# An online readout CMOS detector to measure ion spectra produced by high-
repetition-rate high-power lasers

K. Burdonov  LULI - CNRS, CEA, Sorbonne Université, Ecole Polytechnique,
Institut Polytechnique de Paris - F-91128 Palaiseau cedex, France  Sorbonne
Université, Observatoire de Paris, Université PSL, CNRS, LERMA, F-75005,
Paris, France  IAP, Russian Academy of Sciences, 603155, Nizhny Novgorod,
Russia  JIHT, Russian Academy of Sciences, 125412, Moscow, Russia  R. Lelievre
LULI - CNRS, CEA, Sorbonne Université, Ecole Polytechnique, Institut
Polytechnique de Paris - F-91128 Palaiseau cedex, France  P. Forestier-
Colleoni  Université Paris-Saclay, CEA, CNRS, LIDYL, 91191 Gif-sur-Yvette,
France  T. Ceccotti  Université Paris-Saclay, CEA, CNRS, LIDYL, 91191 Gif-sur-
Yvette, France  M. Cuciuc  ”Horia Hulubei” National Institute for Physics and
Nuclear Engineering, 30 Reactorului Street, RO-077125, Bucharest-Magurele,
Romania  L. Lancia  LULI - CNRS, CEA, Sorbonne Université, Ecole
Polytechnique, Institut Polytechnique de Paris - F-91128 Palaiseau cedex,
France  W. Yao  LULI - CNRS, CEA, Sorbonne Université, Ecole Polytechnique,
Institut Polytechnique de Paris - F-91128 Palaiseau cedex, France  Sorbonne
Université, Observatoire de Paris, Université PSL, CNRS, LERMA, F-75005,
Paris, France  J. Fuchs  LULI - CNRS, CEA, Sorbonne Université, Ecole
Polytechnique, Institut Polytechnique de Paris - F-91128 Palaiseau cedex,
France

###### Abstract

We present the design and the absolute calibration of charged particle high-
repetition-rate online readout CMOS system tailored for high-power laser
experiments. This system equips a Thomson parabola (TP) spectrometer, used at
the Apollon petawatt scale laser facility to measure the spectra of protons
produced by high-intensity laser-target interactions. The RadEye1 CMOS
matrixes array detectors are paired with a custom triggering system for image
grabbing. This allows us to register the proton and ion signals remotely at a
high-repetition-rate. The latter is presently of one shot/min, but the frame
grabbing enables the system to be compatible with modern high-power lasers
running e.g. at 10 Hz. We detail here the implementation, in the harsh
electromagnetic environment of such interactions, of the system, and its
absolute calibration of the RadEye CMOS matrices, which was performed for
proton energies from 4 MeV to 20 MeV.

##  I Introduction

The constant progress of laser technologies in the field of high-peak power
lasers has opened the way to deliver optical pulses to a target chamber for
laser-plasma interaction investigations at unprecedentedly high repetition
rate [1]. For example, the BELLA laser runs at the power of 1 PW and at 1 Hz
repetition rate [2]. At a higher power level of 10 PW, Apollon and ELI-NP will
soon run at one shot per minute [3, 4]. Such increase in the repetition rate
of the lasers needs to be accompanied by a similar capability of detectors
used to measure the secondary ionizing radiations produced in laser-plasma
interactions. Indeed, up to now, in high-energy and high-power laser
experiments, passive detectors, such as Image Plate (IP) [5, 6], CR-39 [7] or
Radiochromic film (RCF) [8], are still mostly used to detect the spectra of
secondary ion beams. These detectors are not compatible with high repetition
rates as they need to be replaced after each shot, which means breaking the
vacuum of the target chamber, which dramatically increases the time required
between each shot.

Efforts have been undertaken in the community to develop in-line detection
systems. These, for example, read, using a CCD/CMOS camera equipped with an
objective, from outside of the vacuum chamber the fluorescence of a
scintillating screen or a micro-channel plate (MCP). Such systems are
routinely used for the detection of hard photons[9], or of relativistic
electrons at conventional and plasma-based electron accelerators [10, 11] and
accelerated positively charged ions [12, 13]. However, the need to transfer
the image from the detector surface to the reading sensor outside is
inevitably accompanied by loss of emitted photon signal, decreasing the
sensitivity of such systems. Thus now arises the need for the development of
fast reusable detectors able to directly detect the incoming radiation or
particles, in order to allow optimizing in real-time their production.

In this paper we present the design and calibration of a new online readout
CMOS system tailored for high-power laser experiments. It equips a Thomson
parabola (TP) spectrometer [14] geared for recording the spectra of protons
and positively charged ions originating from the interaction of multi-PW laser
pulses with solid targets at the Apollon facility [15]. In the sub-PW regime
that was presently commissioned, these ions are produced by target normal
sheath acceleration (TNSA) [16]. The composite detector installed on this
spectrometer uses an array of highly sensitive unprotected RadEye1 CMOS active
sensors[17, 18], Thus, no need of image transfer is required, providing the
maximum sensitivity of the detector. To provide synchronization between the
RadEye’s frame-grabber and the facility trigger, we developed a custom
triggering device. The absolute calibration of the RadEye CMOS response to
protons with energies in the range from 4 MeV to 20 MeV was performed using an
absolutely calibrated [19, 20] image plate grid, placed on top of the TP
spectrometer detector.

The design of the spectrometer and of the CMOS detection system installed on
it will be first described in Section II, followed by the description of the
triggering system in Section III. The results of the spatial proton energy
calibration of the detector and the absolute calibration response of the
RadEye1 CMOS to protons will be presented in Section IV. Finally, we will draw
our conclusions in Section V.

##  II Thomson parabola design

The Thomson parabola spectrometer ANNA (ApolloN thomsoN parabolA) was designed
at CEA for the Apollon short focal area (SFA) vacuum chamber [15] to be used
in laser-plasma interaction experiments for the registration of the energy
spectra of positively charged particles. A photo of the TP inside the chamber
during an experimental campaign is shown in Figure 1(a).

Figure 1: (a) Photography of the Thomson parabola (TP) spectrometer inside the
SFA vacuum chamber of the Apollon facility (within the yellow dashed circle)
(b) Schematic top view of the TP.

Figure 1(b) presents the schematic top-view of the spectrometer. The diameter
of the entrance pinhole may span between 50 µm and 2 mm. Two round magnets
with 70 mm diameter produce 1.05 T magnetic field and two copper slabs with
voltage difference up to 10 kV ($\pm 2$ kV were used for this particular
campaign), serve to disperse in space the particles depending on their energy
and mass-to-charge ratio[14] (see also Figure 2(a)). The spectrometer design
allows to register proton spectra in the range from 1 MeV to 100 MeV.

The TP detector we developed is composed of two highly sensitive RadEye CMOS
matrices that were mounted head to tail. A photo of the TP rear-end, showing
the back of the detectors, is shown in Figure 2(b), demonstrating the mounting
and cabling of the CMOSes inside the TP assembly. The CMOS surface was
impacted by the particles directly. To avoid electromagnetic perturbations
induced onto the readout system by the harsh electromagnetic environment
linked with the laser-target interactions (see Ref. [15] for detailed
measurements), the detectors were enclosed in a Faraday cage formed by the TP
body and the cables were shielded (see Figure 2(b)). This proved very
effective as the detectors were running unperturbed.

The CMOS detectors were triggered 0.4 s before the shot and readout right
after. After each shot the signal was acquired remotely, allowing to use the
spectrometer continuously over a series of shots without the need to access
the detector, oppositely to, for example, commonly used imaging plates [21],
which requires the need to break the vacuum of the target chamber.

Figure 2: (a) A 3D image of the Thomson parabola (TP) spectrometer detector;
(b) photograph of the mounting and cabling the CMOSes on the back of the TP
assembly.

A typical proton spectrum registered by the Thomson parabola spectrometer is
shown in Figure 3. A laser pulse with 9.2 J energy was focused onto the
surface of a 3 $\mu$m aluminum target, providing a bunch of accelerated
protons with a 19.4 MeV cutoff energy[15]. The whole data image consists of
two separated images collected by the two RadEye CMOS matrixes. The so-called
zero-order is the point-projection of the x-rays emitted by the target onto
the detector. It serves as a reference from which the deflection of the
charged particles can be accounted for. A track of deflected protons is seen
on the right-hand side of the image. The proton track is parabolically curved,
as is expected [14]. The low energy part of the spectrum has been
intentionally cut by a thick Al filter since it saturated the detector, as the
number of particles increases exponentially toward the low energies[22]. A
high-energy cut-off and a broad spectrum are clearly seen, which are both
typical signatures of TNSA acceleration [22].

Figure 3: Typical spectral image obtained in a single shot with our detection
system on the back of the Thomson parabola.

##  III External triggering and synchronisation with facility

The RadEye1 is a large area 512$\times$1024 pixel CMOS sensor that can be
triggered by asserting its START signal. The associated Shad-o-Box readout
electronics used in this experiment controls all the signals for the sensor
based on an internal asynchronous frame timing. As this frame timing cannot be
synchronized to an external source, a custom triggering system was developed
that interfaced the incoming trigger signal from the laser to the control
signals for the Shad-o-Box, ensuring that the sensor was not being cleared
when the laser shot arrived and that the correct frame was read out.

The triggering system was built around a PIC16F18446 microcontroller that was
programmed to accept triggers from the laser 400 ms in advance of the shot. A
description of its interface to the rest of the experimental setup is
available in Fig. 4. Frames were acquired remotely after each shot, allowing
us to use the spectrometer continuously over a series of shots without the
need to access the detector by breaking the vacuum of the target chamber.

Figure 4: Integration of the trigger system in the data acquisition setup. The
T line of the Shad-o-Box is asserted in order to inhibit both readout of the
current frame and sensor reset, keeping the sensor in integration mode. The
triggering system asserts this signal based on the laser trigger and the Shad-
o-Box internal asynchronous frame timing, available on the M line. The USB
connection to the PC signals the DAQ software that it should stop acquiring
new frames to prevent overwriting of physics data.

##  IV Absolute calibration of the system

The absolute calibration of the system in energy and sensitivity was performed
during the commissioning experiment session at the Apollon facility [15]. The
spectrometer was placed along the axis perpendicular to the target rear
surface with its entrance pinhole having diameter D = 200 $\mu$m at the
distance of L = 320 mm from the target to the pinhole, giving the spectrometer
solid angle $\Delta\Omega_{TP}=\pi D^{2}/(4L^{2})$ = 0.31 $\mu$sr.

###  IV.1 Energy calibration of the disperson onto the detector

The first part of the spectrometer calibration procedure consisted in
determining whether the real position of a proton of a certain energy on the
detector corresponds to the expected position calculated on the basis of the
calculated proton deflection in the magnetic field:

$E[MeV]=\frac{(q\cdot B\cdot r(L+r/2))^{2}}{200\cdot d[cm]^{2}\cdot m_{p}\cdot
q},$ (1)

where $m_{p}$ = $1.67\times 10^{-27}$ kg is the proton mass, $q$ = $1.6\times
10^{-19}$ C is the proton charge, $B$ = 1.045 T is the nominal magnetic field,
$L$ = 0.252 m is the nominal distance between magnet and detector, $r$ = 0.07
m is the nominal magnetic field length, and $d$ is the distance between zero-
order point on the detector and the point on the detector corresponding to a
certain proton energy.

Aluminum filters with various known thicknesses were placed on top of the
detector in order to block protons with the energy not high enough to go
through the filters. The thickness of aluminum required to stop protons of a
certain energy was calculated using the SRIM (The Stopping and Range of Ions
in Matter) software [23] with a step of 0.5 MeV for proton energies in the
range from 4 MeV to 7 MeV and with a step of 1 MeV in the range from 7 MeV to
18 MeV. Thus, an aluminum plate thickness of 0.2 mm can block proton with
energy below an energy in the range of 5 MeV to 5.5 MeV; and similarly 0.3 mm,
0.4 mm, 1 mm and 1.2 mm thick plates can block protons with energies in the
ranges of 6.5-7 MeV, 7-8 MeV, 13-14 MeV, and 14-15 MeV correspondingly.
Distance from the zero-order point to the proton signal cut-off position can
be obtained from the experimental data for each of mentioned above aluminum
filters. As can be seen in the Figure 5, the red calibration curve
representing the formula 1 fits all the experimentally measured points (black
dots).

Figure 5: Proton energy dispersion calibration of the Thomson parabola.

###  IV.2 Absolute sensitivity calibration of the RadEye1 detector

To have the absolute calibration of the proton energy deposition on the CMOS,
for one shot we placed a grid made of an image plate (IP) with a known
response to the proton impact [24, 20] in front the TP spectrometer detector,
as shown in Figure 6).

Figure 6: Photo of the IP grid that was positioned on top of the RadEye1 CMOS
detector of the TP spectrometer for the absolute calibration measurement of
the sensor sensitivity.

The IP was of the type BAS-TR 2025 (Fuji Photo Film Co. Ltd) with no
protective layer, covering the detection phosphor, and it was scanned using
the Fujifilm BAS-1800 II scanner.

The linear photostimulated luminescence (PSL), which is proportional to the
number of incident protons, can be converted from the raw signal of the
scanner on a logarithmic scale, namely Quantum level (QL), and which is stored
in every single pixel, using the following formula:

$IP_{PSL}=\Big{(}\frac{R}{100}\Big{)}^{2}\times\frac{4000}{S}\times
10^{L\times\big{(}\frac{IP_{QL}}{2^{G}-1}-\frac{1}{2}\big{)}},$ (2)

where $R$ = 50 is the scanner resolution (50 $\mu$m $\times$ 50 $\mu$m square
pixel size); $L$ = 5 is the dynamic range (or Latitude); $G$ = 16 is the
16-bit depth; $S$ = 4000 is the sensitivity [25].

The equation that allows to retrieve the proton spectra, calculated using the
IP scan, is:

$\frac{dN(p^{+})}{dE\cdot
d\Omega}=\int_{-y}^{y}{IP_{PSL}\,dy}\cdot\frac{1}{dE_{IP}\cdot F_{IP}\cdot
CAL_{IP}\cdot\Delta\Omega_{TP}},$ (3)

where $IP_{PSL}$ is the IP scanner data in PSL units (the integral along the
$y$-axis corresponds to integrating all the signal perpendicularly to the
direction of the energy dispersion); $dE$ is the proton energy dispersion;
$F_{IP}=73\pm 3\%$ (40 min after shot) is the IP fading with time from the
exposure [24]; $CAL_{IP}$ = $0.334\cdot E[MeV]^{-0.914}$ is the IP sensitivity
to protons in the range of energies from 2.11 MeV to 20 MeV taken from [24].

Finally, the equation to calculate the number of protons per MeV per
steradians, basing on the data, registered by the RadEye1 CMOS array, is as
follows:

$\frac{dN(p^{+})}{dE\cdot
d\Omega}=\int_{-y}^{y}{RE\,dy}\cdot\frac{1}{dE_{RE}\cdot\Delta\Omega_{TP}\cdot
CAL_{RE}},$ (4)

where $RE$ is the raw data, registered by the CMOSes as TIFF format with 16
bit dynamic range; $dE$ is the proton energy dispersion; and

$CAL_{RE}=(10^{3}\pm 50)\cdot E[MeV]^{-1.5\pm 0.5}$ (5)

is the desired function which allows to fit the absolute proton spectra,
derived from the RadEye, to the one obtained with the help of the IP. This
last calibration factor is obtained by a best fitting of the spectrum obtained
from the RadEye to the one derived from the IP, as is demonstrated in Figure
7.

It should be mentioned that our fitting function differs from the one which
was obtained with with the use of a CR39 detector for the proton energies in
the range from 1 MeV to 3 MeV [17] and which can be written as $(1.09\pm
0.12)\cdot$ E [keV]. Our additional power function coefficient of $-1.5$ can
be neglected for the proton energies around 1 MeV, however for higher energies
our more advanced fitting function has to be used.

Figure 7: Calibration of the RadEye1 detector sensitivity by adjusting the
spectrum that can be derived from it to the one derived from the slotted IP,
and which absolute calibration is known.

##  V Summary

The design of a charged particle high-repetition-rate online readout CMOS
system, tailored for high-power laser experiments, is presented. The system is
installed on the back of an ion spectrometer and consists in RadEye CMOS
matrices that allow online recording of the measured proton spectra, even in
the harsh electromagnetic environment induced by the high-intensity laser
interactions. The calibration of the RadEye CMOS matrices was performed for
proton energies from 4 MeV to 20 MeV, giving the fitting function $CAL_{RE}$ =
$(10^{3}\pm 50)\cdot$E [MeV] -1.5±0.5, allowing to restore the absolute energy
spectrum of the proton signal based on the image recorded by the detector in
TIFF format.

###### Acknowledgements.

This work was supported by funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (Grant
Agreement No. 787539, project GENESIS), and by Grant ANR-17-CE30-
0026-Pinnacle from Agence Nationale de la Recherche.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

## References

  * Danson  _et al._ [2019] C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, and et al.,  _Petawatt and exawatt class lasers worldwide_ , High Power Laser Science and Engineering 7, e54 (2019).
  * Nakamura  _et al._ [2017] K. Nakamura, H.-S. Mao, A. J. Gonsalves, H. Vincenti, D. E. Mittelberger, J. Daniels, A. Magana, C. Toth, and W. P. Leemans,  _Diagnostics, Control and Performance Parameters for the BELLA High Repetition Rate Petawatt Class Laser_ , IEEE Journal of Quantum Electronics 53, 1–21 (2017).
  * Papadopoulos  _et al._ [2019] D. N. Papadopoulos, J. P. Zou, C. L. Blanc, L. Ranc, F. Druon, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, M. Chabanis, C. Bonnin, J. B. Accary, B. L. Garrec, F. Mathieu, and P. Audebert, in  _Conference on Lasers and Electro-Optics_ (Optical Society of America, 2019) p. STu3E.4.
  * Lureau  _et al._ [2020] F. Lureau, G. Matras, O. Chalus, C. Derycke, T. Morbieu, C. Radier, O. Casagrande, S. Laux, S. Ricaud, G. Rey, and et al.,  _High-energy hybrid femtosecond laser system demonstrating 2 × 10 PW capability_ , High Power Laser Science and Engineering 8, e43 (2020).
  * Boutoux  _et al._ [2016] G. Boutoux, D. Batani, F. Burgy, J.-E. Ducret, P. Forestier-Colleoni, S. Hulin, N. Rabhi, A. Duval, L. Lecherbourg, C. Reverdin, K. Jakubowska, C. I. Szabo, S. Bastiani-Ceccotti, F. Consoli, A. Curcio, R. De Angelis, F. Ingenito, J. Baggio, and D. Raffestin,  _Validation of modelled imaging plates sensitivity to 1-100 keV x-rays and spatial resolution characterisation for diagnostics for the “PETawatt Aquitaine Laser”_ , Review of Scientific Instruments 87, 043108 (2016), https://doi.org/10.1063/1.4944863 .
  * Strehlow  _et al._ [2019] J. Strehlow, P. Forestier-Colleoni, C. McGuffey, M. Bailly-Grandvaux, T. S. Daykin, E. McCary, J. Peebles, G. Revet, S. Zhang, T. Ditmire, M. Donovan, G. Dyer, J. Fuchs, E. W. Gaul, D. P. Higginson, G. E. Kemp, M. Martinez, H. S. McLean, M. Spinks, H. Sawada, and F. N. Beg,  _The response function of Fujifilm BAS-TR imaging plates to laser-accelerated titanium ions_ , Review of Scientific Instruments 90, 083302 (2019), https://doi.org/10.1063/1.5109783 .
  * Jeong  _et al._ [2017] T. W. Jeong, P. K. Singh, C. Scullion, H. Ahmed, P. Hadjisolomou, C. Jeon, H. Yun, K. F. Kakolee, M. Borghesi, and S. Ter-Avetisyan,  _CR-39 track detector for multi-MeV ion spectroscopy_ , Scientific Reports 7, 2152 (2017).
  * Bolton  _et al._ [2014] P. Bolton, M. Borghesi, C. Brenner, D. Carroll, C. De Martinis, F. Fiorini, A. Flacco, V. Floquet, J. Fuchs, P. Gallegos, D. Giove, J. Green, S. Green, B. Jones, D. Kirby, P. McKenna, D. Neely, F. Nuesslin, R. Prasad, S. Reinhardt, M. Roth, U. Schramm, G. Scott, S. Ter-Avetisyan, M. Tolley, G. Turchetti, and J. Wilkens,  _Instrumentation for diagnostics and control of laser-accelerated proton (ion) beams_ , Physica Medica 30, 255–270 (2014).
  * Liang  _et al._ [2022] E. Liang, K. Q. Zheng, K. Yao, W. Lo, H. Hasson, A. Zhang, M. Burns, W. H. Wong, Y. Zhang, A. Dashko, H. Quevedo, T. Ditmire, and G. Dyer,  _A scintillator attenuation spectrometer for intense gamma-rays_ , Review of Scientific Instruments 93, 063103 (2022).
  * Buck  _et al._ [2010] A. Buck, K. Zeil, A. Popp, K. Schmid, A. Jochmann, S. D. Kraft, B. Hidding, T. Kudyakov, C. M. S. Sears, L. Veisz, S. Karsch, J. Pawelke, R. Sauerbrey, T. Cowan, F. Krausz, and U. Schramm,  _Absolute charge calibration of scintillating screens for relativistic electron detection_ , Review of Scientific Instruments 81, 033301 (2010), https://doi.org/10.1063/1.3310275 .
  * Downer  _et al._ [2018] M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza,  _Diagnostics for plasma-based electron accelerators_ , Rev. Mod. Phys. 90, 035002 (2018).
  * Nedbailo  _et al._ [2023] R. Nedbailo, J. Park, R. Hollinger, S. Wang, D. Mariscal, J. Morrison, H. Song, G. Zeraouli, G. G. Scott, T. Ma, and J. J. Rocca,  _Compact high repetition rate Thomson parabola ion spectrometer_ , Review of Scientific Instruments 94, 023505 (2023), https://doi.org/10.1063/5.0101859 .
  * Dover  _et al._ [2017] N. P. Dover, M. Nishiuchi, H. Sakaki, M. A. Alkhimova, A. Y. Faenov, Y. Fukuda, H. Kiriyama, A. Kon, K. Kondo, K. Nishitani, K. Ogura, T. A. Pikuz, A. S. Pirozhkov, A. Sagisaka, M. Kando, and K. Kondo,  _Scintillator-based transverse proton beam profiler for laser-plasma ion sources_ , Review of Scientific Instruments 88, 073304 (2017).
  * Thomson [1911] S. J. Thomson,  _Rays of positive electricity_ , The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 21, 225–249 (1911).
  * Burdonov  _et al._ [2021] K. Burdonov, A. Fazzini, V. Lelasseux, J. Albrecht, P. Antici, Y. Ayoul, A. Beluze, D. Cavanna, T. Ceccotti, M. Chabanis, A. Chaleil, S. N. Chen, Z. Chen, F. Consoli, M. Cuciuc, X. Davoine, J. P. Delaneau, E. d’Humières, J.-L. Dubois, C. Evrard, E. Filippov, A. Freneaux, P. Forestier-Colleoni, L. Gremillet, V. Horny, L. Lancia, L. Lecherbourg, N. Lebas, A. Leblanc, W. Ma, L. Martin, F. Negoita, J.-L. Paillard, D. Papadopoulos, F. Perez, S. Pikuz, G. Qi, F. Quéré, L. Ranc, P.-A. Söderström, M. Scisciò, S. Sun, S. Vallières, P. Wang, W. Yao, F. Mathieu, P. Audebert, and J. Fuchs,  _Characterization and performance of the Apollon short-focal-area facility following its commissioning at 1 PW level_ , Matter and Radiation at Extremes 6, 064402 (2021), https://doi.org/10.1063/5.0065138 .
  * Wilks  _et al._ [2001] S. C. Wilks, A. B. Langdon, T. E. Cowan, M. Roth, M. Singh, S. Hatchett, M. H. Key, D. Pennington, A. MacKinnon, and R. A. Snavely,  _Energetic proton generation in ultra-intense laser–solid interactions_ , Physics of Plasmas 8, 542–549 (2001), https://doi.org/10.1063/1.1333697 .
  * Reinhardt  _et al._ [2013] S. Reinhardt, W. Draxinger, J. Schreiber, and W. Assmann,  _A pixel detector system for laser-accelerated ion detection_ , Journal of Instrumentation 8 (03), P03008–P03008.
  * Lindner  _et al._ [2018] F. H. Lindner, J. H. Bin, F. Englbrecht, D. Haffa, P. R. Bolton, Y. Gao, J. Hartmann, P. Hilz, C. Kreuzer, T. M. Ostermayr, T. F. Rösch, M. Speicher, K. Parodi, P. G. Thirolf, and J. Schreiber,  _A novel approach to electron data background treatment in an online wide-angle spectrometer for laser-accelerated ion and electron bunches_ , Review of Scientific Instruments 89, 013301 (2018).
  * Bonnet  _et al._ [2013] T. Bonnet, M. Comet, D. Denis-Petit, F. Gobet, F. Hannachi, M. Tarisien, M. Versteegen, and M. M. Aléonard,  _Response functions of imaging plates to photons, electrons and 4He particles_ , Review of Scientific Instruments 84, 103510 (2013), https://doi.org/10.1063/1.4826084 .
  * Martin  _et al._ [2022] P. Martin, H. Ahmed, D. Doria, A. Alejo, R. Clarke, S. Ferguson, J. Fernández-Tobias, R. R. Freeman, J. Fuchs, A. Green, J. S. Green, D. Gwynne, F. Hanton, J. Jarrett, D. Jung, K. F. Kakolee, A. G. Krygier, C. L. S. Lewis, A. McIlvenny, P. McKenna, J. T. Morrison, Z. Najmudin, K. Naughton, G. Nersisyan, P. Norreys, M. Notley, M. Roth, J. A. Ruiz, C. Scullion, M. Zepf, S. Zhai, M. Borghesi, and S. Kar,  _Absolute calibration of Fujifilm BAS-TR image plate response to laser driven protons up to 40 MeV_ , Review of Scientific Instruments 93, 053303 (2022).
  * Lelasseux and Fuchs [2020] V. Lelasseux and J. Fuchs,  _Modelling energy deposition in TR image plate detectors for various ion types_ , Journal of Instrumentation 15 (04), P04002.
  * Daido  _et al._ [2012] H. Daido, M. Nishiuchi, and A. S. Pirozhkov,  _Review of laser-driven ion sources and their applications_ , Reports on Progress in Physics 75, 056401 (2012).
  * Ziegler  _et al._ [2010] J. F. Ziegler, M. Ziegler, and J. Biersack,  _SRIM – The stopping and range of ions in matter (2010)_ , Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 268, 1818–1823 (2010), 19th International Conference on Ion Beam Analysis.
  * Mančić  _et al._ [2008] A. Mančić, J. Fuchs, P. Antici, S. A. Gaillard, and P. Audebert,  _Absolute calibration of photostimulable image plate detectors used as (0.5–20MeV) high-energy proton detectors_ , Review of Scientific Instruments 79, 073301 (2008).
  * Alejo  _et al._ [2014] A. Alejo, S. Kar, H. Ahmed, A. G. Krygier, D. Doria, R. Clarke, J. Fernandez, R. R. Freeman, J. Fuchs, A. Green, J. S. Green, D. Jung, A. Kleinschmidt, C. L. S. Lewis, J. T. Morrison, Z. Najmudin, H. Nakamura, G. Nersisyan, P. Norreys, M. Notley, M. Oliver, M. Roth, J. A. Ruiz, L. Vassura, M. Zepf, and M. Borghesi,  _Characterisation of deuterium spectra from laser driven multi-species sources by employing differentially filtered image plate detectors in Thomson spectrometers_ , Review of Scientific Instruments 85, 093303 (2014), https://doi.org/10.1063/1.4893780 .


# Some Results on the Improved Bound and Construction of Optimal $(r,\delta)$
LRCs ††thanks: *Corresponding Author

Bin Chen1, Weijun Fang2*, Yueqi Chen3, Shu-Tao Xia3, Fang-Wei Fu4, Xiangyu
Chen5 1 Harbin Institute of Technology, Shenzhen 518055, China.
2School of Cyber Science and Technology, Shandong University, Qingdao,
Shandong, 266237, P.R. China.
3Tsinghua Shenzhen International Graduate School, Tsinghua University,
Shenzhen 518055, China.
4Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071,
China.
5HUAWEI Technologies Co., Ltd., Shenzhen, 518055, P.R. China.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>chen-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by
Prakash _et al._ into distributed storage systems (DSSs) due to their benefit
of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes
among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type
bound is called an optimal $(r,\delta)$ LRC. Constructions of optimal
$(r,\delta)$ LRCs with longer code length and determining the maximal code
length have been an important research direction in coding theory in recent
years. In this paper, we conduct further research on the improvement of
maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq
2\delta+2$, our upper bounds largely improve the ones by Cai _et al._ , which
are tight in some special cases. Moreover, we generalize the results of Chen
_et al._ and obtain a complete characterization of optimal $(r=2,\delta)$-LRCs
in the sense of geometrical existence in the finite projective plane
$PG(2,q)$. Within this geometrical characterization, we construct a class of
optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the
construction and upper bounds are better than previous ones.

###### Index Terms:

Locally repairable codes, finite projective plane, sunflower, Johnson bound

##  I Introduction

In the era of big data, the reliability of large-scale data storage becomes
more and more important. Distributed storage system (DSS) with erasure codes
provides a new paradigm to maintain high reliability and flexibility.
Regenerating codes [1] and locally repairable codes (LRCs) [2] are the two
most representative erasure coding schemes in a DSS, which respectively
optimize the repair bandwidth and repair locality. We mainly focus on LRCs in
this paper, which have been implemented and evaluated in two DSSs, Windows
Azure storage [3] and Hadoop Distributed File System [4]. Next, we formally
give the definition of LRCs in [2]. Let $\mathcal{C}$ be a $q$-ary $[n,k,d]$
linear code with dimension $k$ and minimum distance $d$. We say that a linear
code $\mathcal{C}$ is an $r$-local LRC with _locality_ $r$ if for any
$i\in[n]\triangleq\\{1,2,\dots,n\\}$, there exists a subset
$R_{i}\subset[n]\setminus\\{i\\}$ with $|R_{i}|\leq r$ such that the $i$-th
symbol $c_{i}$ can be recovered by $\\{c_{j}\\}_{j\in R_{i}}$, i.e., $c_{i}$
can be represented as a linear combination of $\\{c_{j}\\}_{j\in R_{i}}$. In
[2], the authors also derived the well-known Singleton-type bound on $d$ for
an $r$-local $q$-ary $[n,k,d]$-LRCs,

$d\leq n-k-\left\lceil\frac{k}{r}\right\rceil+2.$ (1)

Note that bound (1) reduces to the classical Singleton bound when $r=k$.
Optimal $r$-local LRCs achieving the bound (1) with equality have been
obtained in recent years [5, 6, 7, 8, 9, 10, 11, 12, 13]. For general $d$ and
$r$, optimal Reed-Solomon-like LRCs with lengths $n\leq q-1$ [5] are obtained
based on the polynomial evaluation. The reference [6] establishes a connection
between the construction of optimal LRCs and properties of the matroid
represented by its generator matrix. A novel explicit construction of LRCs via
maximum rank distance codes is obtained in [7], which provides new optimal
vector and scalar LRCs. Optimal cyclic or constacyclic LRCs are obtained in
[9, 10, 8]. [11] presents an optimal explicit construction of LRCs with
distance $d=5$ and 6 via binary constant weight codes. For some particular
$r=2,3,5,7,11,23$, optimal LRCs with lengths up to $q+2\sqrt{q}$ are
constructed via elliptic curves [12]. In addition to the progress, the study
of maximal code length [14, 15] and optimal $r$-local LRCs with longer code
length [9, 10, 11, 12, 13] has important value in theory and practice,
especially on the design of LRCs with flexible parameters since shorter
optimal LRCs can be obtained from the longer ones.

To tolerate multiple erasures in each local group, LRCs with $(r,\delta)$
locality were introduced by Prakash _et al_ [16]. Other generalizations
include code with hierarchical locality [17], or with availability [18, 19],
etc. Specifically, the $i$-th symbol $c_{i}$ of a $q$-ary $[n,k]$ linear code
${\mathcal{C}}$ is said to have $(r,\delta)$-locality ($\delta\geq 2$) if
there exists a punctured subcode of ${\mathcal{C}}$ with support containing
$i$, whose length is at most $r+\delta-1$, and whose minimum distance is at
least $\delta$, i.e., there exists a subset
$S_{i}\subseteq[n]\triangleq\\{1,2,\ldots,n\\}$ such that $i\in S_{i}$,
$|S_{i}|\leq r+\delta-1$ and $d({\mathcal{C}}|_{S_{i}})\geq\delta$. A
generalized Singleton-type bound on $d$ was also derived as follows.

$d\leq n-k+1-\left(\left\lceil\frac{k}{r}\right\rceil-1\right)(\delta-1).$ (2)

Following the research line of $r$-local LRCs, various generalized
constructions of optimal $(r,\delta)$ LRCs achieving the bound (2) were
obtained in [20, 21, 22, 23]. When $\delta+1\leq d\leq 2\delta$, optimal
$(r,\delta)$ LRCs with unbounded length were given in [24, 22]. For $d\geq
2\delta+1$, Cai _et al_ [20] first obtain upper bounds on the code length and
constructions of optimal $(r,\delta)$ LRCs, which generalize previously ones
by Guruswami _et al_ [14] under weaker conditions. For $d\geq 3\delta+1$ and
$r\geq d-\delta$, the authors [23] have generalized the results of [13] based
on the sparse hypergraphs to obtain optimal $(r,\delta)$ LRCs with super-
linear in the alphabet size. To the best of our knowledge, known results on
the maximal code length of $(r,\delta)$ LRCs is not tight in general, and the
constructions of $(r,\delta)$ LRCs with longer code length only focus on the
range of larger localities that are less attractive in a practical DSS.

In this paper, we instead focus on the maximum code length of $q$-ary optimal
$(r,\delta)$ LRCs with small localities. To simplify the discussion, we assume
that an $(r,\delta)$ LRC has _disjoint local subcodes_ , i.e,
$(r+\delta-1)\mid n$ and each subcode ${\mathcal{C}}|_{S_{i}}$ is an
$[r+\delta-1,r,\delta]$ MDS code. By generalizing the parity-check matrix
approach ([25, 15]) for $r$-local LRCs, we can similarly obtain the _standard-
form_ parity-check matrix for an $(r,\delta)$ LRCs with disjoint local
subcodes. Within this standard-form parity-check matrix, we can obtain an
improved upper bound on the number $\ell$ of disjoint local subcodes via a
novel linear combination among the columns of each local subcode. Thus we
derive an improved upper bound on the code length compared with that of Cai
_et al_. In addition, we can show that our new bounds are exactly achieved by
some known constructions. Based on the complete characterization of optimal
$r$-local LRCs with $d=6$ and $r=2$ [15], we establish a generalized
characterization of optimal $(r=2,\delta)$ LRCs with $d=2\delta+2$. This
geometrical characterization further help us to verify the existence of
optimal $(r=2,\delta)$ LRCs with $n=(\delta+1)(q+1)$ and $d=2\delta+2$ via the
sunflower structure, which forms a generalization of an optimal quaternary
$(r,\delta)$ LRC in [26]. Following the line-point incidence matrix approach
[15], we further obtain three improved upper bounds on the maximal code length
based on the Johnson bound for constant weight codes. To the best of our
knowledge, our bounds are the best with the lowest order of $q$ for optimal
$(r=2,\delta)$ LRCs with $d=2\delta+2$ until now.

The rest of this paper is organized as follows. In Section II, we introduce
some necessary results on the standard-form parity-check matrix and the
Johnson upper bound for binary constant weight codes. In Section III, we
establish a generalized complete characterization for optimal $(r=2,\delta)$
LRCs with $d=2\delta+2$ and present a class of optimal $(r=2,\delta)$ LRCs via
the sunflower structure. Furthermore, we derive some improved upper bounds
based on the incidence matrix and Johnson bound. Section IV concludes this
paper.

##  II Preliminaries

In this section, we introduce the basic parity matrix structure of optimal
$(r,\delta)$ LRCs with disjoint local groups, and the Johnson bound for binary
constant weight codes that are utilized to obtain our improved upper bounds on
the maximal code length. Let $\mathbb{F}_{q}$ be a finite field with $q$
elements, $\mathbb{F}_{q}^{*}$ denotes the set of its nonzero elements.
Throughout this paper, we assume that $(r+\delta-1)\mid n$ and the LRCs have
_disjoint local repair groups_. By the definition of $(r,\delta)$ locality
[16], we know that an $[n,k,d]$-linear code $C$ has $(r,\delta)$ locality if
there exist $\ell\triangleq\frac{n}{r+\delta-1}$ punctured
$[r+\delta-1,r,\delta]$ MDS subcodes, i.e, there exist $\ell$ submatrices
$\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{\ell}\in\mathbb{F}_{q}^{(\delta-1)\times(r+\delta-1)}$
which are called _locality matrices_ , such that they respectively form a
parity-check matrix of the $[r+\delta-1,r,\delta]$ MDS subcode. Thus an
optimal $(r,\delta)$ LRC with disjoint local repair groups has an equivalent
parity-check matrix $\mathbf{H}$ as the following form:

$\mathbf{H}\\!=\\!\\!\left(\begin{array}[]{cc|cc|c|cc}\mathbf{I}_{\delta-1}&\mathbf{Q}_{1}&&&&&\\\
&&\mathbf{I}_{\delta-1}&\mathbf{Q}_{2}&&&\\\ &&&&\ddots&&\\\
&&&&&\mathbf{I}_{\delta-1}&\mathbf{Q}_{\ell}\\\
\hline\cr\mathbf{0}_{u\times(\delta-1)}&\mathbf{V}_{1}&\mathbf{0}_{u\times(\delta-1)}&\mathbf{V}_{2}&\dots&\mathbf{0}_{u\times(\delta-1)}&\mathbf{V}_{\ell}\end{array}\right),\\!\\!$
(3)

where the upper part of $\mathbf{H}$ contains $\ell$ locality subblocks and
the lower part of $\mathbf{H}$ contains $u\triangleq n-k-\ell(\delta-1)$ rows
which ensure that $rank(H)=n-k$. Note that each submatrix
$\mathbf{H}_{i}=(\mathbf{I}_{\delta-1}\mathbf{Q}_{i})$ exactly forms a
standard generator matrix of a $[r+\delta-1,\delta-1,r+1]$ MDS code, where
$\mathbf{I}_{\delta-1}$ is an identity matrix with unit column vector
$\mathbf{e}_{s}\in\mathbb{F}_{q}^{(\delta-1)}$, $s\in[\delta-1]$, and each
element of $\mathbf{Q}_{i},i\in[\ell]$ is nonzero, i.e.,
$\mathbf{Q}_{i}\in{\mathbb{F}_{q}^{*}}^{{(\delta-1)}\times r}$. The bold-type
letters $\mathbf{V}_{i}$, $i\in[\ell]$, are submatrices in
$\mathbb{F}_{q}^{{u}\times(\delta-1)}$, while $\mathbf{0}_{u\times(\delta-1)}$
represents the all-zero matrix in $\mathbb{F}_{q}^{{u}\times(\delta-1)}$.

Next, we recall Lemma II.2 in [23], which relates the number of rows in the
low part of $H$ with other code parameters.

###### Lemma 1

([23]) Let $n,k,d,r,\delta$ be positive integers with $(r+\delta-1)\mid n$. If
the generalized Singleton-type bound (2) is achieved, then

$n-k=\ell(\delta-1)+d-1-\left(\left\lfloor\frac{d-\delta}{r+\delta-1}\right\rfloor+1\right)(\delta-1),$
(4)
$i.e,\,u=d-1-\left(\left\lfloor\frac{d-\delta}{r+\delta-1}\right\rfloor+1\right)(\delta-1).$
(5)

We denote by $(n,M,d;w)$ a binary constant-weight code of length $n$, size
$M$, minimum distance $d$, and each codeword has a fixed Hamming weight $w$.
The well-known Johnson bound can be summarized as follows.

###### Lemma 2

([27, Johnson Bound]) Let $C$ be a binary $(n,M,d=2\delta;w)$-constant weight
code, then we have

$\displaystyle A(n,2\delta,w)$ $\displaystyle\leq$
$\displaystyle\frac{\binom{n}{w-\delta+1}}{\binom{w}{w-\delta+1}}.$ (6)
$\displaystyle M(w^{2}-wn+\delta n)$ $\displaystyle\leq$ $\displaystyle\delta
n.$ (7)

##  III Main Results

In this section, we always assume that an $(r,\delta)$ LRC has disjoint local
repair groups, and denote

$\displaystyle n_{\max}(q,d,r,\delta)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\max\\{n\mid\text{there exists a $q$-ary}$
$\displaystyle\qquad\qquad\text{optimal $(n,k,d;r,\delta)$ LRC}\\}.$

###  III-A General Improved Upper bound

###### Lemma 3

Let $H$ be an $(\ell(\delta-1)+u)\times$ $\ell(r+\delta-1)$ parity-check
matrix of an $(r,\delta)$ LRC with disjoint local repair groups. If $d\geq
2\delta+1$, then

$\ell\leq\left\lfloor(q^{u}-1)/(q-1)\cdot\left(\begin{array}[]{c}r+\delta-1\\\
\delta\end{array}\right)\right\rfloor$ (8)

###### Proof:

Since $d\geq 2\delta+1$, we know that each column of $\mathbf{V}_{i}$,
$i\in[\ell]$, is a nonzero vector in $\mathbb{F}_{q}^{u}$. For each local
group, we have $\binom{r+\delta-1}{\delta}$ choices of selecting $\delta$
columns. Note that each $\mathbf{H}_{i}$ is an $[r+\delta-1,r,\delta]$ MDS
code, any $\delta$ columns of $\mathbf{H}_{i}$ must be linearly dependent,
i.e., there exists some nonzero linear combination to make these $\delta$
columns contribute to the zero vector. Accordingly, the nonzero linear
combination also produces a nonzero vector in $\mathbb{F}_{q}^{u}$ at the
lower part. In total, we can produce $\ell\cdot\binom{r+\delta-1}{\delta}$
distinct nonzero vectors in $\mathbb{F}_{q}^{u}$ because $d\geq 2\delta+1$.
Then we have
$\ell\leq\left\lfloor(q^{u}-1)/(q-1)\cdot\left(\begin{array}[]{c}r+\delta-1\\\
\delta\end{array}\right)\right\rfloor$. ∎

By Lemma3, we can directly obtain the following improved upper bounds on the
code length.

###### Theorem 1

Let ${\mathcal{C}}$ be a Singleton-optimal $(n,k,d;r,\delta)$ LRC with
disjoint local repair groups.

  1. (i)

If $d=2\delta+1$, then

$n\leq(r+\delta-1)\left\lfloor\frac{q^{u}-1}{(q-1)\cdot\left(\begin{array}[]{c}r+\delta-1\\\
\delta\end{array}\right)}\right\rfloor$ (9)

where
$u=\delta+1-\left(\left\lfloor\frac{\delta+1}{r+\delta-1}\right\rfloor+1\right)(\delta-1)$

  2. (ii)

If $d=2\delta+2$, then

$n\leq(r+\delta-1)\left\lfloor\frac{q^{u}-1}{(q-1)\cdot\left(\begin{array}[]{c}r+\delta-1\\\
\delta\end{array}\right)}\right\rfloor,$ (10)

where
$u=\delta+2-\left(\left\lfloor\frac{\delta+2}{r+\delta-1}\right\rfloor+1\right)(\delta-1)$

  3. (iii)

If $d=3\delta$, then

$n\leq(r+\delta-1)\left\lfloor\frac{q^{u}-1}{(q-1)\cdot\left(\begin{array}[]{c}r+\delta-1\\\
\delta\end{array}\right)}\right\rfloor,$ (11)

where
$u=2\delta-\left(\left\lfloor\frac{2\delta}{r+\delta-1}\right\rfloor+1\right)(\delta-1)$

###### Remark 1

Note that

$2\delta+1\leq d\leq 3\delta\Leftrightarrow
2\leq\frac{d-1}{\delta}<3\Leftrightarrow
t\triangleq\lfloor\frac{d-1}{\delta}\rfloor=2,$

we can directly write out the upper bound of [20, Theorem III.3 ] for
$2\delta+1\leq d\leq 3\delta$:

$n\leq(r+\delta-1)\left\lfloor\frac{q^{u}}{r(q-1)}\right\rfloor.$ (12)

Clearly, our bounds in Theorem 1 are always tighter than the bound (12).
Moreover, our upper bounds (9)-(11) in Theorem 1 can be exactly achieved as
shown below.

###### Corollary 1

Let ${\mathcal{C}}$ be a Singleton-optimal $(n,k,d;r,\delta)$ LRC with
disjoint local repair groups.

  1. (i)

If $d=2\delta+1$ and $r=2$, then

$n\leq q+1.$ (13)

Moreover, $n_{\max}(q,2\delta+1,2,\delta)=q+1$.

  2. (ii)

If $d=2\delta+2$ and $r=2$, then

$n\leq(\delta+1)\cdot\left\lfloor\frac{q^{2}+q+1}{\delta+1}\right\rfloor.$
(14)

Moreover, $n_{\max}(4,8,2,3)=20$.

###### Proof:

  1. (i)

The upper bound (13) can be easily derived by (9). And the constructions of
optimal cyclic and constacyclic $(r,\delta)$ LRCs [9, 10] implies
$n_{\max}(q,2\delta+1,2,\delta)\geq q+1$. Therefore,
$n_{\max}(q,2\delta+1,2,\delta)=q+1$.

  2. (ii)

We can directly obtain the upper bound by (11). On the other hand, the
existence of optimal quaternary $(r=3,\delta=3)$ LRC [26, (36)] implies that
$n_{\max}(4,8,2,3)\geq 20$. Therefore, $n_{\max}(4,8,2,3)=20$.

We finish the proof. ∎

###  III-B Complete Characterization of Optimal
$(n,k,d=2\delta+2;r=2,\delta)$-LRCs

In the following parts, we suppose $r=2$, $(\delta+1)\mid n$ and
$\ell\triangleq n/(\delta+1)$. Let $C$ be an optimal
$(n,k,d=2\delta+2;r=2,\delta)$-LRC with disjoint local repair groups, then
$u\triangleq n-k-\ell(\delta-1)=3$ by Lemma 1. By (3), we may assume that $C$
has a parity-check matrix $H$ as follows:

$\left(\begin{array}[]{ccc|ccc|c|ccc}\mathbf{I}_{\delta-1}&\mathbf{p}_{1}&\mathbf{q}_{1}&0&0&0&\ldots&0&0&0\\\
0&0&0&\mathbf{I}_{\delta-1}&\mathbf{p}_{2}&\mathbf{q}_{2}&\ldots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&0&0&0&\ldots&\mathbf{I}_{\delta-1}&\mathbf{p}_{\ell}&\mathbf{q}_{\ell}\\\
\hline\cr\textbf{0}&\textbf{u}_{1}&\textbf{v}_{1}&\textbf{0}&\textbf{u}_{2}&\textbf{v}_{2}&\ldots&\textbf{0}&\textbf{u}_{\ell}&\textbf{v}_{\ell}\\\
\end{array}\right),$ (15)

where $\textbf{u}_{i},\textbf{v}_{i}\in\mathbb{F}_{q}^{3},$ $i\in[\ell]$.
Denote $\mathcal{V}_{i}=\textnormal{Span}\\{\textbf{u}_{i},\textbf{v}_{i}\\}$
to be a subspace of $\mathbb{F}_{q}^{3}$ spanned by $\textbf{u}_{i}$ and
$\textbf{v}_{i}$ over $\mathbb{F}_{q}$. Since each submatrix
$(\mathbf{I}_{\delta-1},\mathbf{p_{i}},\mathbf{q_{i}})$ forms a standard
generator matrix of an MDS code, we know that
$\mathbf{p}_{i}\triangleq(p_{i,1},p_{i,2},\cdots,p_{i,(\delta-1)})^{T},\mathbf{q}_{i}\triangleq(q_{i,1},q_{i,2},\cdots,q_{i,(\delta-1)})^{T}\in{\mathbb{F}_{q}^{*}}^{(\delta-1)},i\in[\ell]$
by the property of MDS codes [27]. Thus for any fixed $m\in[\delta-1]$, there
must exist $(a_{m},b_{m})\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*}$ such
that

$a_{m}p_{i,m}+b_{m}q_{i,m}=0.$ (16)

We let
$\mathcal{F}_{i}\subseteq\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*},i\in[\ell]$
to be the set containing all of these tuples $(a_{m},b_{m})$ for the $i$-th
repair group. Then we can obtain the following lemma to prove our complete
characterization.

###### Lemma 4

Suppose $q\geq\delta+1$ and $(\delta+1)\mid n$. Let $C$ be a $q$-ary optimal
$(n,k,d=2\delta+2;r=2,\delta)$-LRC with a parity-check matrix $H$ as (15).
Then

  * (i)

$\dim(\mathcal{V}_{i})=2;$

  * (ii)

for any $j\neq i\in[\ell]$, we have
$\textbf{u}_{i},\textbf{v}_{i},a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}\notin\mathcal{V}_{j}$,
where $(a_{m},b_{m})\in\mathcal{F}_{i}$, $m\in[\delta-1]$.

###### Proof:

  1. (i)

If $\dim(\mathcal{V}_{i})\leq 1$ for some $i$, then the $\delta+1$ columns in
the $i$-th repair group of $H$ are linearly dependent, which leads to
$d\leq\delta+1$. This contradicts with $d=2\delta+2$.

  2. (ii)

Suppose for some $j\neq i\in[\ell]$, one of $\textbf{u}_{i},\textbf{v}_{i}$
belongs to $\mathcal{V}_{j}$. Without of loss generality, we assume $i=1$ and
$j=2$. If $\textbf{u}_{1}\in\mathcal{V}_{2}$, then
$\textbf{u}_{1}=a\textbf{u}_{2}+b\textbf{v}_{2}$ for some
$a,b\in\mathbb{F}_{q}$. Then it is easy to verify that the first $\delta$
columns in the first repair group and the $\delta+1$ columns in the second
repair group of $H$ are linearly dependent, which leads to $d\leq 2\delta+1$
and contradicts with the fact that $d=2\delta+2$. Similarly, we can prove
$\textbf{v}_{1}\notin\mathcal{V}_{2}$.

Finally, we assume that for some $m_{0}\in[\delta-1]$, there exists
$(a_{m_{0}},b_{m_{0}})\in\mathcal{F}_{i}$ such that
$a_{m_{0}}\textbf{u}_{1}+b_{m_{0}}\textbf{v}_{1}\in\mathcal{V}_{2}$. Then we
can show that all columns but the $m_{0}$-th one in the first repair group,
and the $\delta+1$ columns in the second repair group are linearly dependent,
which also leads to $d\leq 2\delta+1$.

The proof is completed. ∎

For each 2-subspace $\mathcal{V}_{i}$ of $\mathbb{F}_{q}^{3}$, $i\in[\ell]$,
we define ${\mathcal{V}_{i}\brack 1}_{q}$ used in [28] to be the set of all
1-subspaces of $\mathcal{V}_{i}$. Clearly, $\left|{\mathcal{V}_{i}\brack
1}_{q}\right|=q+1$. To avoid the degenerated case when $\delta=2$ as in [27,
Theorem], we assume that $\delta>2$, i.e, $\delta-1\geq 2$ below. Then we can
obtain the following claim.

_Claim 1:_ For $m,m^{\prime}\in[\delta-1]$, $m\neq m^{\prime}$, the generated
1-subspaces $\textnormal{Span}\\{a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}\\}$
and
$\textnormal{Span}\\{a_{m^{\prime}}\textbf{u}_{i}+b_{m^{\prime}}\textbf{v}_{i}\\}$
in ${\mathcal{V}_{i}\brack 1}_{q}$ are distinct.

###### Proof:

If $\textnormal{Span}\\{a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}\\}$ and
$\textnormal{Span}\\{a_{m^{\prime}}\textbf{u}_{i}+b_{m^{\prime}}\textbf{v}_{i}\\}$
are the same 1-subspace, then there exists $c\in\mathbb{F}_{q}^{*}$ such that

$a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}=c(a_{m^{\prime}}\textbf{u}_{i}+b_{m^{\prime}}\textbf{v}_{i})\Leftrightarrow
a_{m}=ca_{m^{\prime}},b_{m}=cb_{m^{\prime}},$

then we can show that the $(\delta-1)$ columns $\\{\mathbf{e}_{s}\mid
s\in[\delta-1]\setminus\\{m,m^{\prime}\\}\\}\cup\\{\mathbf{p}_{i},\mathbf{q}_{i}\\}$
are linearly dependent, which contradicts with the fact that
$(\mathbf{I}_{\delta-1},\mathbf{p_{i}},\mathbf{q_{i}})$ forms a standard
generator matrix of a MDS code. ∎

Based on Claim 1 and the previous lemma, we can now obtain a complete
characterization of optimal $(n,k,d=2\delta+2;r=2,\delta)$-LRCs as follows.

###### Theorem 2

Suppose $q\geq\delta+1$ and $\delta+1\mid n$. There exists a $q$-ary optimal
$(n,k,d=2\delta+2;r=2,\delta)$-LRC with disjoint repair groups if and only if
there exist $\ell$ distinct 2-subspaces
$\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{\ell}$ of
$\mathbb{F}_{q}^{3}$ such that for each $i\in[\ell]$,

$t_{i}\triangleq\left|\bigcup_{j\in[\ell],j\neq i}\left({\mathcal{V}_{i}\brack
1}_{q}\bigcap{\mathcal{V}_{j}\brack 1}_{q}\right)\right|\leq q-\delta.$

###### Proof:

The proof is divided into two directions as follows.

  * •

Necessity: Let $C$ be an optimal $(n,k,d=2\delta+2;r=2,\delta)$-LRC with a
parity-check matrix $H$. By Lemma 4 and Claim 1, we have
$\dim(\mathcal{V}_{i})=2$ and $\textnormal{Span}\\{\textbf{u}_{i}\\}$,
$\textnormal{Span}\\{\textbf{v}_{i}\\}$,
$\textnormal{Span}\\{a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}\\},m\in[\delta-1]$,
are $(\delta+1)$ distinct 1-subspaces in ${\mathcal{V}_{i}\brack 1}_{q}$,
which do not belong to ${\mathcal{V}_{j}\brack 1}_{q}$ for all $j\neq i$. Thus
$t_{i}\leq q+1-(\delta+1)=q-\delta$.

  * •

Sufficiency: Suppose there exist $\ell$ distinct 2-subspaces
$\mathcal{V}_{1},\mathcal{V}_{2},\ldots,\mathcal{V}_{\ell}$ of
$\mathbb{F}_{q}^{3}$ satisfying the intersection condition in the theorem.
Since $t_{i}\leq q-\delta$, there exist $(\delta+1)$ distinct 1-subspaces of
$\mathcal{V}_{i}$, without loss of generality, we let these $(\delta+1)$
distinct 1-subspaces to be $\textnormal{Span}\\{\textbf{u}_{i}\\}$,
$\textnormal{Span}\\{\textbf{v}_{i}\\}$,
$\textnormal{Span}\\{a_{m}\textbf{u}_{i}+b_{m}\textbf{v}_{i}\\},m\in[\delta-1]$,
$(a_{m},b_{m})\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*}$. Similar to (16),
we can show that there exists
$(p_{i,m},q_{i,m})\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*}$ such that

$a_{m}p_{i,m}+b_{m}q_{i,m}=0.$ (17)

Next, we need to prove that the matrix
$H_{i}=(\mathbf{I}_{\delta-1},\mathbf{p_{i}},\mathbf{q_{i}})$ formed by these
$(p_{i,m},q_{i,m})\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*}$ is a standard
generator matrix of a $[\delta+1,\delta-1,3]$ MDS code, i.e, any $(\delta-1)$
columns of $H_{i}$ are linearly independent. Based on the structure of
$H_{i}$, we only need to prove the last $(\delta-1)$ columns are linearly
independent in the sense of equivalence. By contradiction, we assume that last
$(\delta-1)$ columns are linearly dependent, then there must exist
$(a,b)\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}^{*}$ such that
$ap_{i,1}+bq_{i,1}=0$, $ap_{i,2}+bq_{i,2}=0$, i.e., $a=ca_{1},b=cb_{1}$,
$a=c^{\prime}a_{2},b=c^{\prime}b_{2}$ for some
$c,c^{\prime}\in\mathbb{F}_{q}^{*}$. Thus $a_{1}=c^{-1}c^{\prime}a_{2}$,
$b_{1}=c^{-1}c^{\prime}b_{2}$, which implies
$\textnormal{Span}\\{a_{1}\textbf{u}_{i}+b_{1}\textbf{v}_{i}\\}$ and
$\textnormal{Span}\\{a_{2}\textbf{u}_{i}+b_{2}\textbf{v}_{i}\\}$ contributes
to the same 1-subspace. This makes a contradiction. Therefore, each matrix
$H_{i}=(\mathbf{I}_{\delta-1},\mathbf{p_{i}},\mathbf{q_{i}})$ is indeed a
standard generator matrix of a $[\delta+1,\delta-1,3]$ MDS code, then a linear
code ${\mathcal{C}}$ with parity-check $H$ given as (15) has $(r=2,\delta)$
locality and dimension $k\geq n-\ell(\delta-1)-3=2\ell-3$. By (2), we have
$d\leq 2\delta+2$. We only need to show that $d\geq 2\delta+2$. By
contradiction, we assume that $d\leq 2\delta+1$, i.e., there exist $2\delta+1$
columns of $H$ which are linearly dependent. Due to the structure of $H$, we
only need to prove that $\delta$ columns from one local group and $\delta+1$
columns from another local group are linearly independent. Otherwise, we
assume these $2\delta+1$ columns are linearly dependent and distributed in the
first two repair local groups, then we know that
$\textnormal{Span}\\{\textbf{u}_{1}\\}$,
$\textnormal{Span}\\{\textbf{v}_{1}\\}$, or
$\textnormal{Span}\\{a_{m}\textbf{u}_{1}+b_{m}\textbf{v}_{1}\\},m\in[\delta-1]$
must belong to ${\mathcal{V}_{2}\brack 1}_{q}$ by (17) and the structure of
$H$, which leads to a contradiction.

Thus we finish the proof. ∎

Let $PG(2,q)$ be the projective plane over finite field $\mathbb{F}_{q}$. We
can restate Theorem 2 in an equivalent geometrical language as [15, Theorem
5’].

######  Theorem 2′ (Geometrical Characterization of Optimal
$(n,k,d=2\delta+2;r=2,\delta)$-LRCs)

Suppose $q\geq\delta+1$ and $(\delta+1)\mid n$. Then, there exists an optimal
$(n,k,d=2\delta+2,r=2,\delta)$-LRC with disjoint repair groups if and only if
there exist $\ell$ distinct lines $L_{1},L_{2},\ldots,L_{\ell}$ in $PG(2,q)$,
such that each $L_{i}$ has at most $(q-\delta)$ distinct intersection points.

We refer to the condition in Theorem 2′ as _intersection condition_ in the
sequel. From Theorem 2′, we can employ the theory of finite geometry to study
the construction and bound of optimal $(r,\delta)$ LRCs with minimum distance
$d=2\delta+2$ and locality $r=2$. Note that the _sunflower_ structure [29] of
$PG(2,q)$ naturally satisfies the intersection condition in Theorem 2′ when
$q\geq\delta+1$ since all lines intersect in a common point. Then we can
generalize the $r$-local sunflower construction [15, Theorem 4] to the
$(r,\delta)$ case below.

######  Theorem 3 ($(r,\delta)$ Sunflower Construction)

Let $q\geq\delta+1$ be a prime power, then there exists a $q$-ary optimal
$(n=(\delta+1)(q+1),k=2q-1,d=2\delta+2;r=2,\delta)$-LRC.

###### Remark 2

Note that when $q=4$ and $\delta=3$, the construction (36) of [26] exactly
corresponds to the sunflower constructions and achieves the maximal code
length.

###  III-C Improved Upper Bounds for optimal $(n,k,d=2\delta+2;r=2,\delta)$
LRCs based on Constant Weight Codes

Similar to [15, Theorem 9], we can obtain better upper bounds on the code
length based on binary constant weight codes. By Theorem 2′, we can construct
an $\ell\times(q^{2}+q+1)$-line-point incidence matrix $A$ whose rows
correspond to the $\ell$ lines with Hamming weight $q+1$, and whose columns
correspond to $q^{2}+q+1$ points in $PG(2,q)$. Next, we give an upper bound on
$\ell$ by the intersection condition.

###### Lemma 5

Suppose $q\geq(\delta+1)$. If there exist $\ell$ distinct lines
$L_{1},L_{2},\ldots,L_{\ell}$ in $PG(2,q)$ satisfying the intersection
condition which do not form a Sunflower, then

$\ell\leq\left\lfloor(q^{2}+q+1)/(\delta+2)\right\rfloor.$

###### Proof:

Let $A$ be the $\ell\times(q^{2}+q+1)$-incidence matrix of the $\ell$ lines
defined as above. The intersection condition implies that for each line
$L_{i}$, there exists at least $(\delta+1)$ points on $L_{i}$ not belonging to
any other lines, which implies that there exist at least $\ell(\delta+1)$
columns whose Hamming weight is 1. Equivalently, we assume that $A$ has the
following structure:

$A=\left(\begin{array}[]{c|c}A^{\prime}&\underbrace{\begin{array}[]{cccc}\underbrace{1~{}1~{}\cdots~{}1}_{\delta+1}&0~{}0~{}\cdots~{}0&\ldots&0~{}0~{}\cdots~{}0\\\
0~{}0~{}\cdots~{}0&\underbrace{1~{}1~{}\cdots~{}1}_{\delta+1}&\ldots&0~{}0~{}\cdots~{}0\\\
\vdots~{}\vdots~{}\vdots&\vdots~{}\vdots~{}\vdots&\ddots&\vdots~{}\vdots~{}\vdots\\\
0~{}0~{}\cdots~{}0&0~{}0~{}\cdots~{}0&\ldots&\underbrace{1~{}1~{}\cdots~{}1}_{\delta+1}\\\
\end{array}}_{\ell(\delta+1)~{}\text{columns}}\end{array}\right),$ (18)

where $A^{\prime}$ is an $\ell\times(q^{2}+q+1-\ell(\delta+1))$ matrix. We can
show that the Hamming weight of each column of $A^{\prime}$ is at most
$q-\delta$ since the $\ell$ lines do not form a sunflower and satisfy the
intersection condition. Calculating the number of 1’s in $A^{\prime}$ in two
ways, we obtain that
$\ell\times(q-\delta)\leq(q^{2}+q+1-\ell(\delta+1))\times(q-\delta)$, i.e.,
$\ell\leq\left\lfloor(q^{2}+q+1)/(\delta+2)\right\rfloor.$ ∎

From Theorem 2′ and Lemma 5, we have an improved upper bound as follows.

###### Theorem 4

Suppose $q\geq\delta+1$ and let $C$ be a $q$-ary optimal
$(n,k,d=2\delta+2;r=2,\delta)$ LRC with disjoint local repair groups, then

$n\\!\leq\\!\max\left\\{(\delta+1)(q+1),(\delta+1)\left\lfloor(q^{2}+q+1)/(\delta+2)\right\rfloor\right\\}.$

By employing the Johnson upper bound (Lemma 2), we can obtain two improved
upper bounds that are generalizations of Theorem 9 in [15].

###### Theorem 5

Suppose $q\geq\delta+2$ and a Singleton-optimal $(n,k,d=2\delta+2;r=2,\delta)$
LRC ${\mathcal{C}}$ exists, then

$\displaystyle
n\\!\leq\\!(\delta+1)\left\lfloor\frac{(2\delta+3)q^{2}+q+(\delta+1)^{2}-\sqrt{\Gamma}}{2(\delta+1)^{2}}\right\rfloor\\!\\!=\\!O(q^{2}),$
(19)

where
$\Gamma\triangleq(4\delta+5)q^{4}-(8\delta^{2}+12\delta+2)q^{3}+(4\delta^{3}+6\delta^{2}-1)q^{2}-2(\delta+1)^{2}q+(\delta+1)^{4}$.
Moreover, we have

$\displaystyle n\\!\\!\\!$ $\displaystyle\\!\\!\\!\leq\\!\\!\\!$
$\displaystyle\\!\\!\\!(\delta+1)\left\lfloor\frac{(\delta(q+2)+2)+q\sqrt{4(\delta+1)q-(3\delta^{2}+4\delta)}}{2(\delta+1)}\right\rfloor$
(20) $\displaystyle=$ $\displaystyle O(q^{1.5}).$

###### Proof:

Let $A^{\prime}$ be an $\ell\times(q^{2}+q+1-\ell(\delta+1))$-matrix defined
in (18). By Lemma 5, we know that the Hamming weight of each row of
$A^{\prime}$ is $q-\delta$. Since two lines in $PG(2,q)$ intersect at exactly
one point, the Hamming distance of any two distinct rows of $A^{\prime}$ is
equal to $2(q-\delta)-2=2(q-\delta-1)$. Let $C^{\prime}$ be a binary code
whose codewords are the row vectors of $A^{\prime}$, then $C^{\prime}$ is
exactly a binary $(n,M,d;w)$-constant weight code with
$n=q^{2}+q+1-\ell(\delta+1)$, $M=\ell$, $d=2(q-\delta-1)$ and $w=q-\delta$. By
plugging the parameters into the Johnson bounds (Lemma 2), we can obtain the
desired upper bounds (19)-(20) on $n$ by solving two quadratic inequalities of
$\ell$. ∎

##  IV Conclusion

In this paper, we provide a framework for further research on the maximum code
length and constructions of optimal $(r,\delta)$ LRCs with small localities.
Unlike known upper bounds that are asymptotically achieved, our improved upper
bounds largely close the gap between the bounds and known construction when
$2\delta+1\leq d\leq 3\delta$. It’s worth noting that our bounds can be
exactly achieved in some special cases. Furthermore, we establish a
generalized characterization of optimal $(r=2,\delta)$-LRCs when
$d=2\delta+2$. With this geometrical characterization, we obtain a sunflower
based $(r,\delta)$ LRCs with longest code length $n=(\delta+1)(q+1)$ until
now.

Figure 1: Summary of Related Progress.

We list known results in Figure 1. Despite these advances as shown in Figure
1, it is still a challenging problem to construct optimal $(r,\delta)$ LRCs
achieving the maximal code length when $d\geq 2\delta+1$, especially when
$\delta$ is odd that can not degenerate into the $r$-local case. Our future
work will focus more on this parameter range, and generalize other
combinatorial structures considered in [15] to obtain more optimal
$(r,\delta)$ LRCs with a longer length.

## References

  * [1] A. G Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright, and K. Ramchandran,  “Network coding for distributed storage systems,”  IEEE transactions on information theory, vol. 56, no. 9, pp. 4539–4551, 2010.
  * [2] P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin,  “On the locality of codeword symbols,”  IEEE Transactions on Information theory, vol. 58, no. 11, pp. 6925–6934, 2012.
  * [3] C. Huang, H. Simitci, Y. Xu, A. Ogus, B. Calder, P. Gopalan, J. Li, and S. Yekhanin,  “Erasure coding in windows azure storage,”  in 2012 $\\{$USENIX$\\}$ Annual Technical Conference ($\\{$USENIX$\\}$ $\\{$ATC$\\}$ 12), 2012, pp. 15–26.
  * [4] M. Asteris, D. Papailiopoulos, A. G. Dimakis, R. Vadali, S. Chen, and D. Borthakur,  “Xoring elephants: Novel erasure codes for big data,”  Proceedings of the VLDB Endowment, vol. 6, no. 5, 2013.
  * [5] I. Tamo and A. Barg,  “A family of optimal locally recoverable codes,”  IEEE Transactions on Information Theory, vol. 60, no. 8, pp. 4661–4676, 2014.
  * [6] I. Tamo, D. Papailiopoulos, and A. G Dimakis,  “Optimal locally repairable codes and connections to matroid theory,”  IEEE Transactions on Information Theory, vol. 62, no. 12, pp. 6661–6671, 2016.
  * [7] A. Silberstein, N.and Rawat, O. Koyluoglu, and S. Vishwanath,  “Optimal locally repairable codes via rank-metric codes,”  in 2013 IEEE International Symposium on Information Theory. IEEE, 2013, pp. 1819–1823.
  * [8] Y. Luo, C. Xing, and C. Yuan,  “Optimal locally repairable codes of distance 3 and 4 via cyclic codes,”  IEEE Transactions on Information Theory, vol. 65, no. 2, pp. 1048–1053, 2018.
  * [9] B. Chen, S.-T Xia, J. Hao, and F.-W Fu,  “Constructions of optimal cyclic $(r,\delta)$ locally repairable codes,”  IEEE Transactions on Information Theory, vol. 64, no. 4, pp. 2499–2511, 2017.
  * [10] B. Chen, W. Fang, S.-T Xia, and F.-W Fu,  “Constructions of optimal $(r,\delta)$ locally repairable codes via constacyclic codes,”  IEEE Transactions on Communications, vol. 67, no. 8, pp. 5253–5263, 2019.
  * [11] L. Jin,  “Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes,”  IEEE Transactions on Information Theory, vol. 65, no. 8, pp. 4658–4663, 2019.
  * [12] X. Li, L. Ma, and C. Xing,  “Optimal locally repairable codes via elliptic curves,”  IEEE Transactions on Information Theory, vol. 65, no. 1, pp. 108–117, 2018.
  * [13] C. Xing and C. Yuan,  “Construction of optimal locally recoverable codes and connection with hypergraph,”  arXiv preprint arXiv:1811.09142, 2018.
  * [14] V. Guruswami, C. Xing, and C. Yuan,  “How long can optimal locally repairable codes be?,”  IEEE Transactions on Information Theory, vol. 65, no. 6, pp. 3662–3670, 2019.
  * [15] B. Chen, W. Fang, S.-T Xia, J. Hao, and F.-W Fu,  “Improved bounds and singleton-optimal constructions of locally repairable codes with minimum distance 5 and 6,”  IEEE Transactions on Information Theory, vol. 67, no. 1, pp. 217–231, 2020.
  * [16] N. Prakash, G. M. Kamath, V. Lalitha, and P. V. Kumar,  “Optimal linear codes with a local-error-correction property,”  in 2012 IEEE International Symposium on Information Theory Proceedings. IEEE, 2012, pp. 2776–2780.
  * [17] B. Sasidharan, G. K. Agarwal, and P. V. Kumar,  “Codes with hierarchical locality,”  in 2015 IEEE International Symposium on Information Theory (ISIT). IEEE, 2015, pp. 1257–1261.
  * [18] A. Wang and Z. Zhang,  “Repair locality with multiple erasure tolerance,”  IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 6979–6987, 2014.
  * [19] A. Rawat, D. Papailiopoulos, A. G. Dimakis, and S. Vishwanath,  “Locality and availability in distributed storage,”  IEEE Transactions on Information Theory, vol. 62, no. 8, pp. 4481–4493, 2016.
  * [20] H. Cai, Y. Miao, M. Schwartz, and X. Tang,  “On optimal locally repairable codes with super-linear length,”  IEEE Transactions on Information Theory, vol. 66, no. 8, pp. 4853–4868, 2020.
  * [21] H. Cai and M. Schwartz,  “On optimal locally repairable codes and generalized sector-disk codes,”  IEEE Transactions on Information Theory, vol. 67, no. 2, pp. 686–704, 2020.
  * [22] G. Zhang and H. Liu,  “Constructions of optimal codes with hierarchical locality,”  IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7333–7340, 2020.
  * [23] X. Kong, X. Wang, and G. Ge,  “New constructions of optimal locally repairable codes with super-linear length,”  IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6491–6506, 2021.
  * [24] W. Fang and F.-W Fu,  “Optimal cyclic (r, $\delta$) locally repairable codes with unbounded length,”  Finite Fields and Their Applications, vol. 63, pp. 101650, 2020\.
  * [25] J. Hao, S.-T Xia, K. Shum, B. Chen, F.-W Fu, and Y. Yang,  “Bounds and constructions of locally repairable codes: parity-check matrix approach,”  IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7465–7474, 2020.
  * [26] K. Shum and J. Hao,  “Optimal quaternary $(r,\delta)$-locally repairable codes achieving the singleton-type bound,”  arXiv preprint arXiv:2112.05377, 2021.
  * [27] F. J. MacWilliams and A. Sloane,  The theory of error correcting codes, vol. 16,  Elsevier, 1977.
  * [28] T. Etzion and N. Raviv,  “Equidistant codes in the grassmannian,”  Discrete Applied Mathematics, vol. 186, pp. 87–97, 2015.
  * [29] T. Etzion and L. Storme,  “Galois geometries and coding theory,”  Designs, Codes and Cryptography, vol. 78, no. 1, pp. 311–350, 2016\.