing the same number to the distance in addition to the
$d_{2}=2\big{(}k-\frac{k(k-1)}{v-1}\big{)}$. Therefore,

$\displaystyle
d_{3}=2\left(k-\frac{k(k-1)}{v-1}\right)+\frac{k(k-1)}{2(v-1)}=\frac{4k(v-1)-3k(k-1)}{2(v-1)}.$

This completes the proof. ∎

Theorem 3 is extended to the following. The rows of the derived part of a
balanced weighing matrix form an optimal constant weight ternary code.

######  Theorem 13.

Let $\big{(}v,k,\frac{k(k-1)}{v-1}\big{)}$ be the parameters of a balanced
weighing matrix, and $d=\frac{4k(v-1)-3k(k-1)}{2(v-1)}$. Then

$A_{3}(v-1,d,k-1)=k.$

###### Proof.

Let $W$ be a balanced weighing matrix $BW(v,k,\lambda)$. The rows of the
derived part of $W$ form an optimal constant weight ternary code of length
$n=v-1$, constant weight $w=k-1$ and the constant distance $d$.

The condition $3w^{2}-4nw+2nd=4(v-1)-3(k-1)>0$ by the assumption for the given
parameters. By inequality (1) of Theorem 1

$A_{3}(v-1,d,k-1)\leq\frac{4k(v-1)-3k(k-1)}{4(v-1)-3(k-1)}=k.$

This completes the proof. ∎

A infinite family $BW(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^{m})$
including $BW(19,9,4)$ was constructed in [4]. As a corollary, we obtain:

######  Corollary 14.

$A_{3}(18\cdot\frac{9^{m+1}-1}{8},12\cdot 9^{m},9^{m+1}-1)=9^{m+1}$ for every
non-negative integer $m$.

######  Theorem 15.

Let $\big{(}v,k,\frac{k(k-1)}{v-1}\big{)}$ be the parameters of a balanced
weighing matrix for which $v>\frac{3k-1}{2}$. Then

$v\leq A_{3}(v,d_{3}=\frac{4k(v-1)-3k(k-1)}{2(v-1)},k)<2v.$

###### Proof.

Working out the condition in the restricted Johnson bound Theorem 1

$3k^{2}-4vk+2vd_{3}=3k\bigg{(}k-\frac{v(k-1)}{v-1}\bigg{)}$

which is positive. The upper bound from Theorem 1 is
$\frac{v(4v-3k-1)}{3(v-k)}$. It follows that

$v\leq A_{3}(v,d_{3},k)<2v$

if and only if $v>\frac{3k-1}{2}$. ∎

######  Remark.

There is a $BW(19,9,4)$, see [4]. The rows of the derived part of this
weighing matrix form an optimal constant weight $8$ and equidistant $d=12$
ternary codes of length $18$ by Theorem 13. The condition in Theorem 15 holds
and

$19\leq A_{3}(19,12,9)<38.$

The Johnson upper bound in Theorem 1 provides a smaller number of $30$ for the
possible number of codewords.

## Acknowledgments.

The authors acknowledge many help and guidance from Professor Vladimir
Tonchev. Hadi Kharaghani is supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC). Sho Suda is supported by JSPS KAKENHI
Grant Number 18K03395.

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