# Topological edge states in a Rydberg composite

Matthew T. Eiles<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer
Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany  Christopher W. Wächtler
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187
Dresden, Germany  Department of Physics, University of California, Berkeley,
California 94720, USA  Alexander Eisfeld  Max-Planck-Institut für Physik
komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany  Universität
Potsdam, Institut für Physik und Astronomie, Karl-Liebknecht-Str. 24-25, 14476
Potsdam, Germany  Jan M. Rost  Max-Planck-Institut für Physik komplexer
Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany

###### Abstract

We examine topological phases and symmetry-protected electronic edge states in
the context of a Rydberg composite: a Rydberg atom interfaced with a
structured arrangement of ground-state atoms. The electronic Hamiltonian of
such a composite possesses a direct mapping to a tight-binding Hamiltonian,
which enables the realization and study of a variety of systems with non-
trivial topology by tuning the arrangement of ground-state atoms and the
excitation of the Rydberg atom. The Rydberg electron moves in a combined
potential including the long-ranged Coulomb interaction with the Rydberg core
and short-ranged interactions with each neutral atom; the effective
interactions between sites are determined by this combination. We first
confirm the existence of topologically-protected edge states in a Rydberg
composite by mapping it to the paradigmatic Su-Schrieffer-Heeger dimer model.
Following that, we study more complicated systems with trimer unit cells which
can be easily simulated with a Rydberg composite.

Topological insulators Moore (2009, 2010); Hasan and Moore (2011) describe a
special class of solids exhibiting an insulating bulk but conducting surface
states. These display a surprising immunity to a wide range of local
deformations, inherently avoiding backscattering over broad energy ranges and
circumventing localization in the presence of disorder. Questions about the
existence, behavior, and characterization of topological insulators and the
symmetry-protected edge states that they can host have motivated rapid growth
in this field in recent years Mittal  _et al._ (2014); Lohse  _et al._ (2016);
Nakajima  _et al._ (2016); Lohse  _et al._ (2018); Kraus  _et al._ (2012);
Verbin  _et al._ (2015); Zilberberg  _et al._ (2018); Kane and Lubensky
(2014); Paulose  _et al._ (2015); Salerno and Carusotto (2014); Süsstrunk and
Huber (2015); Ningyuan  _et al._ (2015); Albert  _et al._ (2015); Aspuru-Guzik
and Walther (2012); Kitagawa  _et al._ (2012); Zhao  _et al._ (2018);
Arkinstall  _et al._ (2017); Zhang  _et al._ (2019); Yuen-Zhou  _et al._
(2014). A major effort in this direction is the exploration of well-controlled
systems that exhibit novel topological properties and can be used to clarify
questions about the behavior of topological invariants. Ultracold Rydberg
atoms are promising quantum simulators in this respect due to their high
controllability and exaggerated properties. Two recent examples using very
different approaches illustrate this in the context of the Su-Schriefer-Heeger
(SSH) model Su  _et al._ (1979). In the first, several Rydberg atoms were
arranged in an optical tweezer array. The long-range dipolar interactions
between two Rydberg states of different angular momentum enabled the desired
staggered hopping amplitudes de Léséleuc  _et al._ (2019). In the second
example, multiple Rydberg levels of a single atom were employed to form a
synthetic one-dimensional lattice, with microwave coupling between these
levels setting the hopping amplitudes Kanungo  _et al._ (2022). The design of
simulators of topologically richer systems than the SSH model remains an area
of active research, with most proposals utilizing the long-range interactions
between Rydberg atoms prepared in complicated geometries Yang  _et al._
(2022); Samajdar  _et al._ (2021); Li  _et al._ (2021); Verresen and
Vishwanath (2022); Giudici  _et al._ (2022); Samajdar  _et al._ (2023); Weber
_et al._ (2022).

In this paper, we introduce a different approach utilizing a single Rydberg
atom and an ensemble of trapped ground-state atoms (scatterers) located within
the Rydberg electron’s orbit. Confinement of these scatterers in a particular
arrangement can be provided by an optical lattice or in an array of optical
tweezers. Such a Rydberg composite allows us to design Hamiltonians which
exhibit features associated with symmetry-protected topological insulators. As
we will show, the topological properties can be tuned by the choice of the
scatterer positions and by the principal quantum number $\nu$ of the Rydberg
atom. To demonstrate how to use a Rydberg composite to study topological
physics, we show how to realize three different lattices of increasing
complexity.

Figure 1:  Schematics of the three models: (a) dimer SSH chain, (b) trimer SSH
chain, and (c) triangle chain. In each panel, (i) depicts the desired tight-
binding lattice and defines the couplings between sites, and (ii) sketches the
Rydberg composite corresponding to this lattice. The arc lengths shown in this
panel determine the effective couplings, as illustrated in each panel by
(iii), which shows the relevant interaction curves. The interaction curve $V$
in (a)iii shows the hopping amplitude between site $5$ and neighboring sites
as a function of $D_{5}$, the distance around the circle away from site $5$.
As the position of site 6 moves, so does the energy $V_{56}$ (blue circle
marked 6), modifying the ratio $v_{2}/v_{1}$. Two different interaction curves
are shown in (b)iii. The black (pink) curve shows the interaction $V$ as a
function of the distance $D_{6}$ ($D_{5}$) away from site 6 (5). The blue
markers denote the on-site potential $E_{6}$ and hopping amplitudes $V_{65}$,
$V_{67}$, and $V_{68}$; the pink markers show $E_{5}$, $V_{54}$, $V_{56}$, and
$V_{57}$. The next-nearest-neighbor amplitudes $V_{68}$ and $V_{57}$ are
negligible. By varying $t_{1}$, the distance between sites $6$ and $7$ is
changed, which modifies $V_{67}$ while keeping constant the other
interactions. Panel (c)iii shows these same interaction curves, but now in the
more complicated geometry relevant to the triangle lattice. Careful inspection
of the different points on these two curves shows how the hopping amplitudes
$u$, $a$, $b$, and $c$ in the triangle lattice are realized; the amplitude $u$
is fixed for this geometry, while the other amplitudes vary as a function of
$t_{1}$.

.

The effective Hamiltonian Eiles  _et al._ (2016, 2023); Hunter  _et al._
(2020); Eiles  _et al._ (2020) of a Rydberg composite, with scatterers placed
at positions $\vec{R}_{q}$, is

$H_{\mathrm{e}}=-\frac{\nabla^{2}}{2}-\frac{1}{r}+\sum_{q=1}^{M}2\pi
a_{s}\delta^{3}(\vec{r}-\vec{R}_{q})$ (1)

in atomic units. The first two terms govern the electron’s motion in the
Coulomb field of the Rydberg core, while the last term describes its
interaction with the scatterers using the Fermi pseudopotential, valid in the
low-energy scattering limit. The interaction strength is determined by the
$S$-wave scattering length $a_{s}$ Greene  _et al._ (2000); Eiles (2019). We
neglect the role of quantum defects, as we focus on the perturbation of the
degenerate hydrogen-like states with angular momentum greater than $3$ Eiles
_et al._ (2023). To facilitate the choice of scatterer positions to realize a
particular topologically interesting system, it is convenient to transform the
Hamiltonian (1) to the form of a tight-binding Hamiltonian. This mapping
relies on the fact that the electron-scatterer interaction Fermi (2008); Omont
(1977) is too weak to mix Rydberg states with different principal quantum
numbers $\nu$, but nevertheless it splits the degenerate energy levels with
different angular momentum $l$ but the same $\nu$ into two subspaces. One, of
size $\nu^{2}-M$, remains degenerate and unshifted, while the second, of size
$M$, splits away Eiles  _et al._ (2023); Eiles (2019). The spectrum of the
Hamiltonian $H_{\mathrm{e}}$ restricted to this non-trivial subspace coincides
exactly with that of the Hamiltonian

$H=\sum_{q}^{M}E_{q}|{q}\rangle\langle{q}|+\sum_{q}^{M}\sum_{q^{\prime}\neq
q}^{M}V_{qq^{\prime}}|{q}\rangle\langle{q^{\prime}}|.$ (2)

The states $|{q}\rangle$ describes a wave function which is localized on the
scatterer at position $\vec{R}_{q}$. The matrix elements $E_{q}$ and
$V_{qq^{\prime}}$ are determined by the positions of the scatterers and the
principal quantum number $\nu$ 111 In the high $\nu$ limit where the effect of
quantum-defect-shifted states is negligible, the matrix elements of Eq.2 are
given in closed form:  $\displaystyle V_{qq^{\prime}}$
$\displaystyle=\frac{u_{\nu 0}^{\prime}(x_{-})u_{\nu 0}(x_{+})-u_{\nu
0}(t_{-})u_{\nu 0}^{\prime}(x_{+})}{2(x_{+}-x_{-})}.$ (3) Here,
$x_{\pm}=\frac{1}{2}\left(R_{q}+R_{q^{\prime}}\pm|\vec{R}_{q}-\vec{R}_{q^{\prime}}|\right)$,
$R_{q}=|\vec{R}_{q}|$, and $u_{\nu 0}(r)$ is the $s$-wave reduced hydrogen
radial function defined in the usual way, $\langle\vec{r}|\nu
lm\rangle=\frac{1}{r}u_{\nu l}(r)Y_{lm}(\hat{r})$. The prime denotes the
spacial derivative.. Further details about the connection of this Hamiltonian
with Eq. (1) can be found in Refs. Hunter  _et al._ (2020); Eiles  _et al._
(2023, 2020).

The three composites illustrated in Figure 1 exemplify the ability of
different scatterer arrangements, in combination with the choice of Rydberg
state $\nu$, to design and realize different effective lattice Hamiltonians.
We will use these three composites to demonstrate topological physics in a
Rydberg composite. In each case, we consider atoms arranged in a (broken) ring
around the Rydberg atom, which fixes a common $E_{q}$ for all scatterers. We
select the number of scatterers $M$, the principal quantum number $\nu$, and
radius $R=R_{q}$ such that the desired hopping terms $V_{qq^{\prime}}$ are
realized. To confirm that this leads to a topologically non-trivial
configurations, we theoretically analyze each setup by applying periodic
boundary conditions to the effective Hamiltonian, giving insight into its
topological bulk properties. The bulk-boundary correspondence allows us to
subsequently ascertain its topological aspects in the finite system.

The first model that we consider is the dimer SSH model depicted in Fig.
1a(i): a one-dimensional lattice with staggered nearest-neighbor hopping
amplitudes $v_{1}$ and $v_{2}$ Su  _et al._ (1979). This paradigmatic model
introduces many concepts useful in the analysis of more complicated systems.
Fig. 1a(ii) shows how to design a Rydberg composite which realizes this model.
We set $R=R_{1}=2\nu^{2}$ and place the scatterers on this ring so that they
are separated by arclengths $t_{1}$ and $t_{2}$. By choosing $M<\nu$ we
guarantee that the scatterers are spaced sufficiently far apart that
$V_{qq^{\prime}}$ is negligible when $|q^{\prime}-q|>1$ Eiles  _et al._
(2023). Then, the monotonic dependence of $V_{qq^{\prime}}$ on $t_{1}$ allows
us to stagger the hopping elements simply by staggering the distances $t_{1}$
and $t_{2}$, as is illustrated in Fig. 1(a)iii. For illustrative purposes we
choose $\nu=60$, $M=36$, and fix $t_{2}=2\pi R/45$. These choices yield a
reasonably large lattice size without making the resulting band spectrum and
wave functions overly complicated to visualize. Fig. 2 shows the resulting
eigenspectrum as a function of $t_{1}$. It consists of two distinct bands
separated by an empty band gap when $t_{2}>t_{1}$, i.e. when $v_{1}>v_{2}$
(see Fig. 1a(i)). This band gap closes when $t_{1}=t_{2}$. When $t_{1}>t_{2}$,
$v_{2}>v_{1}$ and the two levels closest to the band gap, highlighted in blue,
split from the bulk and become degenerate at the center of the spectrum.
Investigation of the eigenstates of these levels shows that they evolve from
bulk states spread over the entire composite when $t_{2}<t_{1}$ to edge states
appearing at the boundary of the chain of scatterers when $t_{1}>t_{2}$. In
the two eigenstates depicted in Fig. 2, the black spheres represent the wave
function in the site basis used to describe $H$ in Eq. (2), while the blue
wave function shows the full electronic wave function in position space 222We
plot the absolute values of the wave functions here..

Figure 2:  The energy spectrum of the SSH Rydberg composite, where $\nu=60$
and $R=2\nu^{2}$. The energies are centered around the on-site potential
$E_{q}$, and are plotted as a function of $t_{1}$ for fixed $t_{2}=2\pi R/45$.
The exemplary wave function images shown below the spectrum illustrate the
bulk eigenstate corresponding to the dashed-blue level in the upper band
(left, $t_{1}=3t_{2}/4$) and an edge state (right, $t_{1}=5t_{2}/4$).

The existence of this transition from bulk states living in the energy bands
to edge states situated at the center of the band gap can be understood after
considering the bulk momentum Hamiltonian

$H_{\text{SSH}}(k)=\begin{pmatrix}0&v_{1}+v_{2}e^{-ik}\\\
v_{1}+v_{2}e^{ik}&0\end{pmatrix},$ (4)

which is obtained by applying periodic boundary conditions to the lattice in
Fig. 1(a)i. For the one-dimensional systems studied here, the Zak phase,

$\mathcal{Z}=i\int_{-\pi}^{\pi}\langle\psi_{k}|\partial_{k}\psi_{k}\rangle
dk,$ (5)

where $|{\psi_{k}}\rangle$ is an eigenstate of $H(k)$, is a quantized
topological invariant which can only take the values zero or $\pi$ (modulo
$2\pi$) as long as a symmetry is present Zak (1989). The chiral symmetry of
the SSH model ensures that such a topological invariant exists. Computation of
the Zak phase using the eigenstates of Eq. 4 gives $\mathcal{Z}=\pi$ when
$t_{2}>t_{1}$ and $\mathcal{Z}=0$ when $t_{2}<t_{1}$ Resta (1994, 2000) 333 We
evaluate $\mathcal{Z}$ on a discrete mesh of $k$ points using
$\mathcal{Z}=-\text{Im}\ln\Pi_{k}\langle\psi_{k}|\psi_{k+1}\rangle,$ (6) which
does not require the phase of the eigenstates $|\psi_{k}\rangle$ to be a
continuous function of $k$. .

The Zak phase is intimately related to the existence of edge states through
the bulk-boundary correspondence: the change in the Zak phase at $t_{1}=t_{2}$
reveals a topological phase transition and the appearance of edge states in
the finite-sized system Su  _et al._ (1979); Zak (1989). The implications of
these edge states are manifold. If the scatterers are not perfectly positioned
at their angles on the ring, the hopping amplitudes $V_{qq^{\prime}}$ become
disordered but the chiral symmetry of the lattice is preserved. In such a
case, although states in the bands will become localized due to this disorder,
these edge states will remain unperturbed due to the symmetry protection
afforded by the topology of the system. In contrast, disorder of the atom
positions in the radial direction leads to diagonal disorder and a breakdown
of the chiral symmetry. Consequently, the edge states are no longer symmetry
protected and become indistinguishable from the band. Moving beyond the SSH
model or random disorder, one could modify the on-site potentials in a
controlled fashion by shifting the scatterers to slightly different ring
radii, therefore simulating the Rice-Mele model Rice and Mele (1982).

The second model we consider is a variation on the SSH model where the dimer
unit cells are replaced with trimers. This model, shown in Fig. 1(b)i, was
very recently explored theoretically Martinez Alvarez and Coutinho-Filho
(2019); Anastasiadis  _et al._ (2022) as well as experimentally Guo  _et al._
(2022); Park  _et al._ (2022); Kartashov  _et al._ (2022). The sites within
each trimer unit cell are coupled by $v_{2}$ and $v_{3}$, and the trimers are
coupled via $v_{1}$. We realize this model in our composite setup by keeping
the Rydberg parameters as before, which guarantees nearest neighbor hopping,
but now place the scatterers in a different geometry, as seen in Fig. 1(b)ii.
Applying periodic boundary conditions to the resulting Hamiltonian yields the
bulk momentum Hamiltonian

$H_{\text{trimer}}(k)=\begin{pmatrix}0&v_{2}&v_{1}e^{-ik}\\\ v_{2}&0&v_{3}\\\
v_{1}e^{ik}&v_{3}&0\end{pmatrix}.$ (7)

The topological nature of this model stems from an inversion symmetry when
$v_{2}=v_{3}$. When $v_{1}=v_{2}=v_{3}$, i.e., in the absence of
trimerization, a phase transition occurs. In the Rydberg composite setting, we
ensure the inversion symmetry by setting $t_{2}=t_{3}$. By varying $t_{1}$ to
change $v_{1}$ we sweep the system through the topological phase transition,
as can be seen in the energy spectrum shown in Fig. 3.

Figure 3:  The energy spectrum of the trimer SSH Rydberg composite, where
$\nu=60$ and $R=2\nu^{2}$. The energies are centered around the on-site
potential $E_{q}$, and are plotted as a function of $t_{1}$ for fixed
$t_{2}=t_{3}=2\pi R/45$. One of the edge states is shown in the inset.

The lower band is characterized by a Zak phase $\mathcal{Z}=\pi$ when
$v_{1}>v_{2},v_{3}$ which implies the existence of a pair of topologically
protected edge states in the lower band gap. In the Rydberg composite, this
energy relationship is satisfied when the distance $t_{1}$ is smaller than
$t_{2}$ and $t_{3}$, and we see in Fig. 3 that edge states appear as expected.
The Zak phase of the middle band is zero for all $v_{1}$, and since the
existence of edge modes in the upper band gap is connected to the sum of the
Zak phases of all bands below it, localized boundary modes must also exist in
the upper band gap. One of the edge states is shown in the inset of Fig. 3.
The more complicated case of a non-inversion symmetric lattice, where
$v_{2}\neq v_{3}$, can no longer be characterized using an integer Zak phase.
It has been shown that it supports both topologically-protected and localized,
but not protected, edge states Martinez Alvarez and Coutinho-Filho (2019).
Such features can also be simulated and observed in the Rydberg composite by
setting $t_{2}\neq t_{3}$.

Figure 4:  (a) The energy spectrum of the triangle chain Rydberg composite.
The black curves show the Rydberg energies as a function of the arc-length
$t_{1}$ (see Fig. 1(c)ii) for $\nu=60$, $R=1.74\nu^{2}$, and $M=24$
scatterers. The energies are centered around the on-site potential $E_{q}$,
and are plotted as a function of $t_{1}$ for fixed $t_{2}=2\pi R/45$. The blue
curves show the spectrum for the same lattice parameters but $M=300$.
Topologically protected edge states are found in the middle of the upper band
gap for all $t_{1}$ values, although they disappear when the bands cross as
discussed in the text. One of these edge states, for $t_{1}=0.2$, is shown in
panel (c). The lower band gap has $\mathcal{Z}=\pi$ only within the gray
shaded regions, where degenerate states in the band gap can be seen in the
spectrum of the larger lattice. (b) The Zak phase of the upper and lower band
gaps. Blue regions have $\mathcal{Z}=\pi$; while white regions have
$\mathcal{Z}=0$. The orange curve shows how this parameter space is traversed
as we parameterically change $t_{1}$, as described in the text. Note that
$u\approx-5.5$MHz for the composite studied here.

These two examples have demonstrated that the Rydberg composite can realize
topologically interesting models, exemplified by the dimer and trimer SSH
chains already studied. We now show how to realize a model with richer
topology and complexity. We consider again a system with a unit cell of three
lattice sites, but now connect each pair of sites within the unit cell with an
equal hopping amplitude $u$. Each triangle is coupled to its neighbor by the
hopping amplitudes $a$ (magenta), $b$ (green), and $c$ (cyan, dashed), as
shown in Fig. 1(c)i; in general these amplitudes can all differ. We realize
this Hamiltonian in the Rydberg composite by keeping the same trimer
structure, but now utilizing one of the most appealing features of the Rydberg
composite: the ease with which we can introduce long-range and oscillatory
interactions to the system by changing the overall size of the ring. In
general, as the radius of the ring shrinks, the interactions become longer-
ranged Eiles  _et al._ (2023).

For this example, we select $\nu=60$ and $R_{2}=1.74\nu^{2}$, which leads to
the interaction curves $V_{qq^{\prime}}$ shown in Fig. 1(c)iii. These curves
are approximately sinuisoidal with a rapidly decreasing amplitude as
$|q-q^{\prime}|$ increases. To see how this choice works to engineer the
desired system, consider first the pink curve, which shows the hopping
amplitude from site $5$ to its neighbors as a continuous function of the
distance $D_{5}$ away from this site. The pink markers on this curve show the
hopping amplitudes connecting site $5$ to the labeled site at the specified
positions. The on-site energy $E_{5}=V_{55}$ is not visible on this scale. The
inversion symmetry of the trimer ensures that $V_{56}=V_{54}$; more distant
sites (i.e. $3$ and $7$) are coupled weakly to this site for the chosen value
of $t_{1}$. Now, to make the couplings $V_{46}$ equal to these couplings, we
take advantage of the oscillatory interaction. Looking at the black curve,
which is the same as the pink curve but relative to site $6$, we can read off
the values of $V_{46}$ and $V_{56}$, which are identical as long as the atoms
in each unit cell are an arclength $2\pi R_{2}/45$ apart. In general, larger
unit cells with $n$ participating atoms having identical all-to-all coupling
can be engineered by finding parameters such that $V(t_{2})=V(2t_{2})=\dots
V(nt_{2})$, where $t_{2}$ is the arclength between atoms in a cell and the
$n+1$th atom is placed further at a $t_{1}+nt_{2}$. It is not clear that such
an arrangement can be found for arbitrary $n$, but we verified that the $n=4$
case can be designed, utilizing a smaller ring size where the envelope decays
less quickly.

The band spectrum for this triangle lattice system and $M=24$ atoms, plotted
with black curves as a function of $t_{1}$, is shown in Fig. 4(a). Three bands
are clearly visible. The upper two repeatedly cross one another. Outside of
these crossing regions, a pair of zero-energy modes is clearly present in the
band gap. Between the lower two bands there are no unambiguous edge states for
this small lattice, although inspection of the eigenstates for slightly larger
lattice dimensions (not shown here) suggests that the states in between the
two bands in the region from $0.1<t_{1}<t_{2}$ are indeed edge states.

To analyze this spectrum, we turn again to the bulk Hamiltonian, which is now
given by

$H_{\text{triangle}}^{\text{bulk}}(k)=\begin{pmatrix}2c\cos
k&u+be^{-ik}&u+ae^{-ik}\\\ u+be^{ik}&2c\cos k&u+be^{-ik}\\\
u+ae^{ik}&u+be^{ik}&2c\cos k\end{pmatrix}.$ (8)

Note that the coupling $c$ enters the bulk Hamiltonian only on the diagonal
and therefore has no impact on the gap closing conditions or the topological
phase transitions. The Hamiltonian is inversion-symmetric, and therefore we
can characterize its topological features using, again, the Zak phase as a
quantized topological invariant. In the bottom (top) panel of Fig. 4(c), we
show the phase diagram of $\mathcal{Z}$ as a function of the hopping
amplitudes $a$ and $b$ for the lower (upper) band gap. The blue (shaded) color
denotes the topological phase ($\mathcal{Z}=\pi$) and white the trivial phase
($\mathcal{Z}=0$). The gray dashed lines show the gap closing conditions,
which are sufficient but not necessary for the existence of a topological
phase transition. These occur when $a=b$ or $b=\pm 2u-a$. We have computed
this phase diagram for all $a$ and $b$ values in the plotted range, but as
these depend parametrically on the arclength $t_{1}$, we cannot probe this
full parameter space in the Rydberg composite. The orange curve shows the path
through this parameter space that we can attain by varying $t_{1}$ over the
range shown in Fig. 4a, starting at $t_{1}=0.09$ at the marker and ending at
$t_{1}=0.33$. Using this curve we can analyze the states in panel (a) with
respect to their topological properties. For this choice of parameters, the
orange curve in the upper band gap does not enter the region with a Zak phase
of 0; for this reason there are always edge states visible when the bands are
not overlapping. When the bands overlap, these edge states disappear even
though the Zak phase remains equal to $\pi$. This surprising observation,
contrary to the common belief that a non-trivial Zak phase implies the
existence of localized edge states, has been previously observed and discussed
in Ref. Malki and Uhrig (2019); Malki  _et al._ (2019), but it does not seem
to be widely known. In contrast, the Zak phase in the bottom gap is zero
except for two intervals. These correspond to the regions highlighted in Fig.
4(a) with gray boxes. Since the spectrum in panel (a) is for only a relatively
small number of scatterers, the distinction between band edges and states
within the band gap is not clear. Therefore, we artificially extended the
system to 300 sites and plotted the resulting spectrum in blue. The edge
states in the lower band gap can now clearly be seen, at least in the larger
region with $\mathcal{Z}=\pi$ just below $t_{1}=t_{2}$. We have confirmed that
these are edge states by adding disorder to the system which preserves the
inversion symmetry. Under this disorder, the character of these edge states is
preserved.

The above examples demonstrate that with Rydberg composites one can realize
Hamiltonians that exhibit a variety of interesting topological features. For
the design and interpretation of the Rydberg composite we utilized the useful
link between the Rydberg composite and a tight-binding Hamiltonian, which was
previously elucidated in the context of Anderson localization Eiles  _et al._
(2023). While we did not consider disorder in the present work, defects in the
positions of the ground state atoms would result in disorder in the hopping
terms and on-site energies of the different models studied here, and – as long
as disorder which breaks the chiral or inversion symmetries of our models
could be avoided – the topological protection of the edge states predicted
here could be studied.

Our scheme could be realized using programmable optical tweezers, which have
become a very powerful and versatile tool recently. Although the parameter
regimes used in our examples here are not immediately realizable in current
platforms, primarily due to the difficulties in arranging atoms on the $\sim
100$ nanometer scale, this is not a fundamental obstacle.

###### Acknowledgements.

MTE thanks S. Zhang for pointing out Refs. Malki  _et al._ (2019); Malki and
Uhrig (2019). C.W.W. acknowledges support from the Max-Planck Gesellschaft via
the MPI-PKS Next Step fellowship and is financially supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project No.
496502542 (WA 5170/1-1). AE acknowledges support from the DFG via a Heisenberg
fellowship (Grant No EI 872/10-1).

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#  On Pless symmetry codes, ternary QR codes, and related Hadamard matrices
and designs

Vladimir D. Tonchev  Department of Mathematical Sciences, Michigan
Technological University, Houghton, MI USA 49931<EMAIL_ADDRESS>
###### Abstract

It is proved that a code $L(q)$ which is monomially equivalent to the Pless
symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a
Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard
matrix of type II. Similarly, any ternary extended quadratic residue code
contains the incidence matrix of a Hadamard 3-design associated with a Paley-
Hadamard matrix of type I. If $q=5,11,17,23$, then the full permutation
automorphism group of $L(q)$ coincides with the full automorphism group of
$D(q)$, and a similar result holds for the ternary extended quadratic residue
codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by
codewords of the Pless symmetry code $C(17)$ are enumerated and classified up
to equivalence. There are two equivalence classes of such matrices: the Paley-
Hadamard matrix $H$ of type I with a full automorphism group of order 19584,
and a second regular Hadamard matrix $H^{\prime}$ such that the symmetric
2-$(36,15,6)$ design $D$ associated with $H^{\prime}$ has trivial full
automorphism group, and the incidence matrix of $D$ spans a ternary code
equivalent to $C(17)$.

Keywords: Pless symmetry code, Hadamard matrix, Hadamard 3-design, Hadamard
2-design, Paley-Hadamard matrix.

##  1 Introduction

We assume familiarity with the basic facts and notions from error-correcting
codes and combinatorial designs and Hadamard matrices [2], [4], [7], [11]. All
codes in this paper are ternary. A monomial matrix with entries from $GF(3)$
is a square matrix such that every row and every column contains exactly one
nonzero entry. By an automorphism group of a ternary code we mean monomial
automorphism group, unless specified otherwise. The permutation automorphism
group of a code is the subgroup of its monomial automorphism group that
consists of coordinate permutations only.

A Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ of $1$’s and
$-1$’s such that $HH^{T}=nI$, where $I$ is the identity matrix. It follows
that $n=1,2$, or $n=4t$ for some integer $t\geq 1$. An automorphism of a
Hadamard matrix $H$ is a pair of $\\{0,1,-1\\}$-monomial matrices $L$, $R$
such that $LHR=H$. Two Hadamard matrices $H_{1}$, $H_{2}$ of the same order
are equivalent if there are monomial matrices $L$, $R$ such that
$LH_{1}R=H_{2}$. A Hadamard matrix $H$ is normalized with respect to its $i$th
row and $j$th column if all entries in row $i$ and column $j$ are equal to 1.
If $H$ is a Hadamard matrix of order $n=4t$ that is normalized with respect to
row $i$ and column $j$, deleting the $i$th row and the $j$th column and
replacing all $-1$’s with zeros gives the $(0,1)$-incidence matrix of a
symmetric 2-$(4t-1,2t-1,t-1)$ design $D$ called a Hadamard 2-design, while
deleting the $j$th column of $H$ and the $j$th column of $-H$ from the matrix
$(H,-H)$ gives the point-by-block $(\pm 1)$-incidence matrix of a
3-$(4t,2t,t-1)$ design $D^{*}$, called a Hadamard 3-design obtained from $H$
with respect to column $j$. The design $D$ is the derived design of $D^{*}$
with respect to its $i$th point. A Hadamard matrix $H$ of order $n=4t$ is
regular if all rows of $H$ contain the same number $k$ of $-1$’s. It follows
that $t=m^{2}$ for some integer $m$, $k=2m^{2}\pm m$, and replacing all $-1$’s
with zeros gives the $(0,1)$-incidence matrix of a symmetric
2-$(4m^{2},2m^{2}\pm t,m^{2}\pm m)$ design. For more on Hadamard matrices and
related designs, see, for example, [2, Chapter 7], [7, Chapter 14], [11, Sec.
8.9].

Let $q$ be an odd prime power such that $q\equiv-1\pmod{3}$. The Pless
symmetry code $C(q)$ [22], [23] of length $n=2q+2$ is a ternary self-dual code
with a generator matrix

$G=(I_{q+1},S_{q}),$ (1)

where $I_{q+1}$ is the identity matrix of order $q+1$, and $S_{q}=(s_{i,j})$
is a $(q+1)\times(q+1)$ matrix defined as follows. The rows and columns of
$S_{q}$ are labeled by $\infty$ and the $q$ elements of the finite field
$GF(q)$ of order $q$, where $s_{\infty,\infty}=0$, $s_{a,a}=0$,
$s_{\infty,a}=1$ for $a\in GF(q)$, $s_{a,\infty}=1$ if $-1$ is a square in
$GF(q)$, and $s_{a,\infty}=-1$ if $-1$ is not a square in $GF(q)$ for $a\in
GF(q)$, and $s_{a,b}=1$ for $a,b\in GF(q)$ such that $a\neq b$ and $a-b$ is a
square in $GF(q)$, and $s_{a,b}=-1$ for $a,b\in GF(q)$ such that $a\neq b$ and
$a-b$ is not a square in $GF(q)$. For example, if $q=5$, the rows and columns
of $S_{5}$ are labeled by $\infty,0,1,2,3,4$, and

$S_{5}=\left(\begin{array}[]{rrrrrr}0&1&1&1&1&1\\\ 1&0&1&-1&-1&1\\\
1&1&0&1&-1&-1\\\ 1&-1&1&0&1&-1\\\ 1&-1&-1&1&0&1\\\
1&1&-1&-1&1&0\end{array}\right).$

The main property of $S_{q}$ is that $S_{q}S_{q}^{T}=qI_{q+1}$, which implies
$S_{q}S_{q}^{T}\equiv-I_{q+1}\pmod{3}$; hence $C(q)$ is self-dual. The
symmetry codes $C(5)$111The symmetry code for $q=5$ is equivalent to the
extended ternary Golay code. , $C(11)$, $C(17)$, $C(23)$ and $C(29)$ are
extremal self-dual codes of length $n$ divisible by 12 and minimum distance
$d$ meeting the Mallows-Sloane upper bound $d\leq 3[n/12]+3$ [16]; hence these
codes support 5-designs by the Assmus-Mattson Theorem [3].

Pless [23] proved that in addition to the trivial monomial automorphism of
order 2 corresponding to the negation of all code coordinates, the monomial
automorphism group of $C(q)$ contains a subgroup of order $q(q^{2}-1)$
isomorphic to $PGL_{2}(q)$. In addition, it was proved in [23, Theorem 4.2]
that the symmetry code $C(q)$ contains a set of $2q+2$ codewords of weight
$2q+2$ that form a Hadamard matrix of order $2q+2$. Specifically, if $q\equiv
1\pmod{4}$ the Hadamard matrix formed by codewords of weight $2q+2$ is

$H_{1}(q)=\left(\begin{array}[]{rr}I+S_{q}&-I+S_{q}\\\
-I+S_{q}&-I-S_{q}\end{array}\right),$ (2)

while if $q\equiv-3\pmod{4}$ the Hadamard matrix is

$H_{3}(q)=\left(\begin{array}[]{rr}I+S_{q}&I+S_{q}\\\
I-S_{q}&-I+S_{q}\end{array}\right),$ (3)

where $I$ is the identity matrix of order $q+1$.

The Hadamard matrices (2), (3) are known in the combinatorial literature as
Paley-Hadamard matrices of type II [7, 14.1], [13], [21]. If $q\equiv
3\pmod{4}$, the $(q+1)\times(q+1)$ matrix obtained by bordering the matrix
$S_{q}-I$ with one all-one row and one all-one column is a Hadamard matrix of
order $q+1$, known as a Paley-Hadamard matrix, or a Paley-Hadamard matrix of
type I [7, 14.1], [13], [21]. The unique (up to equivalence) Hadamard matrix
of order 12 is both a Paley-Hadamard matrix of type I for $q=11$ and a Paley-
Hadamard matrix of type II for $q=5$, and its full automorphism group modulo
its center of order 2 is the Mathieu group $M_{12}$ (Hall [8]). The full
automorphism group of a Paley-Hadamard matrix of type I for $q>11$ was
determined by Kantor [12] and is of order $q(q^{2}-1)$, while the full
automorphism group of a Paley-Hadamard matrix of type II for $q>5$ was
determined by de Launey and Stafford [13], and is of order $4fq(q^{2}-1)$ if
$q=p^{f}$, where $p$ is prime.

If $q=5$, 11, or 23, the number of all codewords of full weight $2q+2$ in the
symmetry code $C(q)$ is exactly $4q+4$ [23]. These codewords span the code and
consist of the rows of the Hadamard matrix $H_{1}(q)$ from (2) (resp.
$H_{3}(q)$ from (3)) and its negative, or $2H_{1}(q)$ (resp. $2H_{2}(q)$);
hence the full monomial automorphism group of $C(q)$ coincides with the full
automorphism group of $H_{1}(q)$ (resp. $H_{3}(q)$) [23].

The symmetry code $C(17)$ of length 36 contains 888 codewords of weight 36;
hence it is not clear whether the automorphism group of the Hadamard matrix
$H_{1}(17)$ (2) is the full automorphism group of $C(17)$. In Section 2, we
prove that the full automorphism group of $C(17)$ coincides with the full
automorphism group of $H_{1}(17)$, the latter being a Paley-Hadamard matrix of
type II; hence its order is $4\cdot{17}(17^{2}-1)=19584$. In addition, we
classify all Hadamard matrices of order 36 having as rows codewords of $C(17)$
of weight 36, and show that up to equivalence, there are exactly two such
matrices: $H_{1}(17)$, and a second Hadamard matrix $H^{\prime}$ having the
property that all Hadamard 3-$(36,18,8)$ designs associated with $H^{\prime}$
are isomorphic and have trivial full automorphism group of order 1. The full
automorphism group of $H^{\prime}$ is of order $72$ and is transitive on the
set of 72 rows (as well as the set of 72 columns) of $H^{\prime}$ and
$-H^{\prime}$. The 3-rank of $H^{\prime}$ is 18; thus $C(17)$ is the row space
of $H^{\prime}$. The Hadamard matrix $H^{\prime}$ is regular, and the
symmetric 2-$(36,15,6)$ design with $\pm 1$-incidence matrix $H^{\prime}$ has
trivial full automorphism group.

In Section 3 we discuss Paley-Hadamard matrices of Type I and Hadamard
3-designs arising from extended ternary quadratic residue codes.

##  2 Hadamard matrices and designs arising from symmetry codes

The sum (over $GF(3)$) of all rows of the generator matrix (1) of the symmetry
code $C(q)$ is a vector $v$ of full Hamming weight $2q+2$, with all components
equal to 1 if $-1$ is not a square in $GF(q)$, (that is, $v$ is the constant
all-one vector $\bar{1}=(1,\ldots,1)$), and $v$ has $2q+1$ components equal to
1, and the position labeled by $\infty$ is equal to $-1$ whenever $-1$ is a
square in $GF(q)$.

Next, we consider a code which is monomially equivalent to the Pless symmetry
code $C(q)$, and always contains the all-one vector, namely the code $L(q)$
with a generator matrix $G^{\prime}$ given by

$G^{\prime}=(I_{q+1},U_{q}),$ (4)

where $U_{q}$ is a $(q+1)\times(q+1)$ matrix obtained from $S_{q}$ by
replacing every nonzero entry in the column labeled by $\infty$ with $-1$.
Clearly, the generator matrix $G^{\prime}$ (4) is identical with the generator
matrix $G$ (1) if $-1$ is not a square in $GF(q)$, and is obtained by negating
one column of $G$ if $-1$ is a square in $GF(q)$. Thus, the matrices (1) and
(4) generate monomially equivalent ternary codes.

Using (4), we obtain a parity check $P$ matrix of $L(q)$ given by

$P=(-U_{q}^{T},I_{q+1}).$ (5)

Note that since $L(q)$ is self-dual, the rows of $P$ are codewords of $L(q)$.
It is easy to check that the matrix $H$ given by

$H=\left(\begin{array}[]{c}G^{\prime}+P\\\
G^{\prime}-P\end{array}\right)=\left(\begin{array}[]{cc}I_{q+1}-U_{q}^{T}&U_{q}+I_{q+1}\\\
I_{q+1}+U_{q}^{T}&U_{q}-I_{q+1}\end{array}\right)$ (6)

is a Hadamard matrix of order $2q+2$, with rows being codewords of $L(q)$.

###### Theorem 2.1

The code $L(q)$ contains a set of $4q+2$ (0,1)-codewords of weight $q+1$ that
form the block-by-point incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$
design $D(q)$ associated with a Paley-Hadamard matrix of type II.

Proof. All entries in the first row of the Hadamard matrix $H$ (6) are equal
to 1; that is, $H$ is normalized with respect to its first row, and
consequently, all entries in the first row of $-H$ are equal to $-1$. Adding
the constant codeword $\bar{2}=(2,\ldots,2)$ with all entries equal to 2 to
every row of the matrix

$\left(\begin{array}[]{rr}H\\\ -H\end{array}\right)$

gives a $(0,1)$-matrix $M$ with all-zero first row, and all-one row labeled by
the first row of $-H$. Deleting the all-zero row and the all-one row from $M$
gives a $(4q+2)\times(2q+2)$ $(0,1)$-matrix $A$, being the block-by-point
incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design associated with
the first row of $H$. Clearly, $H$ is equivalent to the corresponding matrix
(2) or (3); hence $H$ is equivalent to a Paley-Hadamard matrix of type II.
$\Box$

###### Theorem 2.2

If $q=5,11,17,23$, the code $L(q)$ contains exactly $4q+2$ (0,1)-codewords of
weight $q+1$, and every such codeword is the incidence vector of a block of
the Hadamard 3-design $D(q)$ from Theorem 2.1.

Proof. Let $m$ denote the total number of (0,1)-codewords of weight $q+1$ in
$L(q)$. By Theorem 2.1, $m\geq 4q+2$. If $v\in L(q)$ is a (0,1)-codeword of
weight $q+1$ then $v+\bar{1}$ is a codeword of full weight $2q+2$, having
$q+1$ components equal to 1, and $q+1$ components equal to 2. Adding the
codewords $\bar{1}$ and $\bar{2}=2\cdot\bar{1}$ gives $m+2\geq 4q+4$ codewords
of weight $2q+2$. Since $C(q)$ contains exactly $4q+4$ codewords of weight
$2q+2$ for $q=5,11,23$ [23], the statement is true in these cases.

The case $q=17$ needs additional analysis because the symmetry code $C(17)$ ,
as well as its equivalent code $L(17)$, contains 888 codewords of weight 36
[23]. The set of all codewords of weight 36 is easily computed with Magma [6].
This set comprises of the following codewords:

  * •

the 36 rows of the Hadamard matrix $H$ (6), one of the rows being $\bar{1}$,
and 35 rows with 18 components equal to 1, and 18 components equal to $-1$
(note that $-1\equiv 2\pmod{3}$);

  * •

the 36 rows of $2H$ that include $\bar{2}$ and 35 rows with 18 components
equal to 1, and 18 components equal to 2;

  * •

a set $T$ of 408 codewords having 15 components equal to 1 and 21 components
equal to 2;

  * •

a set $2T$ of 408 codewords obtained by multiplying every codeword from $T$ by
2.

Note that adding $\bar{2}$ to any $(0,1)$-codeword of weight 18 gives a
codeword of weight 36 with 18 $1$’s and 18 $2$’s; hence the code $L(17)$
contains exactly 70 $(0,1)$-codewords of weight 18 obtained by adding the
codeword $\bar{2}$ to the rows of $H$ and $2H$, and these 70 $(0,1)$-codewords
form the incidence matrix of the 3-design $D(17)$ from Theorem 2.1. $\Box$

###### Note 1

The code $L(29)$ contains 19606 (0,1)-codewords of weight 30. It is an open
question whether this set contains the incidence matrices of any Hadamard
3-$(60,30,14)$ designs that are not isomorphic to $D(29)$. The number of
codewords of weight 60 in $L(29)$ is 41184. It seems likely that there may be
Hadamard matrices of order 60 formed by codewords of weight 60 that are not
equivalent to the Paley-Hadamard matrix of type II.

###### Corollary 2.3

If $q=5,11,17,23$, the full permutation automorphism group of $L(q)$ coincides
with the full automorphism group of the Hadamard 3-design $D(q)$ from Theorem
2.1.

Proof. The results of De Launey and Stafford [13] and Norman [20] imply that
the full automorphism group of $D(q)$ has order $q(q-1)$ if $q>5$ is prime.
The full automorphism group of $D(5)$ has order 7920.

Any derived design with respect to a point of a Hadamard
3-$(2q+2,q+1,(q-1)/2)$ design $D$ is a symmetric Hadamard 2-$(2q+1,q,(q-1)/2)$
design $D^{\prime}$ of order $q-(q-1)/2=(q+1)/2$. Since $q\equiv-1\pmod{3}$, 3
divides $(q+1)/2$. If 9 does not divide the order $(q+1)/2$ (which is true if
$q=5$, 11, or 23), the rank of the incidence matrix if $D^{\prime}$ over
$GF(3)$ (or the 3-rank of $D^{\prime}$) is equal to $q+1$ (see, for example,
Assmus and Key [1], [2]), hence the 3-rank of $D$ is $q+1$ and the code $L(q)$
is spanned by the incidence matrix of $D$. If $q=17$, a direct computation
shows that the 3-rank of $D(17)$ is 18, hence $D(17)$ spans the code $L(17)$.

$\Box$

###### Theorem 2.4

(i) The code $L(17)$ contains two equivalence classes of Hadamard matrices of
order 36 having as rows codewords of weight 36, with representatives the
Hadamard matrix $H$ (6), which is equivalent to a Paley-Hadamard matrix of
type II and has full automorphism group of order $19584$, and a second
Hadamard matrix $H^{\prime}$, being a regular Hadamard matrix such that the
symmetric 2-$(36,15,6)$ design $D^{\prime}$ with $(0,1)$-incidence matrix
$(H^{\prime}+J)/2$, where $J$ is the $36\times 36$ all-one matrix, has a
trivial automorphism group.
(ii) The row span of the incidence matrix of the 2-$(36,15,6)$ design
$D^{\prime}$ is an extremal ternary $[36,18,12]$ code equivalent to the
symmetry code $C(17)$.
(iii) The full automorphism group of the code $L(17)$ coincides with the full
automorphism group $H$.

Proof. (i) In the context of Hadamard matrices, we consider the element 2 of
$GF(3)$ as $-1$. Using the notation from the proof of Theorem 2.2, we define a
graph $\Gamma$ having as vertices the 408 codewords from $T$, where two
codewords $u,v\in T$ are adjacent in $\Gamma$ if and only if the Hamming
distance between $u$ and $v$ is 18, or equivalently, the intersection of the
supports of the (0,1)-vectors $\bar{2}-u$ and $\bar{2}-v$ is of size 6.
Replacing all entries equal to 2 by zero in every vector from $T$ gives a set
$T(0,1)$ of (0,1)-vectors of weight 15. Using the restricted Johnson bound, it
is easy to verify that the maximum number codewords in a binary constant
weight code of length 36 with codewords of weight 15 and minimum distance 18,
is 36. Every set $K$ of 36 vectors from $T(0,1)$ that meets the Johnson bound
corresponds to a clique of size 36 in $\Gamma$, and the $36\times 36$ matrix
having as rows the vectors from $K$ is the incidence matrix $N$ of a symmetric
2-$(36,15,6)$ design (see [29, Theorem 2.4.12, page 99] or [31, Sec. 3]).
Replacing all zeros in $N$ with $-1$’s gives a regular Hadamard matrix of
order 36. Using the clique finding algorithm Cliquer [19], a quick computer
search shows that the graph $\Gamma$ contains exactly 272 cliques of size 36,
or in other words, there are 272 collections of 36 codewords from $T$ that
form a Hadamard matrix of order 36. Further analysis with Magma shows that all
272 Hadamard matrices are equivalent to a matrix $H^{\prime}$ with a full
monomial automorphism group of order 72 that acts transitively on the set of
size 72 being the union of the rows of $H^{\prime}$ and the rows of
$-H^{\prime}$.

The incidence matrix of the symmetric 2-$(36,15,6)$ design $D^{\prime}$
obtained by replacing all $-1$-entries of $H^{\prime}$ with zeros is listed in
the Appendix. The design $D^{\prime}$ has a trivial full automorphism group of
order 1.

(ii) The 3-rank of the incidence matrix of $D^{\prime}$ is 18, and its row
span over $GF(3)$ is a ternary $[36,18,12]$ code equivalent to the Pless
symmetry code.

Parts (iii) was verified by computer using Magma. The full automorphism group
of $L(17)$ partitions the set of the 888 codewords of weight 36 into two
orbits, of length 72 and 816 respectively, the orbit of length 72 comprised of
the rows of $H$ (6) and $-H$. Thus, the full automorphism group of the code
$L(17)$ coincides with the full automorphism group of $H$, and is of
order222This is the order of the Paley-Hadamard matrix of Type II for $q=17$
[13]. $19584=2^{7}\cdot 3^{2}\cdot 17.$ $\Box$

###### Note 2

Up to equivalence, there are exactly 11 Hadamard matrices of order 36 with
automorphism groups of order divisible by 17 (Tonchev [28]). Each of these
matrices spans a ternary self-dual code of length 36, but only the symmetry
code $C(17)$ spanned the Paley-Hadamard matrix of type II is extremal, that
is, has minimum distance 12, and supports 5-designs. A stronger
characterization of the Pless symmetry code $C(17)$ was proved by Huffman
[10], namely that up to equivalence, $C(17)$ is the only extremal ternary
self-dual code of length 36 that admits a monomial automorphism of order $17$.

###### Note 3

Hadamard matrices and designs are used for the construction of self-orthogonal
and self-dual codes over other finite fields. A classical example is the
extended binary Golay code generated by a bordered incidence matrix of a
symmetric Hadamard 2-$(23,11,5)$ design associated with a Paley-Hadamard
matrix of type I. Hadamard matrices of order 28 with an automorphism of order
7 [27] were used by Pless and Tonchev [24] for the classification of self-
orthogonal codes over $GF(7)$. The Paley-Hadamard matrix of type II of order
28 is the only Hadamard matrix of this order that admits an automorphism of
order 13 and yields an extremal binary self-dual code of length 56 [26], [30].
More extremal binary self-dual codes derived from Hadamard matrices of order
28 were found in [5].

##  3 Hadamard matrices and designs arising from ternary QR codes

The symmetry codes $C(11)$, $C(23)$, and $C(29)$ have siblings with the same
parameters and weight distribution, being ternary extended quadratic-residue
codes that support 5-designs by the Assmus-Mattson theorem. If $q\equiv
3\pmod{4}$ is a prime power, a quadratic residue (QR) code of length $q$ is a
code spanned by the (0,1)-incidence matrix $A$ of a symmetric Hadamard
2-$(q,(q-1)/2,(q-3)/4)$ design obtained from the Paley-Hadamard matrix of type
I, and its extended code is spanned by a matrix obtained by adding one all-one
column to $A$. If, in addition, $q\equiv-1\pmod{3}$, that is, $q$ is of the
form $q=12s+11$ for some integer $s\geq 0$, the ternary extended QR code is
self-dual.

###### Theorem 3.1

Let $q=12s+11$ be a prime power, and let $QR_{q}$ be the ternary extended QR
code of length $q+1$.
(i) $QR_{q}$ contains a Paley-Hadamard matrix of type I having as rows
codewords of weight $q+1$.
(ii) $QR_{q}$ contains a set of $2q$ (0,1)-codewords of weight $(q+1)/2$ that
form the incidence matrix of a Hadamard 3-$(q+1,(q+1)/2,(q-3)/4)$ design
associated with the Paley-Hadamard matrix of type I of order $q+1$.
(iii) If $q=11$, 23 or 47, $QR_{q}$ contains exactly $2q$ (0,1)-codewords of
weight $(q+1)/2$, and the permutation automorphism group of the code coincides
with the full automorphism group of the Hadamard 3-$(q+1,(q+1)/2,(q-3)/4)$
design from part (ii).

Proof. (i) The statement (i) is implicit in [3]. The column sum of the
(0,1)-incidence $A$ of the Hadamard 2-$(q,(q-1)/2,(q-3)/4)$ design obtained
from the Paley-Hadamard matrix of type I is

$(q-1)/2=(12s+10)/2\equiv-1\pmod{3},$

and the sum of all $q$ entries of the all-one column is $q\equiv-1\pmod{3}$,
hence the sum over $GF(3)$ of all rows of the $q\times(q+1)$-matrix $B$
obtained by bordering $A$ with one all-one column, is equal to the constant
vector $\bar{2}$. Since $QR_{q}$ is the row span of $B$, the constant vectors
$\bar{1}$ and $\bar{2}$ belong to the code. Let $E$ be the $(q+1)\times(q+1)$
matrix obtained from $B$ by adding one extra all-one row. The matrix
$H_{q+1}=2J-E$, where $J$ is the $(q+1)\times(q+1)$ all-one matrix, is a
Paley-Hadamard matrix of type I. Every row of $H_{q+1}$ is the difference of
the codeword $\bar{2}$ and a row of $B$, hence the rows of $H_{q+1}$ belong to
the code $QR_{q}$.

(ii) Adding the codeword $\bar{2}$ to every row of

$\left(\begin{array}[]{rr}H_{q+1}\\\ -H_{q+1}\end{array}\right)$ (7)

gives a $(2q+2)\times(q+1)$ matrix with one all-zero row, one all-one row, and
$2q$ (0,1)-rows of weight $(q+1)/2$ that form the incidence matrix of a
Hadamard 3-$(q+1,(q+1)/2,(q-3)/4)$ design associated with $H_{q+1}$.

(iii) The proof is similar to that of Corollary 2.3. $\Box$

###### Note 4

The number of codewords of weight 60 in $QR_{59}$ is 41184. It seems likely
that there may be Hadamard matrices of order 60 formed by codewords of weight
60 that are not equivalent to the Paley-Hadamard matrux of type I from Theorem
3.1.

##  4 Concluding remarks

The extended ternary Golay code of length 12, the Pless symmetry codes $C(q)$
($q=11,17,23$ and 29), of lengths 24, 36, 48 and 60, the extended ternary QR
codes of lengths 24, 48 and 60, and an extremal code of length 60 discovered
by Nebe and Villard [18] as an analogue of the Pless symmetry code $C(29)$,
are the only known extremal ternary self-dual codes of length divisible by 12
that support 5-designs. It is known that the symmetry code of length 84
($q=41$), as well as the extended QR code of this length are not extremal.
Extremal ternary self-dual codes of length $n$ divisible by 12 do not exist
for $n=72,96,120$, and all $n\geq 144$, because then the extremal Hamming
weight enumerator contains a negative coefficient [25].

All ternary self-dual codes of length 24 have been classified up to
equivalence (Harada and Munemasa [9]), and the symmetry code $C(11)$ and the
extended QR code are the only extremal codes of this length. Nine of the self-
dual ternary codes of length 24 are spanned by Hadamard matrices of order 24
[14], [15], but only two codes, $QR_{23}$ and $C(11)$, that are spanned by the
Paley-Hadamard matrices of type I and II respectively, are extremal.

It is an interesting open question whether the Pless symmetry codes of length
36, 48, and 60, the extended QR codes of lengths 48 and 60, and the extremal
code of length 60 found by Nebe and Villard [18] are the only extremal self-
dual codes of these lengths. The results from Section 2 show that the symmetry
code of length 36 can be obtained from a Hadamard matrix that is not a Paley-
Hadamard matrix of type II, and a natural question that arises is whether any
other extremal codes of length 36, 48, or 60 can be obtained form Hadamard
matrices that are not of Paley type.

The extremal ternary self-dual codes of lengths $n\geq 36$ have not been
classified up to equivalence. A partial classification of such codes of length
$n\leq 40$ admitting automorphisms of prime order $p\geq 5$ was given by
Huffman [10]. In addition, it was proved by Nebe [17] that, up to equivalence,
the only extremal ternary self-dual codes of length 48 that admit an
automorphism of a prime order $p\geq 5$, are the Pless symmetry code and the
extended QR code.

## References

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##  5 Appendix

$\begin{array}[]{c}110110001101001100001010100000110001\\\
000010101000001100100111001011001101\\\
011000100010001100011001110011010010\\\
010010000000010011110100110010110101\\\
001011110100100010011111100000000001\\\
011111000110011000000110000110000110\\\
000000001111110000110110100001011010\\\
000010010111101001000000111111000001\\\
010001000101111110010001001001100100\\\
100011000110000110111000001110011000\\\
000111111000000000010000101101110110\\\
000111101011110000101001010010100000\\\
010100110010101110100100100100101000\\\
100101000000101001011100000011101011\\\
101001001001001010101100110101000100\\\
001001100011001011100010001000110011\\\
010001111100011001001100011000011000\\\
100100010010011001111011101000000100\\\
100010101000111111010010010100000010\\\
000100010001100110001110011010010110\\\
111110001010100011000101001001010000\\\
001010000001010101001101101100101010\\\
111001011000110100100000101010000011\\\
110100100101000000110101011100000011\\\
000101001110000111000001110000001111\\\
101000111111000101010100000010100100\\\
010000011100000011101011000111100010\\\
001100100100110101101000000101010101\\\
001100011001011010010001000110011001\\\
011000001010100000011010011100101101\\\
100001010010010100000111010101110001\\\
101010010100101000100001010000111110\\\
011111010001000101110010010001001000\\\
101100100100010010000010111011101000\\\
110010110011010010001000000001001111\\\
110001100001100001000011100110011100\\\ \end{array}$

A $2$-$(36,15,6)$ design associated with the Pless symmetry code of length 36