∎

e1e-mail<EMAIL_ADDRESS>(corresponding author)

11institutetext: Max Planck Institut für Physik, Föhringer Ring 6, 80805
Munich, Germany

# Identification and simulation of surface alpha events on passivated surfaces
of germanium detectors and the influence of metalisation

I. Abtaddr1  C. Goochaddr1  F. Hagemannaddr1  L. Hauertmanne1,addr1  X.
Liuaddr1  O. Schulzaddr1  M. Schusteraddr1  A. J. Zsigmondaddr1

(Received: date / Accepted: date)

###### Abstract

Events from alpha interactions on the surfaces of germanium detectors are a
major contribution to the background in germanium-based searches for
neutrinoless double-beta decay. Surface events are subject to charge trapping,
affecting their pulse shape and reconstructed energy. A study of alpha events
on the passivated end-plate of a segmented true-coaxial n-type high-purity
germanium detector is presented. Charge trapping is analysed in detail and an
existing pulse-shape analysis technique to identify alpha events is verified
with mirror pulses observed in the non-collecting channels of the segmented
test detector. The observed radial dependence of charge trapping confirms
previous results. A dependence of the probability of charge trapping on the
crystal axes is observed for the first time. A first model to describe charge
trapping effects within the framework of the simulation software
_SolidStateDetectors.jl_ is introduced. The influence of metalisation on
events from low-energy gamma interactions close to the passivated surface is
also presented.

††journal: Eur. Phys. J. C

##  1 Introduction

High-purity germanium (HPGe) detectors are well suited to search for physics
beyond the Standard Model such as neutrinoless double-beta ($0\nu\beta\beta$)
decay LEGEND:2017cdu ; 0bnn:Review2019 ; Majorana:2019nbd ; GERDA:2020xhi ;
LEGEND:2021bnm  and dark matter CoGeNT:2013 ; SuperCDMS:2014 ;
PhysRev:LimitWIMPS2018 . In such rare-event searches, it is essential to
suppress background events as much as possible.

The LEGEND experiment LEGEND:2017cdu ; LEGEND:2021bnm  will search for
$0\nu\beta\beta$ decay in 76Ge using HPGe detectors based on the experience of
the preceding experiments Gerda GERDA:2012qwd ; GERDA:2020xhi  and Majorana
Demonstrator Majorana:2013cem ; Majorana:2019nbd . In these experiments, a
substantial part of the background events with observed energies close to the
$Q$-value of the $0\nu\beta\beta$ decay ($Q_{\beta\beta}=2039$ keV)
0bnn:Review2019  originates from events near the surface of the detectors
GERDA:2019cav . Pulse-shape discrimination techniques have been developed to
reject such surface events GERDA:2022ixh , including a dedicated method to
identify events originating from the large passivated surfaces of the point-
contact detectors of the Majorana Demonstrator Majorana:2020xvk . A
substantial further reduction of background is essential for LEGEND to reach
its goal of increasing the half-life sensitivity to $0\nu\beta\beta$ decay of
76Ge to $10^{28}$ years and beyond LEGEND:2021bnm . A deep understanding of
the HPGe detectors and their signal pulses is essential to achieve this.

A study of alpha events on the passivated surface of the segmented true-
coaxial n-type HPGe detector, "Super-Siegfried" Abt:2016trw , was performed in
the GALATEA test facility Abt:2014bpa  at the Max Planck Institute for Physics
in Munich. Such events are an important background to $0\nu\beta\beta$ decay
searches when charge trapping reduces the observed energy to values within the
region of interest around $Q_{\beta\beta}$ GERDA:2019cav .

For the first time, the dependence of charge trapping on the orientation of
the crystal axes is presented. First attempts to simulate surface effects with
the open-source software package _SolidStateDetectors.jl_ Abt:2021mzq  are
also shown.

In general, all contacts of germanium detectors are completely coated with a
thin metal layer, i.e. are fully metalised. However, the segments of the
detector Super-Siegfried were originally not fully metalised but were only
partially metalised on a dedicated contact area. The results of studies
Abt:2016trw  where data were taken with partially metalised contacts were
compared to results from data taken after the segments became fully metalised.
The influence of the extent of the metalisation is demonstrated.

##  2 Experimental setup and data taking

GALATEA is based on a vacuum chamber in which the top and side surfaces of a
detector can be scanned using a system of three motorised stages, see Fig. 1.

Figure 1: a) An illustration adapted from Abt:2014bpa  and b) a photo of the
vacuum chamber of GALATEA with the main components marked in a). c) Schematic
of the central part of GALATEA with the infrared (IR) shield and the top
collimator and an illustration of the beam-spot.

The HPGe detector is mounted on a holder, which is cooled via a copper cooling
finger submerged in an integrated liquid-nitrogen tank. The fill level of the
tank is monitored and the internal tank is automatically refilled through a
connected dewar.

The special feature of GALATEA is that the path between the detector and the
source is free of material, facilitating the use of alpha sources. Two sources
can be mounted in two tungsten collimators focusing the radiation onto the
mantle and top plate of the detector, respectively. The collimators are moved
by linear stages and slide along slits on the side and top of a silver-coated
copper hat serving as an infrared shield. A rotational stage moves these
components around the detector, such that almost any point on the detector top
surface or mantle can be irradiated. Significant improvements of the setup
since previous studies Abt:2016trw , such as stable long-time operation,
automatised data taking and the use of open alpha sources, allowed for more
detailed studies of charge trapping.

Two open alpha 241Am sources, each with an activity of 74 kBq, were installed
in the two collimators allowing for simultaneous scans of the side and the top
of the detector. The isotope 241Am decays into 237Np with dominant alpha
energies of $Q_{\alpha_{1}}=5485.6$ keV, $Q_{\alpha_{2}}=5442.8$ keV and
$Q_{\alpha_{3}}=$ 5388.2 keV NuclearDataBase:1999 . In addition, 237Np emits
59.5 keV gammas. The alphas lose their energy within 20 µm of germanium with
the Bragg peak at 17 µm Hauertmann2021:PhDThesis  and create events directly
underneath the surface of the detector. Figure 2 shows the observed spectrum
in response to irradiation from the side. The reduction of the observed
energies with respect to the emitted energies corresponds to the contact
thickness on the side of the detector. The width of the observed lines arises
from self-absorption in the source.

Figure 2: $Q$-values (vertical lines) and measured alpha energy spectrum
(histogram) in response to 241Am irradiation onto the side of the Super-
Siegfried detector for one hour. The height of the vertical lines represent
the branching ratio of the respective decay.

Super-Siegfried is a cylindrical true-coaxial n-type segmented HPGe detector
produced by Canberra France, now Mirion Technologies PhD:Irlbeck2014 . The
detector has a height of 70 mm, an outer radius of 37.5 mm and a borehole with
a radius of 5 mm. At the top and the bottom, the borehole widens to a radius
of 10 mm within about 3 mm. The impurity density as stated by the manufacturer
of the crystal is $0.44\times 10^{10}$ cm-3 at the top and $1.30\times
10^{10}$ cm-3 at the bottom. The recommended operational voltage is 3000 V.

Figure 3: a) Side view: The segment structure of the mantle of Super-Siegfried
with dimensions and reference coordinate system. b) Top view: The positions of
the measurements on the top surface are marked. Also shown are the crystal
axes in a) and b). The framed numbers denote the segments numbers, in b) for
the segments underneath segment 19.

The lithium-drifted inner surface of the borehole (not including the area of
the widening) is the only n+ contact, the so-called core contact. The outer
mantle is segmented into 19 p+ contacts produced via boron implantation, see
Fig 3a. The 18 lower segments are ordered in a $6\times 3$ geometry in
$\varphi$ and $z$. These segments are read-out at their centres via one Kapton
printed-circuit-board Abt:2007rf . An additional segment located above these
18 segments, segment 19, has a height of 5 mm and is not segmented in
$\varphi$. It is read out via a cable at $\varphi_{\mathrm{ro}}\approx
135^{\circ}$, see Fig. 3. Prior to the full metalisation, the read-out cable
for segment 19 was connected at about the same location. The detector, as
ready to be mounted on the cooling finger in GALATEA, is depicted in Fig. 4.
The fairly thin segment 19 was designed to study events on the passivated
surface of the top end-plate. Charges drifting towards the segment 19 contact
also create strong so-called mirror pulses in the segments underneath, which
facilitate a detailed pulse-shape analysis.

Figure 4: Picture of the germanium detector Super-Siegfried taken after all
segments were fully metalised. The Kapton printed-circuit-board and the
holding structure are also visible.

The positions of the two sources were calibrated by moving the collimators and
identifying the segment boundaries and the crystal axes as well as the
relative position of the holding structure Hauertmann2021:PhDThesis . Figure
3b shows the position of the $\langle 100\rangle$ and $\langle 110\rangle$
crystal axes, referred to as fast and slow axes, respectively, as the charge
carriers drift faster and slower along these axes in germanium. All events
recorded for a given position of the sources are from here on called a
measurement. All positions refer to the positions of the centre of the beam-
spot on the detector surface. The individual measurement positions on the top
and side of the detector are also shown in Fig. 3. They comprise two radial
scans, RSF along a fast axis and RSS along a slow axis, and two rotational
(azimuthal) scans, AST from the top at a fixed radius of $23.8$ mm and from
the side at $z=40$ mm and ASS only from the side on segment 19. The index $j$
is used throughout this paper to refer the $j$-th measurement of the
respective scan. Each measurement of these four scans lasted one hour. In
addition, a five hour long background measurement (BG) was performed where the
sources did not irradiate the detector.

The alpha beam from the top as provided by the setup was simulated with Geant4
Geant4:2003  to investigate the incident spectrum and beam-spot on the top
surface. A simulation of monoenergetic 5.485 MeV alphas from the source in the
top collimator showed that the resulting spectrum has a tail towards low
energies, see Fig. 5. This tail is populated by alphas which loose energy
through interactions with the wall of the collimator or the edge of the slit
in the infrared shield. The shape and population of the beam-spot is shown in
Fig. 6. The sharp cut-off, perpendicular to the radial direction, is caused by
the additional collimation provided by the infrared shield.

For a single event, 20 pulses (from the core and all 19 segments), each with
5000 samples, were recorded with a sampling rate of 250 MHz by two 16-channel
Struck SIS3316 STRUCK  analog-to-digital converter units. The index
$i\in[0,19]$ is used throughout this paper to specify the channel. All devices
of GALATEA were controlled and monitored via one software package, which
allowed for automated scans of the detector. The trigger was provided by the
core channel connected to the first unit which forwarded its trigger signal to
the second unit via a short cable. The constant time delay of the second unit
was taken into account in the offline pulse processing
Hauertmann2021:PhDThesis .

Figure 5: Energy spectrum of the alphas reaching the detector as predicted by
Geant4 for monoenergetic 5.485 MeV alphas from the source volume directed
downwards with an opening angle of 3 degrees. Figure 6: Distributions of
energy-weighted energy depositions of the alphas reaching the detector as
predicted by GEANT4 for monoenergetic 5.485 MeV alphas from the source volume
directed downwards with an opening angle of 3 degrees. The y-axis is parallel
to the radial direction in the cylindrical coordinate system of the detector,
see Fig. 3b.

A typical raw (prior to any processing) pulse, from the core channel is shown
in Fig. 7a.

Figure 7: a) Typical raw pulse from the core channel of Super Siegfried. The
three windows of the pulse are indicated by arrows and labelled. b) The same
pulse after calibration and cross-talk correction. The horizontal line in the
inset in b) indicates the mean of the tail of the pulse.

This included a correction for the decay of charge in the preamplifier of each
channel $i$ Hauertmann2017:MasterThesis . After baseline subtraction, an
exponential function, $e^{-t/\tau_{i}}$, was fitted to the part of each pulse
following the rise, the so-called tail. For all measurements, the
characteristic decay constants of each preamplifier, $\tau^{c}_{i}$, were
determined by fitting a Cauchy function to the $\tau_{i}$-distributions for
bulk events Hauertmann2017:MasterThesis . The pulse shown in Fig. 7a is shown
in Fig. 7b after cross-talk correction and calibration. All pulses were
corrected for linear cross-talk and calibrated for each individual measurement
with an automated procedure Hauertmann2017:MasterThesis .

##  3 Alpha event characteristics and selection

The spectra of the calibrated core and segment 19 energies, $E_{0}$ and
$E_{19}$, are shown in Fig. 8 for a measurement from AST with both 241Am
sources

Figure 8: a) Core, $E_{0}$, and b) segment 19, $E_{19}$, spectra of the
background measurement BG and a typical 241Am measurement from AST with the
top beam-spot at ($\varphi=257.9^{\circ}$, $r=23.8$ mm) and the side beam-spot
at ($\varphi=167.9^{\circ}$, $z=40.0$ mm). The spectrum of the background
measurement is normalised in time to the 241Am measurement.

irradiating the detector. The signal of the alphas hitting the top and the
side of the detector are well separated due to the different structure of the
surface layers. The respective backgrounds from natural radioactivity are also
shown for comparison.

Alphas deposit their energy in a small volume ($<1$ mm3) and create so-called
single-site events. Thus, the collecting segment for alpha events from the top
is always segment 19 and $E_{0}$ and $E_{19}$ should be equal to one of the
$Q_{\alpha_{k}}$, $k\in\nolinebreak\\{1,2,3\\}$. However, Fig. 8 shows that
the observed energies, $E_{0}$ and $E_{19}$, of these top alpha events are
massively reduced compared to $Q_{\alpha_{k}}$ and that these observed alpha
energies form wide distributions instead of three peaks. In addition, the
shape and central position of the distributions are slightly different in the
core and segment 19. The reduction of energy, $\mathcal{O}$(2 MeV), is
partially caused by a so-called dead layer on top of the detector, where the
created electron-hole pairs recombine immediately and, thus, do not contribute
to the signal pulses. However, a dead layer alone can neither explain the
widths of the distributions nor the non-zero difference between $E_{0}$ and
$E_{19}$. Both observations can only be explained by the trapping of charge
carriers during their drift through the detector.

In this case, electrons or holes are trapped in the crystal and do not reach a
contact. After the other charge carriers are collected, the trapped charge
carriers still induce signals in the contacts. This results in reduced and
unequal values of $E_{0}$ and $E_{19}$, depending on the position of the
trapping. In non-collecting segments, so-called truncated mirror pulses
Abt:2016trw  are observed where the signal does not return to the baseline.

The characteristic pulses from an event with net hole trapping,
$E_{19}<E_{0}$, are shown in Fig. 9. The collecting segment 19 shows a lower
energy than the core and the two segments underneath, segments 9 and 10, show
positive truncated mirror pulses indicating net hole trapping. The
characteristic pulses from an event with net electron trapping,
$E_{0}<E_{19}$, are shown in Fig. 10, where negative truncated mirror pulses
are observed in segments 9 and 10.

Due to the feature of truncated mirror pulses, alpha events affected by charge
trapping can also appear as so-called multi-segment events even though they
are single-site events. Thus, a strict single-segment 19 cut, $E_{0}\approx
E_{19}$, cannot be used to filter out events induced by environmental gammas,
which are often multi-segment events as the gammas predominantly perform
Compton scattering at these energies. However, in contrast to gamma induced
events, the energies measured in the other segments should never be larger
than $E_{19}$ for events induced by alphas from the top. Therefore, a softer
single-segment 19 cut was defined to filter out gamma events:

  * $\bullet$

S-cut: $E_{19}>\sum_{i=1}^{18}E_{i}$ .

Alpha events affected by charge trapping feature another characteristic: After
the correction for the specific decay of charge in the preamplifiers, pulses
from "normal" events have a flat tail, see Fig. 7b. However, it has been shown
that the pulses of alpha events on passivated surfaces have positive tail-
slopes Majorana:2020xvk ; Hauertmann2021:PhDThesis  after the preamplifier
specific correction using the decay constants $\tau^{c}_{i}$. These positive
tail-slopes are visible in Fig. 9a and Fig. 10a.

Figure 9: Pulses of a typical event with net hole trapping of the a) core and
segment 19 and b) the two closest segments 9 and 10 of a measurement from RSS
at $\varphi=180.6^{\circ}$ and $r=18.8$ mm. The observed energies, $E_{9}$ and
$E_{10}$, are also indicated. The horizontal line in the inset in a) is at the
mean of the tail of the core pulse.

The possible physical origins of the non-zero tail slopes of alpha-event
pulses are a slow release of trapped charge carriers or a very slow drift of a
part of the charge cloud due to a very low mobility near the surface. A
positive tail-slope corresponds to a seemingly increased decay constant of the
pulse, $\tau_{i}$, in comparison to $\tau^{c}_{i}$. Thus, alpha events can be
identified prior to any pulse processing or cross-talk correction by comparing
$\tau_{i}$ with $\tau^{c}_{i}$.

Typical $\tau_{i}$ distributions for events passing the S-cut are shown in
Fig. 11 for a measurement from AST and BG. The $\tau_{i}$ distributions can be
modelled by a superposition of Cauchy functions ($\mathcal{C}$) via the
function $\mathcal{T}(\mathcal{P}_{\mathcal{T}})$ with the parameters
$\mathcal{P}_{\mathcal{T}}$

$\displaystyle\mathcal{P}_{\mathcal{T}}=$
$\displaystyle~{}\\{\mu^{\tau},\sigma^{\tau}_{1},\sigma^{\tau}_{2},A^{\tau}_{1},A^{\tau}_{2},A^{\tau}_{\alpha},\mu^{\tau}_{\alpha},\sigma^{\tau}_{\alpha}\\}\,,$
(1) $\displaystyle\mathcal{T}(\mathcal{P}_{\mathcal{T}})=$
$\displaystyle~{}A^{\tau}_{1}\cdot\mathcal{C}(\mu^{\tau},\sigma^{\tau}_{1})+A^{\tau}_{2}\cdot\mathcal{C}(\mu^{\tau},\sigma^{\tau}_{2})\,+$
(2)
$\displaystyle~{}A^{\tau}_{\alpha}\cdot\mathcal{C}(\mu^{\tau}_{\alpha},\sigma^{\tau}_{\alpha})\,,$

where $\mu^{\tau},\sigma^{\tau}_{1},\sigma^{\tau}_{2},A^{\tau}_{1}$ and
$A^{\tau}_{2}$ model the underlying background from "normal" events and
$A^{\tau}_{\alpha},\mu^{\tau}_{\alpha}$ and $\sigma^{\tau}_{\alpha}$ model the
second peak which is associated with alpha events.

In order to study charge trapping in detail, the events induced by alphas from
the top source are selected with the S-cut and a second cut:

  * $\bullet$

$\tau$-cut: $\tau_{0}\in[\mu^{\tau}_{\alpha,0}\pm 2\sigma^{\tau}_{\alpha,0}$]
and $\tau_{19}\in[\mu^{\tau}_{\alpha,19}\pm 2\sigma^{\tau}_{\alpha,19}$] .

Figure 10: Pulses of a typical event with net electron trapping of the a) core
and segment 19 and b) the two closest segments 9 and 10 of the measurement
from RSS at $\varphi=180.6^{\circ}$ and $r=32.8$ mm. The observed energies,
$E_{9}$ and $E_{10}$, are also indicated. The horizontal line in the inset in
a) is at the mean of the tail of the core pulse.

Figure 11: Distributions of a) $\tau_{0}$ and b) $\tau_{19}$ for events
selected with the S-cut from AST at $\varphi=257.9^{\circ}$ and $r=23.8$ mm
and from BG. The solid line is the function $\mathcal{T}$, see Eq. (2), fitted
to the distributions of the 241Am measurement. The $\pm 2\sigma$ interval,
used for the $\tau$-cut, is indicated by the vertical dashed lines.

Figure 12 shows the $E_{0}$ and $E_{19}$ spectra before the alpha selection
cuts, after applying the S-cut and after applying the additional $\tau$-cut
(S$\tau$-cut).

Figure 12: Spectra of a) $E_{0}$ and b) $E_{19}$ before and after the
application of the S-cut and S$\tau$-cut for a measurement of the scan AST
with the top beam-spot at ($\varphi=257.9^{\circ}$, $r=23.8$ mm) and the side
beam-spot at ($\varphi=167.9^{\circ}$, $z=40.0$ mm).

The S-cut removes events with alphas entering the side of the detector and
suppresses gamma events, for which most of the energy is deposited outside of
segment 19. After the S$\tau$-cut, only alpha events on the top remain. Above
1000 keV, they form a broad peak. It has been cross-checked that the events in
the energy region of the broad alpha peak indeed form the secondary peak
observed in the $\tau$ distribution, confirming that the $\tau$-cut selects
alpha events Hauertmann2021:PhDThesis . Similarly, analysing the mirror pulses
from the neighbouring segments also shows that the S$\tau$-cut is effective in
selecting alpha events with charge trapping. These checks verify that the core
pulse alone is sufficient to select alpha events. Thus, the rejection of
events based only on positive tail slopes, is confirmed to be an effective way
to reduce the background for $0\nu\beta\beta$ decay searches. It should be
noted that signal events very close to the surface might also be rejected.

##  4 Location dependence of charge trapping

The correlation between $E_{0}$ and $E_{19}$ is shown in Fig. 13 for the
selected alpha events from the measurement at $\varphi=\nolinebreak
257.9^{\circ}$ and $r=23.8$ mm from AST. Events at this radial position are
subject to more electron trapping and higher energies are observed in segment
19 than in the core. The $E_{0}$ and $E_{19}$ distributions are fitted via
maximum likelihood estimation to find the most likely $E_{0}$ and $E_{19}$.
The function $\mathcal{M}(E|\mathcal{P}_{\mathcal{M}})$, with the parameters
$\mathcal{P}_{\mathcal{M}}$, defined as

$\displaystyle\mathcal{P}_{\mathcal{M}}=$
$\displaystyle~{}\\{A,R,E_{\mu},\sigma_{1},\Delta E_{\mu},\sigma_{2},s\\}\,,$
(3) $\displaystyle\mathcal{M}(E|\mathcal{P}_{\mathcal{M}})=$
$\displaystyle~{}A\cdot[(1-R)\cdot\mathcal{N}(E|E_{\mu},\sigma_{1})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,+R\cdot\mathcal{N}(E|E_{\mu}-\Delta
E_{\mu},\sigma_{2})$ (4)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\cdot(\textrm{erf}(-s\cdot(E-(E_{\mu}-\Delta
E_{\mu})))+1)]$

was chosen to model the distribution of observed energies of the selected
events.

Figure 13: Correlation between core and segment 19 energies of alpha events
selected as described in Sec. 3 from a measurement of the scan AST at
$\varphi=232.9^{\circ}$ and $r=23.8$ mm. Also shown are the two
marginalisations. The dashed (black) lines mark the position of the parameter
$E^{0/19}_{\mu,j,\mathrm{obs}}$ for this measurement $j$ of the respective
fitted function $\mathcal{M}$, see Eq. (4), shown as a solid (red) line.

It consists of two Normal distributions $\mathcal{N}$: the first,
$\mathcal{N}(E_{\mu},\sigma_{1})$, modelling the main peak and the second,
$\mathcal{N}(E_{\mu}-\Delta E_{\mu},\sigma_{2})$ modulated by an error
function, modelling the low-energy tail of the distribution. The parameter
$E_{\mu}$ corresponds to the most likely observed energy of the alphas.

The obtained values for the core, $E_{\mu,j,\mathrm{obs}}^{0}$ and segment 19,
$E_{\mu,j,\mathrm{obs}}^{19}$, for the different measurements in the radial
scans RSF (fast axis) and RSS (slow axis) are shown in Fig. 14 as a function
of $r$.

Figure 14: The fitted $E_{\mu,j,\mathrm{obs}}^{0}$ and
$E_{\mu,j,\mathrm{obs}}^{19}$ of the measurements from a) RSF and b) RSS.
Error bars indicating uncertainties would be hidden by the markers.

The uncertainties on the fitted values of $E_{\mu}$ are only a few keV. These
results confirm the trends observed previously Abt:2016trw  that for events at
smaller radii hole trapping dominates and moving towards segment 19 electron
trapping becomes dominant. This can be understood as a higher probability of
charge trapping when the charge carriers have a longer drift path towards the
contacts. The result of the two scans reveal a dependence of the amount of
charge trapping on the crystal axes orientation. The energies observed in the
scan along the fast axis are up to 1 MeV higher than those in the scan along
the slow axis.

The fitted values of $E_{\mu,j,\mathrm{obs}}^{0}$ and
$E_{\mu,j,\mathrm{obs}}^{19}$ over $\varphi$ are shown in Fig. 15 for the
measurements of the rotational scan AST at $r=23.8$ mm. There is a clear
effect of the crystal axes on $E_{\mu,j,\mathrm{obs}}^{0}$ and
$E_{\mu,j,\mathrm{obs}}^{19}$.

Figure 15: The fitted $E_{\mu,j,\mathrm{obs}}^{0}$ and
$E_{\mu,j,\mathrm{obs}}^{19}$ of the measurements from AST at $r=23.8$ mm.
Error bars indicating uncertainties would be hidden by the markers. Data
points taken four months later are shown with open markers.

Near a fast axis, more energy is recorded, which would imply a thinner dead
layer or less net trapping of charge carriers. The latter is reasonable as the
mobility for drifts along the fast axis is higher. An effect of the crystal
axes on the observed alpha energies is seen for the first time in GALATEA.
This was possible due to the increased amount of data compared to previous
studies Abt:2016trw .

The open markers in Fig. 15 show data points taken four months after the
original scan. Several warming-cooling cycles probably changed the surface
conditions influencing the amount of charge trapping.

##  5 Simulation of charge trapping

A dead layer alone can only equally reduce the alpha energy observed in the
core and segment 19. It can neither explain the observed differences between
$E_{\mu,j,\mathrm{obs}}^{0}$ and $E_{\mu,j,\mathrm{obs}}^{19}$ nor the
observed truncated mirror pulses in non-collecting segments. The observed
alpha energies can only be explained by taking charge trapping effects into
account. There is no established microscopic model available which includes
charge trapping. Here, a model consisting of three different parts is
introduced to describe the radial dependence of $E_{\mu,j,\mathrm{obs}}^{0}$
and $E_{\mu,j,\mathrm{obs}}^{19}$. The model was implemented using
_SolidStateDetectors.jl_ (SSD) Abt:2021mzq , which is an open-source package
to simulate solid-state detectors such as germanium detectors. It is written
in the Julia programming language and it comprises the calculation of the
electric field, $\mathbfcal{E}$, the weighting potentials Shockley ;
Abt:2021mzq  of the contacts, $W_{i}$, the drift of the charge carriers
through the semiconductor and the induced signals on the contacts. The drift
of the charge carriers is calculated in time steps by deriving a drift vector,
$\mathbf{v}_{d}$, from the mobility, $\mu^{e/h}$, and the electric field at
the current position of the respective charge carrier:
$\mathbf{v}^{e/h}_{d}=\mu^{e/h}\cdot\mathbfcal{E}$. In each time step, the
signal in each channel is determined by evaluating the weighting potential of
all charge carriers at their current position.

The three parts of the model describing the physics underneath the passivated
surface are:

  1. 1.

a so-called dead layer,

  2. 2.

a so-called surface channel,

  3. 3.

probabilistic charge trapping.

The dead layer varying with $r$, $DL(r)$, is implemented for a thickness
modelled as a quadratic function with three parameters, $DL_{1}$, $DL_{2}$ and
$DL_{3}$, which are the dead-layer thicknesses at the radii 12.8 mm, 23.0 mm
and 34.8 mm. The allowed parameter space for these three parameters was chosen
as [0, 20] µm which is motivated by the location of the Bragg peak for alphas
as described in Sec. 2. For all $r$, $DL(r)$ is forced to be $\geq 0$ µm. The
initial energy, $Q_{\alpha}^{*}(r)$, of the events is the reduced energy based
on the thickness of the dead layer at the respective radial position. The
reduction is calculated by integrating the energy loss of alphas in germanium
over the dead-layer thickness Hauertmann2021:PhDThesis .

The surface channel is implemented in the simulation as a virtual volume, in
which $\mathbf{v}^{e/h}_{d}$ is modulated to ensure a drift parallel to the
passivated top surface,

$\mathbf{v}^{e/h}_{d,\mathrm{m}}=(\mathbf{v}^{e/h}_{d}\cdot\mathbf{e}_{r})\cdot\mathbf{e}_{r}+(\mathbf{v}^{e/h}_{d}\cdot\mathbf{e}_{\varphi})\cdot\mathbf{e}_{\varphi},$
(5)

where $\mathbf{e}_{r}$ and $\mathbf{e}_{\varphi}$ are the radial and azimuthal
unit vectors. This virtual volume is a tube directly underneath the top
surface with a height of 100 µm, ranging from $r=10.05$ mm to $r=37.45$ mm.
The height of the virtual volume is motivated by the fact that charge trapping
was not observed for 59.5 keV gammas also emitted by the 241Am source
Hauertmann2021:PhDThesis . Most of these gammas interact with the germanium
within about 1 mm. Thus, only events very close, $\mathcal{O}$(20 µm), to the
surface seem to be affected by surface effects. The modulation of the drift
vector, which corresponds to a modulation of the mobility tensor, is justified
as the assumption of an infinite crystal made in the derivation of the
mobility tensors for electrons and holes is not valid this close to the
surface.

It is also clear that the environment of the detector has to be taken into
account in the simulation when handling the drift of charge carriers close to
surfaces. If the field simulation is limited to the volume of the detector,
reflecting boundary conditions are typically used at surfaces like the
passivated top plate. This, however, forces the $z$-component of the electric
field, $\mathcal{E}_{z}$, to be zero at the surface, which is shown in Fig.
16a. Even though this actually leads to the desired drift along the surface,
it would incorrectly shift the cause for it from the mobility tensor to the
electric field. Thus, in order to learn something about the mobility tensor
close to surfaces, one has to use the correct electric field, which can have a
non-zero $z$-component at these surfaces as shown in Fig. 16b in which also
the unmodulated drift paths for one event at $r=24.8$ mm just underneath the
top surface are shown. The modulated drift paths of the event are shown in
Fig. 16c.

Figure 16: The $z$-component of the electric field, $\mathcal{E}_{z}$, in the
top part of Super-Siegfried. The dashed (dotted) lines are the drift
trajectories of the electrons (holes) of an event simulated at $r=24.8$ mm and
$\varphi=135.6^{\circ}$ corresponding to one measurement position of the scan
RSF marked by the star. In a), the field simulation is limited to the volume
of the detector. In b) and c), the field simulation includes the infrared
shield and detector base plate of GALATEA. In c), the drift model is modified,
forcing the charge carriers to drift along the surface.

The probabilistic trapping of charge carriers is introduced in each step of
the drift inside the virtual volume underneath the top plate as a probability
for the charge carrier (electron or hole) to get trapped, $p_{t}^{e/h}$. It is
calculated based on the step duration, $\Delta t=1$ ns, on the current
modulated drift velocity, $|\mathbf{v}^{e/h}_{d,\mathrm{m}}|$, and on the
probability parameters $p_{t,0}^{e}$ and $p_{t,0}^{h}$ for electrons and
holes:

$p_{t}^{e/h}(\Delta
t,|\mathbf{v}^{e/h}_{d,\mathrm{m}}|)=1-\exp(-p_{t,0}^{e/h}\cdot\Delta
t/|\mathbf{v}^{e/h}_{d,\mathrm{m}}|)\hskip 10.0pt.$ (6)

The allowed parameter space for $p_{t,0}^{e}$ and $p_{t,0}^{h}$ is [0,
$\infty$] µm/ns2 ensuring $p_{t}^{e/h}(\Delta t,|v_{d}|)\in[0,1]$. In each
step, a random number between 0 and 1 is generated. If the generated number is
below the $p_{t}^{e/h}$ of that step, the charge carrier is permanently
stopped (trapped) at its current position.

A Bayesian fit was performed with the software _BAT.jl_ Schulz:2021BAT  to the
data points of RSF to determine the best values for the five parameters
$DL_{1}$, $DL_{2}$, $DL_{3}$, $p_{t,0}^{e}$ and $p_{t,0}^{h}$. Flat priors
were assumed for all five parameters.

For the fit, one event per set of parameters was simulated for each location.
Each event was simulated with 2000 electron-hole pairs,
$N_{\mathrm{ep}}=2000$, and the signal induced by each charge carrier was
weighted with $Q_{\alpha}^{*}(r)/N_{\mathrm{ep}}$. Each of those charge
carriers could get trapped in each step of the drift as described earlier. In
the fit, the seed for the random numbers was fixed. Thus, the same random
numbers were used for each set of parameters and, therefore, each specific set
of parameters always resulted in the same simulated energies. It was tested
and verified that the large number of charge carriers ensured that the
simulated energies of the events were basically independent, $\mathcal{O}$(1
keV), of the chosen seed.

The likelihood, $\mathcal{L}$, that a set of parameters represents the data
was calculated using simulated energies, $E_{\mu,j,\mathrm{sim}}^{0/19}$, in
the core and segment 19 for events spawned at all measurement positions of the
scan RSF. The data are the most likely observed alpha energies,
$E_{\mu,j,\mathrm{obs}}^{0/19}$, as shown in Fig. 14a for each position of the
scan RSF for the core and segment 19.

The simulation does not take certain systematic effects into account111E.g.
possible inhomogeneities of the crystal and passivation layer, varying
temperatures $\mathcal{O}$(5 K) in GALATEA, details of the impurity profile,
charge cloud effects, no (simulated) trapping along the widening of the
borehole.. Therefore, the simulated core and segment 19 energies were
associated with an estimated uncertainty of $\sigma_{\mathrm{sim}}=100\,$keV.
The individual likelihood, $l_{j}^{0/19}$, for each core and segment 19 energy
at each radial position is calculated as

$l_{j}^{0/19}=\mathcal{N}(E_{\mu,j,\mathrm{sim}}^{0/19},\sigma_{\mathrm{sim}})(E_{\mu,j,\mathrm{obs}}^{0/19})\hskip
10.0pt,$ (7)

where $\mathcal{N}(E_{\mu,j,\mathrm{sim}}^{0/19},\sigma_{\mathrm{sim}})$ is a
Normal distribution with mean $E_{\mu,j,\mathrm{sim}}^{0/19}$ and a standard
deviation of $\sigma_{\mathrm{sim}}$. The overall $\mathcal{L}$ is given as
the product over all positions:

$\mathcal{L}=\prod_{j}l_{j}^{0}\cdot l_{j}^{\mathrm{19}}\hskip 10.0pt.$ (8)

The result of the fit is shown in Fig. 17. The simulation follows the trend of
the data very well. The marginalised posterior distributions of the parameters
$DL_{1}$, $DL_{2}$ and $DL_{3}$ are shown in Fig. 18 and of the parameters
$p_{t,0}^{e}$ and $p_{t,0}^{h}$ in Fig. 19. The fit suggests that the dead-
layer thickness increases towards the mantle of the detector reducing the
amount of energy deposited in the active volume. It also suggests that the
trapping probability is about a factor of three higher for electrons than for
holes.

Figure 17:  a) Dead-layer thickness as given by the fit to the data of the
scan RSF, see Fig. 14a. b) Deposited energy $Q_{\alpha}^{*}$ corresponding to
the depth of the dead layer and simulation results in form of (1 $\sigma$, 2
$\sigma$, 3 $\sigma$) posterior-predictive credibility intervals (solid,
dashed and dotted lines) for $E_{\mu,j,\mathrm{sim}}^{0}$ and
$E_{\mu,j,\mathrm{sim}}^{19}$ compared to $E_{\mu,j,\mathrm{obs}}^{0}$ and
$E_{\mu,j,\mathrm{obs}}^{19}$ of the scan RSF. The error bars do not show the
uncertainty on the data points but $\sigma_{\mathrm{sim}}=$ 100 keV.  Figure
18: Marginalised posterior distribution, $p(DL_{k})$, of the thickness of the
dead layer at $r=$ 12.8 mm ($DL_{1}$), $r=$ 23.0 mm ($DL_{2}$) and $r=$ 34.8
mm ($DL_{3}$). Figure 19: Marginalised posterior distribution,
$p(p_{t,0}^{e/h})$, of the two parameters describing the charge-trapping
probability.

The parameters of the global mode of the fit were used to simulate 10000
events at each radial position of the scan RSF. Each event was simulated with
$N_{\mathrm{ep}}=30$ and no seed was set. The distributions of
$E_{\mathrm{sim}}^{0}$ and $E_{\mathrm{sim}}^{19}$ and measured
$E_{\mathrm{data}}^{0}$ and $E_{\mathrm{data}}^{19}$ are shown in Fig. 20 for
the three radial positions 14.8 mm, 24.8 mm and 34.8 mm. Different values for
$N_{\mathrm{ep}}$ were tried and $N_{\mathrm{ep}}=30$ was found to produce as
broad energy distribution like seen in the data due to statistical
fluctuations. The relatively low value of $N_{\mathrm{ep}}=30$ resulting in
reasonably broad distributions could mean that the trapping mechanism actually
happens in some coherent way, such that always a group of electrons or holes
gets trapped during the drift. Another explanation might be that there are
certain locations at the surface where charge carriers are more likely to get
trapped such that, again, a group of electrons or holes is trapped at the same
time or, respectively, at the same location.

Figure 20: Correlation between core and segment 19 energies and their
marginalisations next to them. On the left (a, b, c): Alpha events selected as
described in Sec. 3 of three measurements from RSF. On the right (d, e, f):
Simulated alpha events spawned at the position of the respective measurement
shown on the left. The histograms obtained from the simulations are scaled to
the respective histogram on the left.

Even though the model was not tuned to predict correlations, correlations
similar to the observed correlations are produced. For events at lower and
larger radii, see Fig. 20a and Fig. 20c, the measured 2D-distributions are
narrow and stretched. This is also predicted by the simulation, see Fig. 20d
and Fig. 20f. It can be explained by the shapes of the weighting potentials of
the contacts, see Fig. 21. The gradient of the weighting potential increases
close to the respective contact.

Figure 21: Weighting potentials of the core and segment 19 along $r$ at the
top surface of the detector at $\varphi=135.6^{\circ}$ (fast axis) and $z=70$
mm as calculated with SSD.

For events at larger radii, the holes have only a short drift path towards
segment 19 and, thus, less holes are trapped during the drift. In addition,
trapped holes still contribute significantly to the segment 19 signal as they
are close to the contact. The electrons, however, are trapped at lower radii
where the gradient of the weighting potential of segment 19 is basically zero.
Thus, the exact location of the trapping of electrons does not influence the
signal of segment 19 after the electrons moved inwards below a certain radial
position, where the weighting potential of segment 19 is almost constant.
Therefore, for events at small (large) radii the spread of $E_{19}$ ($E_{0}$)
for a given $E_{0}$ ($E_{19}$) is smaller than the spread of $E_{0}$
($E_{19}$) for a given $E_{19}$ ($E_{0}$).

At medium radii, see Fig. 20b, a half-moon shaped distribution is observed.
This is not reproduced by the simulation, see Fig. 20e. However, this is
probably because the spread of the beam-spot was not taken into account in the
simulation. All simulated events were spawned at the center of the beam-spot.
Taking the beam-spot into account, see Fig. 6, would result in a superposition
of the distributions for the different $r$ in the beam-spot.

The simulation does not include the low-energy tails in the data which are due
to collimation effects, see Fig. 5. It produces normally shaped distributions
for $E_{0}$ and $E_{19}$, see Figs. 20d-f. It is, however, interesting that
the low-energy alphas seem to be affected by charge trapping in the same way
as the high-energy alphas. The trapping mechanisms seem to depend only on the
location and not on the total energy deposited.

##  6 Effect of metalisation

In previous studies Abt:2016trw , a strong dependence of the rise time of the
pulses on $\varphi$ was observed for both alpha and gamma induced events in
segment 19. For these studies, segment 19 was only metalised in a small area
where the read-out cable was, and is, connected to the segment at
$\varphi_{\mathrm{ro}}=135^{\circ}$ over a range of about 20∘. Since then, the
detector was reprocessed and segment 19 became fully metalised.

The rise time, $\mathcal{T}_{10}^{90}$, defined as the time in which the pulse
rises from 10% to 90% of its maximum amplitude, is used for the rise-time
studies presented here. The distribution of $\mathcal{T}_{10}^{90}$ of the
core and segment 19 pulses of alpha induced events is shown in Fig. 22 for one
position of the scan AST. The histograms were fitted with scaled Normal
distributions for each position. The fitted means of these distributions as a
function of $\varphi$ are shown in Fig. 23 for both the core and segment 19.

The core pulses show the expected modulation due to the crystal axes and a
small dependence on $\Delta\varphi_{j}=|\varphi_{j}-\varphi_{\mathrm{ro}}|$,
where $\varphi_{j}$ is the azimuthal position of the measurement $j$. This is
quantified by comparing the rise times at two positions close to different
fast axes of the crystal. At positions close to the fast axis at
$\varphi\approx 225^{\circ}$, the rise time $\mathcal{T}_{10}^{90}$ is about
15 ns longer than at positions close to the fast axis near near
$\varphi_{\mathrm{ro}}$.

Figure 22: Distribution of the rise time $\mathcal{T}_{10}^{90}$ for the core
and segment 19 of the measurement at $\varphi=262.9^{\circ}$ from the scan
AST. Also shown are scaled Normal distributions fitted to the histograms.
Figure 23: Fitted means of the distribution of $\mathcal{T}_{10}^{90}$ for the
core and segment 19 of alpha measurements from AST. The error bars indicate
systematic uncertainties due to the slightly fluctuating temperature of the
detector in GALATEA Hauertmann2017:MasterThesis .

For segment 19, this effect is more pronounced and the influence of the
crystal axes becomes less observable. Here, $\mathcal{T}_{10}^{90}$ is about
205 ns at $\varphi\approx 225^{\circ}$ and only about 135 ns at
$\varphi_{\mathrm{ro}}$. This difference of about 70 ns is, however, strongly
reduced compared to the previous measurements Abt:2016trw , where a variation
of the rise times of about 730 ns in segment 19 and about 50 ns in the core
was observed.

The dependence of $\mathcal{T}_{10}^{90}$ on $\Delta\varphi_{j}$ is not only
present for alpha induced events on the top surface but also for events
induced on the top by 59.5 keV gammas emitted from the Americium source
Hauertmann2021:PhDThesis .

Figure 24:  Averaged rise times $\mathcal{T}_{10}^{90}$ of events induced by
collimated 121.8 keV gammas from 152Eu entering segment 19 from the side for
a) segment 19 and b) the core, adapted from PhD:Lenz2010 ; Abt:2016trw . The
$\varphi$-axes were modified to match the coordinate system chosen for this
paper. c): Rise times $\mathcal{T}_{10}^{90}$ determined from the superpulses
formed from the gamma events induced by 59.5 keV gammas from the 241Am source
from the rotational side scan ASS. The error bars indicate systematic
uncertainties due to the slightly fluctuating temperature of the detector in
GALATEA Hauertmann2017:MasterThesis .

Prior to full metalisation, not only the rise times of events induced from the
top were, but also events induced from 121.8 keV gammas222Alpha events could
not be induced on the side of segment 19 because it was, and is, covered with
Kapton tape., entering segment 19 from the side of the detector from a 152Eu,
were affected. The values of $\mathcal{T}_{10}^{90}$ for segment 19 showed a
very strong dependence on $\Delta\varphi_{j}$ PhD:Lenz2010 ; Abt:2016trw  as
shown in Fig. 24a. No such effect was observed for core pulses, see Fig. 24b.
After the full metalisation, only a very small dependence of
$\mathcal{T}_{10}^{90}$ on $\Delta\varphi_{j}$ is observable as shown in Fig.
24c where the rise times of superpulses333Superpulses are the averaged pulses
of all selected gamma events of a measurement. They are used at these low
energies because the electronic noise is too large to determine
$\mathcal{T}_{10}^{90}$ for individual pulses. Hauertmann2021:PhDThesis  of
59.5 keV gamma events from the scan ASS are depicted. In contrast to the
previous measurements, the modulation reflecting the crystal axes is now also
visible in segment 19. This indicates that details of contacting and
metalisation can influence pulse shapes significantly. Such considerations
could help to design germanium detectors such that events close to passivated
surfaces could be easily identified. One idea is to leave a gap between the
passivation and the beginning of the metalisation.

##  7 Summary and outlook

Charge trapping was observed in alpha events on the passivated surface of the
end-plate of an n-type true-coaxial segmented HPGe detector. Such events occur
with energies in the region around 2 MeV where the signal for $0\nu\beta\beta$
decay in 76Ge would appear. The trapping of holes and electrons was clearly
identified by truncated mirror pulses observed on the segmented mantle of the
detector. This identification confirmed that alpha events can be selected
effectively by analysing the tail slope of the core pulses. Thus, the
background due to alphas on the passivated surfaces can be efficiently
suppressed for detectors as used in the first phase of the LEGEND experiment
currently under construction at the Gran Sasso underground laboratory.

Radial scans of the detector top surface showed that charge trapping is
dominated by hole trapping for events at small radii and by electron trapping
for events at large radii as expected from the length of the respective drift
paths. The trapping probability per µm/ns2 was found to be three times larger
for electrons than for holes for drifts directly underneath the passivated top
surface. A dependence of the probability of charge trapping on the crystal
axes has been observed for the first time in GALATEA. Near a fast axis,
$\langle 100\rangle$, less net charge trapping is observed than near a slow
axis, $\langle 110\rangle$. This can be understood as a consequence of higher
mobilities along the fast axis.

Charge trapping was simulated with the open-source software package
_SolidStateDetectors.jl_. The radial dependence of the probability of charge
trapping can be described by a model including a radius-dependent dead layer,
a modulation of the charge-carrier mobility and probabilistic charge trapping
of electrons and holes. In the future, the model can be extended to take
crystal axes effects into account in order to describe the azimuthal
dependence. In addition, new versions of _SolidStateDetectors.jl_ can take
charge-cloud effects like diffusion and self-repulsion into account.
Furthermore, the non-zero tail slope, used to identify surface alpha events,
could be added to the simulation by introducing a probabilistic release of
trapped charge carriers in each step similar to the trapping probability, or
by letting the trapped charge carriers still drift with a very low velocity.

The influence of the segment metalisation on the pulse shapes of events close
to the passivated top surface of the detector was studied. In previous
studies, very long pulses were observed close to the top for a metalisation
scheme where only small areas were metalised. For a full metalisation, normal
(faster) pulses and a much weaker dependence of the rise times on the event
position with respect to the position of the read-out contact were observed.
However, a small $\varphi$ dependence beyond the crystal axes modulation was
observed even for full metalisation. Metalisation schemes could be used to
influence the pulses of surface events to facilitate easier identification of
such events.

## References

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# Classifying toric 3-fold codes of dimensions 4 and 5

Tori Braun  Tori Braun
Ripon College
Ripon, Wisconsin<EMAIL_ADDRESS>,  James Carzon  James Carzon
University of Michigan–Dearborn
Dearborn, Michigan<EMAIL_ADDRESS>,  Jenna Gorham  Jenna Gorham
University of Arizona
Tucson, Arizona
<EMAIL_ADDRESS>and  Kelly Jabbusch  Kelly Jabbusch
Department of Mathematics & Statistics
University of Michigan–Dearborn
Dearborn, Michigan
<EMAIL_ADDRESS>
###### Abstract.

A toric code is an error-correcting code determined by a toric variety or its
associated integral convex polytope. We investigate $4$\- and $5$-dimensional
toric $3$-fold codes, which are codes arising from polytopes in
$\mathbb{R}^{3}$ with four and five lattice points, respectively. By computing
the minimum distances of each code, we fully classify the $4$-dimensional
codes. We further present progress toward the same goal for dimension $5$
codes. In particular, we classify the $5$-dimensional toric $3$-fold codes
arising from polytopes of width 1.

###### Contents

  1. 1 Introduction
    1. 1.1 Statements of our theorems
  2. 2 Minimum distances for dimension 4
    1. 2.1 Formula for 4 dimensions
  3. 3 Classification of toric 3-fold codes of dimension 4
  4. 4 On dimension 5 codes
    1. 4.1 Minimum distances for dimension 5
    2. 4.2 Classification for dimension 5

##  1\. Introduction

It is unknown in general whether an error correcting code will be “good” -
say, in terms of the number of errors which it can dependably handle - until
its parameters are computed, leading to an interest in classifying all the
possible distinct codes. Some impressive results have come from particular
toric codes, which Hansen introduced as the natural extension of Reed-Solomon
codes in 1998 [Hansen98]. By the correspondence between toric varieties and
integral convex polytopes, one obtains a toric code $C_{P}$ determined by a
polytope $P$. It is our imperative to be able to determine in general whether
a code $C_{P}$ is the same as a code $C_{P^{\prime}}$, where $P\neq
P^{\prime}$, in an essential way.

Many previous authors have written toward answering this question in
particular cases. Little and Schwarz classified toric surface codes - toric
codes corresponding to polytopes in $\mathbb{R}^{2}$ \- of dimensions up to 5
[LSchwarz]. Luo, Yau, Zhang, and Zuo furthered this to toric surface codes up
to 6 dimensions [LYZZShort], and later these same authors further classified
most of the 7 dimension codes with Hussain [HLYZZ]. The classification of 7
dimension codes was independently carried out by Cairncross, Ford, Garcia, and
Jabbusch [REU2019]. Departing slightly from the trend of these investigations,
we present original theorems disambiguating the toric codes which are
furnished by polytopes in $\mathbb{R}^{3}$ with up to 4 lattice points. We
refer to these as toric 3-fold codes of dimension 4 (as compared to toric
surface codes) to avoid confusion. We then make progress toward a
classification of toric 3-fold codes of dimension 5 by considering those
polytopes structurally very similar to those of dimension 4.

At a high level, a toric code $C$ is constructed by selecting a finite field
$\mathbb{F}_{q}$ and a convex lattice polytope $P$ in
$[0,q-2]^{m}\subset\mathbb{R}^{m}$. Codewords are obtained from evaluating
linear combinations of monomials arising from the lattice points of $P$ at
points in $(\mathbb{F}_{q}^{*})^{m}$, and so one can represent the code by a
generator matrix whose row space is the dictionary of codewords. A code can be
further described by parameters such as length, the number of points in
$(\mathbb{F}_{q}^{*})^{m}$, denoted $n$; dimension, the number of lattice
points $k=\lvert P\cap\mathbb{Z}^{m}\rvert$; and minimum distance, the minimum
Hamming distance between codewords, denoted $d$. Note that we refer to $k$ as
the dimension of $C$ and denote it $dim(C)$, which is not to be confused with
the dimension $m$ of the space in which $P$ is embedded. Two codes will be
considered the same if they are monomially equivalent, a term defined shortly.

Note that all toric codes obtained from polytopes in $\mathbb{R}^{3}$
containing fewer than 4 lattice points are trivially classified because they
can also be embedded in $\mathbb{R}^{2}$ and have been previously studied.
Likewise, for the purposes of this paper we will also overlook those codes
furnished by polytopes of 4 or 5 lattice points in $\mathbb{R}^{3}$ which can
be embedded in $\mathbb{R}^{2}$ since they are also well known.

More explicitly, the construction of a toric code is as follows. Selecting a
finite field of order $q$ a prime power and a lattice convex polytope $P$ in
$[0,q-2]^{m}$ with $k$ lattice points, one obtains a toric code $C_{P}$ by the
vector space $Im(\epsilon)$, where the map
$\epsilon:\mathcal{L}(P)\rightarrow(\mathbb{F}_{q})^{n}$ evaluates polynomials
$f$ in

$\mathcal{L}(P)=\text{Span}_{\mathbb{F}_{q}}\\{\textbf{x}^{p}:p\in
P\cap\mathbb{Z}^{m}\\}$

at each point in $(\mathbb{F}_{q})^{m}$. For example, let $q=11$ and let $P$
be the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,0,1)$, and
$(1,1,1)$. (The significance of this choice is explained shortly.) Then we
construct the collection of polynomials.

$\mathcal{L}(P)=\\{ax^{0}y^{0}z^{0}+bx^{1}y^{0}z^{0}+cx^{0}y^{0}z^{1}+dx^{1}y^{1}z^{1}:a,b,c,d\in\mathbb{F}_{11}\\}.$

We may then select a polynomial $f$ in $\mathcal{L}(P)$ and evaluate it at
each of the $n$ triples $(x,y,z)\in\left(\mathbb{F}_{11}^{*}\right)^{3}$. Our
resulting codeword is the $n$-tuple $(f(1,1,1),f(2,1,1),\ldots,f(11,11,11))$.
This is more general than Reed-Solomon because the polynomials
$f\in\mathcal{L}(P)$ may be in more than one indeterminate. The generator
matrix $G$ of $C_{P}$ is the matrix $(\textbf{x}^{p})$ with
$\textbf{x}\in(\mathbb{F}_{q}^{*})^{m}$ and $p\in P\cap\mathbb{Z}^{m}$, where
each column corresponds to a different point x in the field. Then

$C_{P}=\\{uG:u\in(\mathbb{F}_{q})^{k}\\}.$

Let $C_{P_{1}}$ and $C_{P_{2}}$ be toric codes with generator matrices $G_{1}$
and $G_{2}$ respectively.

######  Definition 1.0.1.

We say $C_{P_{1}}$ and $C_{P_{2}}$ are monomially equivalent if there exists
an invertible $n\times n$ diagonal matrix $\Delta$ and an $n\times n$
permutation matrix $\Pi$ such that

$G_{2}=G_{1}\Delta\Pi.$ (1)

As a remark, if there exists an affine unimodular transformation
$t:\mathbb{Z}^{n}\rightarrow\mathbb{Z}^{n}$ such that $t(P_{1})=P_{2}$, then
we say that $P_{1}$ and $P_{2}$ are lattice equivalent, which implies that
$C_{P_{1}}$ and $C_{P_{2}}$ are monomially equivalent [LSchwarz, Theorem 4].
For the toric 3-fold codes of dimension 4, we may make ready use of a classic
theorem by White.

######  Theorem 1.0.2 ([white_1964]).

Every 3-polytope with 4 integral lattice points of volume $t$ is lattice
equivalent to an empty tetrahedron

$T(s,t)=\text{Conv}\\{(0,0,0),(1,0,0),(0,0,1),(s,t,1)\\},$

for some $s\in\mathbb{Z}$ where $\gcd(s,t)=1$. Moreover, $T(s,t)$ is lattice
equivalent to $T(s^{\prime},t)$ if and only if $s^{\prime}=\pm s^{(\pm 1)}$
(mod $t$).

Figure 1. Monomials from the vertices of the empty tetrahedron $T(s,t)$.

Thus if two 3-fold toric codes of dimension 4 are not monomially equivalent,
then we know that they can be furnished by two empty tetrahedra with different
values of the parameters $s$ or $t$.

###  1.1. Statements of our theorems

Therefore an outline of our approach to classifying four dimensional codes is
as follows. Firstly, we provide some preliminary machinery in Section 3, most
notably a formula for the minimum distances of four dimensional codes.

######  Theorem 1.1.

For any four dimensional toric code on an empty tetrahedron $T(s,t)$, the
minimum distance is

$d(C_{T(s,t)})=\begin{cases}(q-1)^{3}-(q-1)^{2}&\mbox{if }\gcd(t,q-1)=1\\\
(q-1)^{3}-(q-1)(q-3)-q\gcd(t,q-1)&\mbox{else.}\end{cases}$

Secondly, we inquire about the equivalence classes which we obtain when we fix
$s$. To answer for this case, we prove the first classification theorem:

######  Theorem 1.2.

Let $C_{1}$ and $C_{2}$ be four dimensional toric codes on empty tetrahedra
$T(s,t_{1})$ and $T(s,t_{2})$, respectively, over the field $\mathbb{F}_{q}$.
Then $C_{1}$ and $C_{2}$ are monomially equivalent if and only if
$\gcd(t_{1},q-1)=\gcd(t_{2},q-1)$.

Lastly, if we conversely fix the parameter $t$, then we determine the
equivalence classes in this case by proving the second classification theorem:

######  Theorem 1.3.

Let $C_{1}$ and $C_{2}$ be four dimensional toric codes on empty tetrahedra
$T(s_{1},t)$ and $T(s_{2},t)$, respectively, over the field $\mathbb{F}_{q}$.
If either of the following conditions hold true, then $C_{1}$ and $C_{2}$ are
monomially equivalent:

  * •

$s_{1}\equiv s_{2}\bmod\gcd(t,q-1)$

  * •

$s_{1}\equiv\pm s_{2}^{\pm 1}\bmod t$

Furthermore, if neither of these hold, then the codes are not monomially
equivalent.

The proofs of these theorems are undertaken in Section 4, and we thereby
obtain a full classification of these toric codes.

Section 5 consists of describing the codes of dimension 5 and proving several
results. In particular, we identify what monomial equivalence classes exist
for codes arising from polytopes with 5 lattice points, all of which can be
found in the union of two consecutive hyperplanes. (In other words, these are
the polytopes of width 1 per the terminology used by Blanco and Santos
[BS4and5].) However, we stop short of a full classification of these toric
codes by not addressing those codes of width 2.

Throughout, much of our geometric notation and terminology will respect that
of White and of Blanco and Santos. For example, the toric code furnished by
the empty tetrahedron $T(s,t)$ will be written as $C_{T(s,t)}$, and in Section
5 we will use the invariant of a polytope with 5 lattice points which they
call the signature \- an unordered integer pair - to refer to it and its
furnished code. For example, if $P$ has signature $(i,j)$ and parameter values
$s$ and $t$, then from it we obtain the code $C_{P^{(i,j)}(s,t)}$. We will
then refer to the signature of the polytope furnishing a code as the signature
of the code as well.

##  2\. Minimum distances for dimension 4

A premium tool for evidencing the monomial inequivalence of two toric codes is
the minimum distance invariant. If two codes are monomially equivalent, then
they must have the same minimum distance. We now formally treat the dimension
4 case’s minimum distances.

######  Definition 2.0.1.

The minimum distance of a code $C$ is

$d(C)=\underset{f\neq
g\in\mathcal{L}(P)\setminus\\{0\\}}{min}h(\epsilon(f),\epsilon(g)),$

where $h(\cdot,\cdot)$ is the Hamming metric.

Recall that the Hamming metric $d$ counts the number of components which are
distinct between two tuples of equal length. Alternatively, we will use the
following fact:

######  Remark 2.0.2.

An equivalent expression for the minimum distance of a toric code $C_{P}$
corresponding to a polytope $P\subseteq\mathbb{R}^{m}$ is

$d(C)=(q-1)^{m}-\underset{0\neq f\in\mathcal{L}(P)}{max}Z(f),$ (2)

where $Z(f)$ is the number of zeros of $f$.

Computing different maximum numbers of zeros for polynomials furnished by
codes from different polytopes is a simple test that the codes are not
monomially equivalent. We also make use of a theorem by Soprunov and
Soprunova.

######  Theorem 2.0.3 ([SS2]).

Let $P\subseteq\mathbb{R}^{m}$ and $Q\subseteq\mathbb{R}^{n}$. Then

$d(C_{P\times Q})=d(C_{P})d(C_{Q}).$

For example, if $P$ is a polytope in $\mathbb{R}^{2}$ and we take $Q$ to be
$\\{\textbf{0}\\}$ in $\mathbb{R}^{1}$, then the product $P\times Q$ is the
embedding of $P$ in $\mathbb{R}^{3}$. We can thereby determine the minimum
distances of the degenerate cases of 3-fold toric codes. In particular, the
four lattice equivalence classes of polygons with 4 lattice points represented
in Figure 2 (each denoted $P^{(i)}_{4}$ for $i=1,2,3,4$ as in [LSchwarz]) can
be embedded in $\mathbb{R}^{3}$ to obtain these pedantic exceptions. We will
denote the $\mathbb{R}^{3}$ embedding of $P^{(i)}_{4}$ as $P^{\prime(i)}_{4}$.
Their resulting minimum distances are as listed:

  * •

$d(C_{P^{\prime(1)}_{4}})=(q-1)^{3}-3(q-1)^{2}$

  * •

$d(C_{P^{\prime(2)}_{4}})=(q-1)^{3}-2(q-1)^{2}$

  * •

$d(C_{P^{\prime(3)}_{4}})=(q-1)^{3}-(2q-3)(q-1)$

  * •

$d(C_{P^{\prime(4)}_{4}})>(q-1)^{3}-(1+q+2\sqrt{q})(q-1)$

The fourth of these polygons is often called the “exceptional triangle” in
this setting, possibly because its minimum distance formula remains elusive.

1$x$$x^{2}$$x^{3}$$P^{(1)}_{4}$

1$x$$x^{2}$$y$$P^{(2)}_{4}$

1$x$$y$$xy$$P^{(3)}_{4}$

1$x$$x^{-1}y^{-1}$$y$$P^{(4)}_{4}$

Figure 2. Lattice equivalence class representatives of polygons with 4 lattice
points.

We turn now to a formula for the minimum distance in dimension 4 of the
nondegenerate cases.

###  2.1. Formula for 4 dimensions

In order to find the minimum distance of $C_{T(s,t)}$, we use (2). Then the
polynomial with the maximum number of zeros will return us the minimum
distance. According to the shape of $T(s,t)$, we know that such an $f\neq 0$
must be of the form

$f(x,y,z)=a+bx+cz+dx^{s}y^{t}z,$ (3)

where $a$, $b$, $c$, and $d$ are not all $0$ in $\mathbb{F}_{q}$. For
simplicity, define

$f_{x}(y,z)=a+bx+cz+d_{x}xy^{t}z$

when $x$ is fixed and $d_{x}=dx^{s-1}$. By doing so, we are able to count the
vanishing points of $f$ for each $x\in\mathbb{F}_{q}^{*}$ without concern for
the value of $s$ since we know that each will also be a vanishing point for
$f_{x}$ for some $x$ as well. Furthermore, each vanishing point of $f_{x}$
corresponds to one of $f$. It follows that

$Z(f)=\sum_{x}Z(f_{x}).$

See 1.1

To prove the theorem, note that if $f(x,y,z)=1+x$ for example, then so long as
$x=-1$ there are $q-1$ choices for each of $y$ and $z$ to obtain a vanishing
point and so $\underset{f}{\max}\,Z(f)\geq(q-1)^{2}$ for any $s$ and $t$. We
will first show that only when $t$ and $q-1$ are not relatively prime can we
find another nonzero polynomial with more zeros than $1+x$.

######  Proposition 2.1.1.

If $f\in\mathcal{L}(T(s,t))$ has no zero coefficients, then

$Z(f)\leq q\gcd(t,q-1)+(q-1)(q-3).$

Furthermore, there exists an $f$ for which this count is attained.

###### Proof.

Let $x=-ab^{-1}$. Then

$f_{-ab^{-1}}(y,z)=z(c+d_{-ab^{-1}}(-ab^{-1})y^{t}).$

Write $N(x)$ for the number of values of $y$ for which $c+d_{x}xy^{t}=0$. In
this case, we count $(q-1)N(-ab^{-1})$ zeros since $z$ is free to be any unit.

On the other hand, we also must consider points $(x,y,z)$ at which $f$
vanishes for which $x\neq-ab^{-1}$. Fix $x$. There are $(q-1)-N(x)$ choices of
$y$ for which $c+d_{x}xy^{t}\neq 0$. Therefore we have $\underset{x\neq-
ab^{-1}}{\sum}[(q-1)-N(x)]$ combinations of $x$ and $y$ which allow us to set

$z=-(a+bx)(c+dx^{s}y^{t})^{-1}.$

Considering both of these subcases, then, the total number of zeros is

$\displaystyle(q-1)N(-ab^{-1})$ $\displaystyle+\underset{x\neq-
ab^{-1}}{\sum}[(q-1)-N(x)]$
$\displaystyle=qN(-ab^{-1})+(q-1)(q-2)-\underset{x}{\sum}N(x)$
$\displaystyle=qN(-ab^{-1})+(q-1)(q-3)$ (4)

since certainly every value of $y$ will satisfy $c+d_{x}xy^{t}=0$ for some
$x$, and so the summation term is equal to $q-1$. We obtain the most zeros of
$f$ when we maximize $N(-ab^{-1})$ which occurs when $y^{t}=\frac{bc}{ad_{x}}$
has solutions. In particular, if $t$ and $q-1$ are not relatively prime, there
are as many as $\gcd(t,q-1)$ solutions. Using this fact with (2.1), we
conclude that if $f$ has no zero coefficients, then

$Z(f)\leq q\gcd(t,q-1)+(q-1)(q-3).$

∎

So more generally we have

$\underset{0\neq f\in\mathcal{L}(P)}{\max}Z(f)\geq q\gcd(t,q-1)+(q-1)(q-3).$
(5)

As we will show, if $\gcd(t,q-1)>1$, then (5) gives the highest zero count; if
else, then no polynomial vanishes at more than $(q-1)^{2}$ points. As with our
example of $f=1+x$, the next proposition covers the other cases.

######  Proposition 2.1.2.

If $f\in\mathcal{L}(T(s,t))$ has at least one zero coefficient, then

$Z(f)\leq(q-1)^{2}.$

###### Proof.

We will show that any polynomial $f$ with at least one zero coefficient has at
most $(q-1)^{2}$ vanishing points, using again the form of $f$ in (3) for
consistency. The subcases this time are as follows:

  * •

Suppose $d=0$. Then $f(x,y,z)=a+bx+cz$. If $a+bx=0$, then $cz=0$ implies that
$c=0$. Otherwise $a+bx\neq 0$, so we can let $z=-(a+bx)c^{-1}$. In either case
we have $Z(f)=(q-1)^{2}$.

  * •

Suppose $c=0$. (Note that because (3) has the aforementioned aesthetic
symmetry between $x$ and $z$ after fixing $x$, this subcase is treated the
same way as the subcase $b=0$.) Fixing $x=-ab^{-1}$, we have
$f_{-ab^{-1}}(y,z)=d_{-ab^{-1}}(-ab^{-1})y^{t}z$. We obtain
$d_{-ab^{-1}}(-ab^{-1})y^{t}z=0$, a contradiction, so for whatever choice of
$y$ we can choose $z=-(a+bx)(d_{x}xy^{t})^{-1}$ and have $Z(f)=(q-1)(q-2)$.

  * •

Suppose $a=0$. Then $f(x,y,z)=bx+cz+dx^{s}y^{t}z$. Fix $y$. If
$c+dx^{s}y^{t}=0$, then $bx=0$, a contradiction. Instead, so long as
$x\neq-c(dy^{t})^{-s}$, we can set $z=-bx(c+dx^{s}y^{t})^{-1}$ and have
$Z(f)=(q-1)(q-2)$.

By exhausting these three subcases we find that if $f$ has at least one zero
coefficient, it has at most $(q-1)^{2}$ vanishing points; furthermore, we see
that regardless of whether $t$ and $q-1$ are relatively prime we have

$\underset{0\neq f\in\mathcal{L}(P)}{\max}Z(f)\geq(q-1)^{2}.$ (6)

Thus we have our lower bound on $Z(f)$. ∎

What remains is a compilation of our results from the two cases in order to
have our formula.

See 1.1

###### Proof of Theorem 1.1.

The maximum number of vanishing points is equal to either the expression on
the right of (5) or the one on the right of (6) since both expressions give
upper bounds on the maximum number of vanishing points in the two distinct
cases. It follows that if $t$ and $q-1$ are relatively prime and $q>2$, then
(6) becomes an equality since

$q+(q-1)(q-3)<2(q-1)+(q-1)(q-3)=(q-1)^{2}.$

If $\gcd(t,q-1)\geq 2$, then

$q\gcd(t,q-1)\geq 2q>2(q-1),$

so (5) becomes an equality. By use with (2), we conclude the proof of the
proposition. ∎

##  3\. Classification of toric 3-fold codes of dimension 4

In order to compare generator matrices, we introduce the following notation.
Consider a partition of $(\mathbb{F}_{q}^{*})^{3}$ according to ordered pairs
of the first and last coordinates and an additional condition involving the
parameter $t$; in particular, define the partition

$\mathscr{P}_{t}=\\{P(x_{0},z_{0},A)\ni(x,y,z)\in(\mathbb{F}_{q}^{*})^{3}:(x,z)=(x_{0},z_{0}),y^{t}=A\\}.$

A cell in $\mathscr{P}_{t}$ may be empty, namely one for which $y^{t}=A$ has
no solutions. The columns of $G_{1}$ and $G_{2}$ are indexed by the elements
of $(\mathbb{F}_{q}^{*})^{3}$, so let $I_{i}(x_{0},z_{0},A)$ denote the set of
column indices in $G_{i}$ which correspond to points in a cell in
$\mathscr{P}_{t_{i}}$. For example, a representative column would be

$G_{1}e_{m}=\begin{bmatrix}1\\\ x_{0}\\\ z_{0}\\\
x_{0}^{s}Az_{0}\end{bmatrix},$

so $m\in I_{1}(x_{0},z_{0},A)$.

See 1.2

###### Proof of Theorem 1.2.

$(\Rightarrow)$ By contrapositive, supposing that
$\gcd(t_{1},q-1)\neq\gcd(t_{2},q-1)$ immediately implies that $d(C_{1})\neq
d(C_{2})$, so the codes are monomially inequivalent.

$(\Leftarrow)$ Let $l$ be the common $\gcd$, and let $\alpha$ be a primitive
element of $\mathbb{F}_{q}$. Denote by $G_{1}$ and $G_{2}$ generator matrices
of $C_{1}$ and $C_{2}$, respectively, for which each column in either is led
by a $1$ so that we need only worry to reorder the columns to determine
equivalence. In other words, we know that equivalence means that there exist
matrices - an invertible diagonal $\Delta$ and permutation $\Pi$ \- for which
(1) holds. However, at least the first rows are easily matched because they
are all $1$s, so $\Delta=I$ and we need only confirm that there exists a
satisfactory permutation $\Pi$.

We will show that $I_{1}$ and $I_{2}$ have the same cardinality for each
$x_{0},z_{0},A\in\mathbb{F}_{q}^{*}$. There exist integers $k_{1}$, $k_{2}$,
and $k_{3}$ such that

$\displaystyle t_{1}$ $\displaystyle=k_{1}l$ $\displaystyle t_{2}$
$\displaystyle=k_{2}l$ $\displaystyle q-1$ $\displaystyle=k_{3}l$

Suppose $\alpha^{m}$ is a solution to $y^{t_{1}}=A$. Then
$(\alpha^{m})^{k_{1}l}=A$. Every solution can be found from $\alpha^{m}$ to be
of the form $\alpha^{m+jk_{3}}$, $j=0,1,\ldots,l-1$. Similarly, each solution
to $y^{t_{2}}=A$ is of the form $\alpha^{n+jk_{3}}$ for some $n$, so the
solutions to either equation differ by multiplication by $\alpha^{n-m}$ and
are thereby subject to a bijective correspondence. It follows that if $\lvert
I_{1}\rvert>0$, then $\lvert I_{1}\rvert=\lvert I_{2}\rvert=l$. Supposing, on
the other hand, that $y^{t_{1}}=A$ has no solutions in $\mathbb{F}_{q}^{*}$
implies that $A$ is not in the range of the Frobenius automorphism $a\mapsto
a^{l}$, so it is also the case that neither does $y^{t_{2}}=A$ have solutions.
In any case, therefore, $\lvert I_{1}\rvert=\lvert I_{2}\rvert$.

Finally, we let $\Pi$ be any permutation on $(q-1)^{3}$ letters such that for
all $x_{0},z_{0},A\in\mathbb{F}_{q}^{*}$ we obtain

$\sum_{m\in I_{1}(x_{0},z_{0},A)}e_{m}=\sum_{n\in I_{2}(x_{0},z_{0},A)}\Pi
e_{n}.$

Then $\Pi$ is sufficient to satisfy our relation in (1) to confirm that
$C_{1}$ and $C_{2}$ are monomially equivalent. ∎

Note that our approach to proving the Theorem 1.2 is followed very closely
again in Section 5 with Proposition 4.2.1. Now we turn to proving our second
classification theorem.

See 1.3

###### Proof of Theorem 1.3.

$(\Rightarrow)$ Let $G_{1}$ and $G_{2}$ be generator matrices of $C_{1}$ and
$C_{2}$, respectively. We will first show that either of the conditions
implies monomial equivalence between $C_{1}$ and $C_{2}$.

Firstly, let $l=\gcd(t,q-1)$ and suppose $s_{1}\equiv s_{2}\bmod l$. Then
there exist $j,k\in\mathbb{Z}$ such that $t=jl$ and $s=s_{2}-s_{1}=kl$. We
seek to show that for any $x$ and $y_{2}$ there exists $y_{1}$ to satisfy

$x^{s_{1}}y_{1}^{t}=x^{s_{2}}y_{2}^{t}.$ (7)

For simplicity, let $n,i_{2}\in\mathbb{Z}$ give $x=\alpha^{n}$ and
$y_{2}=\alpha^{i_{2}}$, where $\alpha$ is a primitive element of the field.
Then by setting

$i_{1}\equiv i_{2}+nkj^{-1}\bmod\left(\frac{q-1}{l}\right),$

we get that

$i_{1}(jl)\equiv i_{2}(jl)+n(kl)\bmod(q-1).$

(Note that $j^{-1}\bmod\left(\frac{q-1}{l}\right)$ exists because
$\gcd(j,\frac{q-1}{l})=1$.) This is equivalent to

$(\alpha^{i_{1}})^{t}=(\alpha^{n})^{s}(\alpha^{i_{2}})^{t}$

which is in turn equivalent to (7), as desired. Using the same ordering as
above, we have that $C_{1}$ and $C_{2}$ are monomially equivalent.

The second condition for monomial equivalence follows from White’s theorem
classifying four point polytopes in $\mathbb{R}^{3}$. These two conditions are
sufficient for monomial equivalence.

$(\Leftarrow)$ We now turn to prove that these conditions are necessary.
Suppose $T(s_{1},t)$ and $T(s_{2},t)$ are not lattice equivalent but $C_{1}$
and $C_{2}$ are monomially equivalent. That is, the second condition does not
hold. This supposition amounts to a stipulation that if $G_{2}=G_{1}\Delta\Pi$
as in Definition 1.0.1, then $\Delta=I$. (That is, a nontrivial diagonal
matrix amounts to a reordering of the rows of $G$ (or lattice points in $P$).
See [LSchwarz, Theorem 4].) Furthermore, assume that $s_{2}-s_{1}\not\equiv
0\bmod\gcd(t,q-1)$. That is, the first condition does not hold. We proceed
toward a contradiction. We obtain $s_{2}-s_{1}=kl+r$ for some $k$ and $0<r<l$,
where again $l=\gcd(t,q-1)$.

Because there exists a permutation between the generator matrices of $C_{1}$
and $C_{2}$, we may define a map
$\mathbb{F}_{q}^{*}\rightarrow\mathbb{F}_{q}^{*}$ which sends $y_{2}$ to the
value of $y_{1}$ satisfying

$y_{1}^{t}=\alpha^{s_{2}-s_{1}}y_{2}^{t}$

where $\alpha$ is a primitive element of $\mathbb{F}_{q}^{*}$. Let $\phi$
denote the Frobenius automorphism $y\mapsto y^{t}$. Since $1\in Im(\phi)$ but
$1\notin\alpha^{r}Im(\phi)$ for $r<l$, we obtain a contradiction. We conclude
that the two conditions stated in Theorem 1.3 are both sufficient and
necessary for monomial equivalence. ∎

For completeness, we will exhibit that $C_{1}$ and $C_{2}$ are monomially
equivalent if $s_{1}\equiv\pm s_{2}^{\pm 1}\bmod t$ by explicit affine
unimodular transformations.

Case 1. Suppose $s_{1}\equiv s_{2}\bmod t$. Then
$\frac{s_{1}-s_{2}}{t}\in\mathbb{Z}$. We can make use of the assignment rule

$\begin{bmatrix}x\\\ y\\\
z\end{bmatrix}\mapsto\begin{bmatrix}1&\frac{s_{1}-s_{2}}{t}&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}$

which maps each of the vertices of $T(s_{2},t)$ to those of $T(s_{1},t)$. Thus
the polytopes are lattice equivalent.

Case 2. Suppose $s_{1}\equiv s_{2}^{-1}\bmod t$. Then
$\frac{-s_{1}s_{2}+1}{t}\in\mathbb{Z}$. This time we will use the assignment
rule

$\begin{bmatrix}x\\\ y\\\
z\end{bmatrix}\mapsto\begin{bmatrix}s_{1}&\frac{-s_{1}s_{2}+1}{t}&0\\\
t&-s_{2}&0\\\ 0&0&-1\end{bmatrix}\begin{bmatrix}x\\\ y\\\
z\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\ 1\end{bmatrix}.$

Once again, our map is an affine unimodular transformation and sends the
vertices of $T(s_{2},t)$ to those of $T(s_{1},t)$, so again we have lattice
equivalence.

Case 3. Suppose $s_{1}\equiv-s_{2}$. Then
$\frac{s_{1}+s_{2}}{t}\in\mathbb{Z}$. Thus we can define a similar
transformation mapping vertices of $T(s_{2},t)$ to those of $T(s_{1},t)$ as
follows:

$\begin{bmatrix}x\\\ y\\\
z\end{bmatrix}\mapsto\begin{bmatrix}-1&\frac{s_{1}+s_{2}}{t}&-1\\\ 0&1&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}+\begin{bmatrix}1\\\
0\\\ 0\end{bmatrix}.$

Thus, once again, the polytopes are lattice equivalent.

Case 4. Suppose $s_{1}\equiv-s_{2}^{-1}$. Then a transformation can be
constructed from lattice points of $T(s_{2},t)$ to those of $T(s_{1},t)$ by a
composition of those mappings defined in cases 2 and 3.

The first of these affine unimodular transformations was also presented by
Blanco and Santos [BS4and5]. The following corollary is a consequence of the
first condition listed in Theorem 1.3, namely that $s_{1}\equiv
s_{2}\bmod\gcd(t,q-1)$.

######  Corollary 3.0.1.

If $\gcd(t,q-1)=1$, then $C_{T(s_{1},t)}$ and $C_{T(s_{2},t)}$ are monomially
equivalent.

###### Proof.

Since $s_{1}\equiv s_{2}\bmod 1$ trivially (regardless of $s_{1}$ and
$s_{2}$), the codes are monomially equivalent by Theorem 1.3. ∎

##  4\. On dimension 5 codes

There are many more polytopes up to lattice equivalence in $\mathbb{R}^{3}$
with 5 lattice points than there are with 4. Two new invariants which we use
to distinguish between those which are not lattice equivalent are the width
and signature. The width of a polytope $P\subseteq\mathbb{R}^{3}$ is the
minimum number of consecutive hyperplanes in $\mathbb{R}^{3}$ which contain
$P$. The signature of $P$ is a statement about its affine dependence. There
are four signatures of width 1 polytopes with 5 lattices, listed in Table 1.

Sig. | Volume | Representative
---|---|---
(2,1) | (-2$t$, $t$, 0, $t$, 0) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (-1, 0, 0), ($s$, $t$, 1)
| $0\leq s\leq\frac{t}{2}$ |
(2,2) | (-1, 1, 1, -1, 0) | (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1)
(3,1) | (-3, 1, 1, 1, 0) | (0, 0, 0), (1, 0, 0), (0, 1, 0), (-1, -1, 0), (0, 0, 1)
(3,2) | ($-s-t$, $s$, $t$, 1, -1) | (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), ($s$, $t$, 1)
| $0<s\leq t$ |
Table 1. Five-point polytopes of width 1 [BS4and5].

Refer to the lattice points of $P$ by $p_{i}$ and the volumes of the convex
hulls of $\\{p_{j}\\}_{j\neq i}$ by $w_{i}$, $i=1,\ldots,5$. The $w_{i}$ are
possibly negative depending on orientation. Then there is an affine dependence

$\sum_{k=1}^{5}(-1)^{k-1}w_{k},\hskip 5.69054pt\text{ and }\hskip
5.69054pt\sum_{k=1}^{5}(-1)^{k-1}w_{k}p_{k}.$ (8)

The signature of $P$ is $(i,j)$ if the dependence (8) has $i$ positive
coefficients and $j$ negative ones. (Thus the signature is unordered because
the equations in (8) may be multiplied by $-1$.)

We summarize the lattice equivalence classes of polytopes in this case with
Table 2. Regarding the rows for signatures $(2,1)$ and $(3,2)$, note that $s$
and $t$ must be relatively prime. Thus there are an infinite number of
equivalence classes with these signatures. However, we consider only finitely
many when observing the toric codes over the field $\mathbb{F}_{q}$ since
there come constraints on the size of $P$ with the choice of $q$.

Sig. | Volume | Representative
---|---|---
(3,1) | (-9, 3, 3, 3, 0) | (0, 0, 0), (1, 0, 0), (0, 1, 0), (-1, -1, 0), (1, 2, 3)
(4,1) | (-4, 1, 1, 1, 1) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (1, 1, 1), (-2, -1, -2)
(4,1) | (-5, 1, 1, 1, 2) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (1, 2, 1), (-1, -1, -1)
(4,1) | (-7, 1, 1, 2, 3) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (1, 3, 1), (-1, -2, -1)
(4,1) | (-11, 1, 3, 2, 5) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (2, 5, 1), (-1, -2, -1)
(4,1) | (-13, 3, 4, 1, 5) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (2, 5, 1), (-1, -1, -1)
(4,1) | (-17, 3, 5, 2, 7) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (2, 7, 1), (-1, -2, -1)
(4,1) | (-19, 5, 4, 3, 7) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (3, 7, 1), (-2, -3, -1)
(4,1) | (-20, 5, 5, 5, 5) | (0, 0, 0), (1, 0, 0), (0, 0, 1), (2, 5, 1), (-3, -5, -2)
Table 2. Five-point polytopes of width 2 [BS4and5].

Figure 3. Five point polytopes with signatures $(2,1)$ (left) and $(3,2)$
(right).

###  4.1. Minimum distances for dimension 5

We now offer and prove the following statements on the minimum distances of
dimension 5 toric codes from only the polytopes of width 1.

######  Theorem 4.1.1 (Minimum Distances for Five Dimensions).

For the five point polytopes of width 1, we obtain the following minimum
distance formulae and bounds:

$\displaystyle d(C_{P^{(2,1)}(s,t)})$ $\displaystyle=(q-1)^{3}-2(q-1)^{2}$ (9)
$\displaystyle d(C_{P^{(2,2)}})$ $\displaystyle=(q-1)^{3}-(2q^{2}-5q+3)$ (10)
$\displaystyle d(C_{P^{(3,1)}})$
$\displaystyle\geq(q-1)^{3}-(q-1)(1+q+2\sqrt{q})$ (11) $\displaystyle
d(C_{P^{(3,2)}(s,t)})$ $\displaystyle\geq(q-1)^{3}-(q-2)^{2}-(s+t)q$ (12)
$\displaystyle d(C_{P^{(3,2)}(s,t)})$
$\displaystyle\leq(q-1)^{3}-(q-1)(q-3)-q\gcd(s+t,q-1)$ (13)

###### Proof.

The cases (10) and (11) are the simplest cases. Recall that the minimum
distance of the toric code from $P^{\prime(3)}_{4}$, the unit square in
$\mathbb{R}^{3}$, is $(q-1)^{3}-(2q-3)(q-1)$ and that from the exceptional
triangle $P^{\prime(4)}_{4}$ is bounded by
$d\geq(q-1)^{3}-(1+q+2\sqrt{q})(q-1)$. Since the five point polytopes with
signatures $(2,2)$ and $(3,1)$ are exactly the unit pyramids over the unit
square and the exceptional triangle, respectively, we have that their minimum
distances are increased by a factor of $(q-1)$ [SS2, Theorem 2.3].

Consider now (9). Note that $(x+1)(x^{-1}-1)$ has $2(q-1)^{2}$ zeros. We claim
that the polynomial

$f=A+Bx+Cx^{-1}+Dz+Ex^{s}y^{t}z$

has no more than $2(q-1)^{2}$ zeros. If $E=0$, then any polynomial
$f=A+Bx+Cx^{-1}+Dz$ will have no more than $2(q-1)^{2}$ zeros, since we have
the unit pyramid over the line in the plane with three lattice points, which
gives a code of minimum distance $(q-1)^{3}-2(q-1)^{2}$. So we may assume that
$E\neq 0$. Call $P=A+Bx+Cx^{-1}$ and $Q=D+Ex^{s}y^{t}$.

We reduce this to two cases.

Case 1. Suppose $P$ has no solutions. Then $f=0$ only if $Q\neq 0$ so that we
may select $x$ and $y$ freely and let $z=-PQ^{-1}$. Using the notation $N(x)$
similarly to its use in the proof of Theorem 1.2, giving the number of values
of $y$ for which $Q_{x}=0$ for a fixed $x$, we see that there are

$\displaystyle Z(f)$ $\displaystyle=\sum_{x\in\mathbb{F}_{q}^{*}}[(q-1)-N(x)]$
$\displaystyle=(q-1)^{2}-\sum_{x\in\mathbb{F}_{q}^{*}}N(x)$
$\displaystyle=(q-1)^{2}-(q-1)$ $\displaystyle<2(q-1)^{2}$

zeros of $f$, proving this case of the claim.

Case 2. Suppose $P$ has two solutions. Call them $x_{1}$ and $x_{2}$. Then
$f=0$ if $x=x_{1}$, $y$ is one of $N(x_{1})$ choices so that $Q=0$, and $z$ is
anything at all or likewise if $x=x_{2}$ and $y$ is one of $N(x_{2})$ choices
for the same reason; or if $Q\neq 0$ as in Case 1. Therefore we have

$\displaystyle Z(f)$ $\displaystyle=(q-1)(N(x_{1})+N(x_{2}))+\sum_{x\neq
x_{1},x_{2}}[(q-1)-N(x)]$ $\displaystyle=q(N(x_{1})+N(x_{2}))+(q-1)(q-4)$

zeros. Recall that $N(x)=0$ or $d=\gcd(t,q-1)$. Note also that since $d<q-1$
and $d\mid(q-1)$, we know that $d\leq\frac{q-1}{2}$ and so

$\displaystyle Z(f)$ $\displaystyle=q(N(x_{1})+N(x_{2}))+(q-1)(q-4)$
$\displaystyle\leq 2qd+(q-1)(q-4)$ $\displaystyle\leq q(q-1)+(q-1)(q-4)$
$\displaystyle=2(q-1)(q-2)$ $\displaystyle<2(q-1)^{2}.$

This proves our claim for this case as well.

We turn to (12). We consider a polynomial of the form

$f=A+Bx+Cy+Dz+Ex^{s}y^{t}z$

and let $P=A+Bx+Cy$ and $Q=D+Ex^{s}y^{t}$ be the same as above. Let $N$ denote
the number of pairs $(x,y)$ for which $P$ and $Q$ are both $0$.

Case 1. Suppose $A\neq 0$. There are $(q-2)-N$ pairs $(x,y)$ which give
$P=0\neq Q$ (since we cannot have $x=AB^{-1}$). There are $(q-1)-N$ pairs
which give $P\neq 0=Q$. Therefore the number of zeros of $f$ are the sum of
those giving $P\neq 0\neq Q$, which are

$(q-1)^{2}-[(q-1)-N]-[(q-2)-N]-N=(q-2)^{2}+N,$

and those giving $P=0=Q$, which are $N(q-1)$. Thus there are $(q-2)^{2}+Nq$
zeros.

Case 2. Suppose $A=0$. This time, $(q-1)-N$ pairs $(x,y)$ give either $P=0\neq
Q$ or $P\neq 0=Q$, and so there are $(q-1)(q-3)+Nq<(q-2)^{2}+Nq$ zeros.

Note that $N$ can be found as the number of solutions $x$ to the equation
$(A^{\prime}+x)^{t}x^{s}=B^{\prime}$ (where $A^{\prime}=AB^{-1}$ and
$B^{\prime}=(-1)^{t+1}(CB^{-1})^{t}DE^{-1}$) which has degree $s+t$. It
follows that $d\leq(q-1)^{3}-(q-2)^{2}-(s+t)q$.

Lastly, we turn to (13). Using the same counts of zeros in cases 1 and 2 from
(12), we have

$\underset{0\neq f\in\mathcal{L}(P)}{\max}Z(f)\geq(q-1)(q-3)+Nq.$

Since $N\leq\gcd(s+t,q-1)$ is an attainable bound, we obtain our upper bound
on the minimum distance. Thus concludes the proof of all formulae and bounds
on the minimum distance. ∎

Proceeding as we have before, we use Theorem 4.1.1 to complete a
classification of codes furnished by the polytopes with which the theorem is
concerned. We begin by providing propositions akin to our main classification
theorems to treat codes furnished by polytopes of signature $(2,1)$. Then we
classify codes furnished by polytopes of signature $(3,2)$. Finally, we show
that codes furnished by polytopes of width 1 and different signatures are not
monomially equivalent.

###  4.2. Classification for dimension 5

######  Proposition 4.2.1.

Let $C_{1}$ and $C_{2}$ be toric codes on five point polytopes
$P^{(2,1)}(s,t)$ and $P^{(2,1)}(s,t^{\prime})$, respectively, over the field
$\mathbb{F}_{q}$. Then $C_{1}$ and $C_{2}$ are monomially equivalent if and
only if $\gcd(t,q-1)=\gcd(t^{\prime},q-1)$.

###### Proof.

($\Leftarrow$) As was the case with Theorem 1.2, we have monomial equivalence
when we find a permutation matrix $\Pi$ which matches the columns
corresponding to $I_{1}(x_{0},z_{0},A)$ with those of $I_{2}(x_{0},z_{0},A)$
for all $(x_{0},z_{0},A)\in(\mathbb{F}_{q}^{*})^{3}$. Indeed, since we are
only including in the generator matrices the additional row determined by
$\epsilon(x^{-1})$, we may repeat that previous argument exactly to conclude
the sufficiency of this condition.

($\Rightarrow$) We don’t have the luxury of a discerning minimum distance
formula in this case, so instead suppose $A$ is in the image of both of the
automorphisms $a\mapsto a^{t}$ and $a\mapsto a^{t^{\prime}}$. We cannot create
the desired permutation $\Pi$ matching the $\gcd(t,q-1)$ columns
$I_{1}(x_{0},z_{0},A)$ to the $\gcd(t^{\prime},q-1)$ columns
$I_{2}(x_{0},z_{0},A)$, and so $C_{1}$ and $C_{2}$ are not monomially
equivalent. ∎

######  Proposition 4.2.2.

Let $C_{1}$ and $C_{2}$ be toric codes on five point polytopes
$P^{(2,1)}(s,t)$ and $P^{(2,1)}(s^{\prime},t)$, respectively, over the field
$\mathbb{F}_{q}$. Then $C_{1}$ and $C_{2}$ are monomially equivalent if and
only if $s\equiv s^{\prime}\bmod\gcd(t,q-1)$.

###### Proof.

Unlike with Theorem 1.3, there are no nontrivial lattice equivalences between
signature $(2,1)$ polytopes. Otherwise this proposition is proved identically
to that of the first condition in the statement of that theorem. ∎

######  Proposition 4.2.3.

Let $C_{1}$ and $C_{2}$ be toric codes on five point polytopes
$P^{(3,2)}(s,t)$ and $P^{(3,2)}(s^{\prime},t^{\prime})$, respectively, over
the field $\mathbb{F}_{q}$. Then $C_{1}$ and $C_{2}$ are monomially equivalent
if and only if $(s,t)=(s^{\prime},t^{\prime})$.

###### Proof.

Suppose $C_{1}$ and $C_{2}$ are monomially equivalent so that their generators
$G_{1}$ and $G_{2}$ are chosen such that $G_{2}=G_{1}\Pi$. Thus
$G_{2}e_{m}=G_{1}\Pi e_{m}$ for each $m=1,\ldots,(q-1)^{3}$. In particular, we
know that

$G_{2}e_{m}=\begin{bmatrix}1\\\ x\\\ y\\\ z\\\
x^{s}y^{t}z\end{bmatrix}=G_{1}\Pi e_{m}$

for some $(x,y,z)\in\mathbb{F}_{q}^{*}$. It follows that
$x^{s}y^{t}z=x^{s^{\prime}}y^{t^{\prime}}z$; likewise for all $(x,y,z)$. Of
course, by choosing $(x,1,1)$ we see that $x^{s}=x^{s^{\prime}}$ for all $x$.
Similarly, by choosing $(1,y,1)$ we see that $y^{t}=y^{t^{\prime}}$ for all
$y$. It follows that $s=s^{\prime}$ and $t=t^{\prime}$. ∎

######  Proposition 4.2.4.

Let $P_{1}$ and $P_{2}$ be five point polytopes of width 1. Then $C_{P_{1}}$
and $C_{P_{2}}$ are monomially equivalent only if $P_{1}$ and $P_{2}$ have the
same signature.

Note well that having the same signature is not sufficient for monomial
equivalence.

###### Proof.

This proposition amounts to stating that we obtain distinct codes when
choosing polytopes of different signatures at the outset. We conclude this by
first considering the minimum distance information that we have about all of
these cases. Firstly, there is no risk of equivalence between polytopes of
signatures $(2,1)$ and $(2,2)$ since for a fixed $q$ the furnished codes have
different minimum distances given by the formulas.

If $q\geq 8$, then there can be no equivalence between $(2,1)$ and $(3,1)$
since $2(q-1)^{2}>(q-1)(1+q+2\sqrt{q})$ in that case. To fully justify that
codes from these polytopes will never be equivalent, a manual verification
that minimum distances (or another invariant) differ for $q<8$ will suffice.
Comparing $(2,1)$ and $(3,2)$, we require matching shape geometric parameters
for both polytopes so that $x^{s}y^{t}z=x^{s^{\prime}}y^{t^{\prime}}z$ in
general. However, since $x^{-1}\neq y\in\mathbb{F}_{q}^{*}$ in general, we can
have no equivalent codes. Comparing $(2,2)$ and $(3,1)$, if $q\geq 11$, then
$(q-1)(2q-3)>(q-1)(q+q+2\sqrt{q})$, so we can have no equivalent codes. If
$q<11$, a manual check shows that the codes are distinct. Comparing $(2,2)$
and $(3,2)$, since $x^{s}y^{t}z\neq xy$ in general, we can have no equivalent
codes. Finally, comparing $(3,1)$ and $(3,2)$, since $x^{s}y^{t}z\neq
x^{-1}y^{-1}$ in general, we can have no equivalent codes. ∎

In the cases where signature $(3,2)$ occurs in comparison with codes of
another signature, note that our minimum distance intervals are not tight
enough to conclude monomial inequivalence. However, it is easy to check that
the generator matrices are distinct because $P^{(3,2)}(s,t)$, $P^{(3,1)}$, and
$P^{(2,2)}$ all share four lattice points and $P^{(3,2)}(s,t)$ and
$P^{(2,1)}(s,t)$ share four lattice points.

The patterns one will observe in the minimum distances of codes furnished by
five point polytopes of width 2 are somewhat less tractable. It would be
interesting to uncover what common and salient features there are of those
polynomials’ vanishing points in general. We end our classification of 3-fold
toric codes at the injunction demarcated by this nontrivial question.

## Acknowledgements

This research was completed at the REU Site: Mathematical Analysis and
Applications at the University of Michigan-Dearborn. We would like to thank
the National Science Foundation (DMS-1950102); the National Security Agency
(H98230-19); the College of Arts, Sciences, and Letters; and the Department of
Mathematics and Statistics for their support.

## References