# Derivation of a statistical model for classical systems obeying the
fractional exclusion principle

Projesh Kumar Roy<EMAIL_ADDRESS>The Institute of Mathematical
Sciences, 4th cross road, C.I.T. Campus, Taramani, Chennai – 600113, Tamil
Nadu, India  Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai – 400094, Maharashtra, India

###### Abstract

The violation of the Pauli principle has been surmised in several models of
the Fractional Exclusion Statistics and successfully applied to several
quantum systems. In this paper, a classical alternative of the exclusion
statistics is studied using the maximum entropy methods. The difference
between the Bose-Einstein statistics and the Maxwell-Boltzmann statistics is
understood in terms of a separable quantity, namely the degree of
indistinguishability. Starting from the usual Maxwell-Boltzmann microstate
counting formula, a special restriction related to the degree of
indistinguishability is incorporated using Lagrange multipliers to derive the
probability distribution function at equilibrium under NVE conditions. It is
found that the resulting probability distribution function generates real
positive values within the permissible range of parameters. For a dilute
system, the probability distribution function is intermediate between the
Fermi-Dirac and Bose-Einstein statistics and follows the exclusion principle.
Properties of various variables of this novel statistical model are studied
and possible application to classical thermodynamics is discussed.

Fractional Exclusion statistics, Haldane-Wu statistics, CFES, Polychronakos
statistics, Effective temperature in equilibrium

##  I Introduction

It is known for a long time that, the Pauli exclusion principle is essentially
a limiting case in the quantum theory of fundamental particles and not always
valid in certain scenarios Milotti  _et al._ (2007). It was understood,
several systems in nature, e.g. 2D-anyone, follow a mixed statistics which can
not be explained using any existing fundamental statistical theories. In the
past century, various attempts were made to unify the three fundamental
statistical theories present in nature – namely Maxwell-Boltzmann statistics
(MB), Fermi-Dirac statistics (FD), and Bose-Einstein statistics (BE) – under a
single statistical theory which will be able to explain the mixed statistics
phenomenon. Gentile statistics Gentile j. (1940), Tsallis statistics Tsallis
(1988), collision model March  _et al._ (1993); March (1997), Acharya-Swamy
statistics Acharya and Swamy (1994) etc. were some of the early attempts to
unify all fundamental statistical theories. In this context, the theory of
fractional exclusion statistics (FES) Khare (2005) was developed based on
fractional exclusion principle (FEP), which was designed to be intermediate
between FD and BE statistics. FEP refers to the multiple-but-finite occupancy
of the building blocks at a certain state. In this paper, the term particle is
used to describe the building blocks of any realistic natural system. It has
important applications in the many quantum systems like quantum computation
Nayak  _et al._ (2008), anyone statistics Dasnières de Veigy and Ouvry (1994,
1995); de Veigy and Ouvry (1995); Bartolomei  _et al._ (2020), fractional
quantum hall effect Stern (2008), high temperature superconductivity Laughlin
(1988a, b), even black holes Aghaei Abchouyeh  _et al._ (2018). Most of the
early versions of FES models (described above) were not derived from first-
principle-based reasoning, but mostly based on empirical extrapolations
between FD and BE statistics. In 1994, Wu Haldane (1991); Wu (1994); Nayak and
Wilczek (1994); Anghel (2007, 2009, 2013) and Isakov Isakov (1994a, b)
independently proposed a novel FES model – namely Haldane-Wu statistics (HW) –
which was derived directly from counting the number of microstates for a
strongly correlated quantum system via maximum entropy methods Jaynes (1957a,
b) (MaxEnt). HW statistics aims to compute the correct number of conformations
of a system evolving under FEP. It provides a microscopic view of the particle
distribution at various energy levels and generalizes the Pauli exclusion
principle to multiple occupation numbers. It should be noted, a similar (but
not identical) model to HW statistics was proposed by Ramanathan Ramanathan
(1992) in 1992, also based on modified microstates counting.

HW statistics has been successful in predicting the statistics of 2D-anyone
systems projected at lowest Landau level under a strong magnetic field
Dasnières de Veigy and Ouvry (1995); de Veigy and Ouvry (1995) and excitation
in pure Laughlin liquids Camino  _et al._ (2005); Arovas  _et al._ (1984). Not
just quantum systems, HW statistics have significant applications in realistic
classical systems as well, e.g., adsorption of polyatomic molecules onto
substrates Riccardo  _et al._ (2004, 2005); Cerofolini (2006); Dávila  _et
al._ (2009a, b); Matoz-Fernandez  _et al._ (2011); Matoz-Fernandez and
Ramirez-Pastor (2014); Riccardo  _et al._ (2019), yielding phenomenon under
mechanical stress Mezzasalma (2019), calculation of entropy in ice Ihm (1996),
coil-helix transition transition of polypeptide Sharma  _et al._ (2015), etc.
However occurrence of negative probabilities is a problem in HW statistics
Nayak and Wilczek (1994); Polychronakos (1996a, b); Chaturvedi and Srinivasan
(1997a, b); Murthy and Shankar (1999). As pointed out by A. Polychronakos
Polychronakos (1996a), this problem is inherent in HW statistics and cannot be
avoided via any approximation. An alternate statistical model was suggested by
him using path integral formulation – namely Polychronakos statistics (AP)
Polychronakos (1996a, b); Chung and Hassanabadi (2018); Hoyuelos (2018) –
which avoids the problem of negative probabilities under certain conditions.
Following the success of HW and AP statistics, several other FES models based
on microstate counting have also been suggested, e.g. Pólya-urn model Niven
(2009); Niven and Grendar (2009), Fractional Superstatistics Ourabah and
Tribeche (2018) model, and other mixed statistics models Abutaleb (2014); Yan
(2021).

Parallel to Wu, Kaniadakis et al. Kaniadakis and Quarati (1993, 1994);
Kaniadakis (1995); Kaniadakis  _et al._ (1996) developed a classical analog of
FES – namely the Kinetic model – by including the FEP in the non-linear
Fokker-Plank equation. In this kinetic model, only classical particles are
considered; however, their hopping into different states was constrained using
the FEP. In this way, quantum effects were introduced into the dynamics of a
classical system. This model ultimately evolved into $\kappa$-deformed
statistics Kaniadakis (2001); Kaniadakis  _et al._ (2002); Kaniadakis and
Scarfone (2019), which was able to reproduce both HW and AP type statistics
depending on the particular choice of the transition parameters Kaniadakis
_et al._ (1996); Kaniadakis (2001).

All FES models described above have originated from quantum statistics.
Although quantum-FES (QFES) models have been widely used in certain classical
systems (described above), such strict validity of FEP in classical systems
can not be ascertained a priori. By definition, the microstate counting
formula of the classical systems cannot be different from the MB statistics,
but the total entropy of the system can be amended using additional
constraints. Using this idea, an alternate FES-type model for classical
systems is provided – namely the classical fractional exclusion statistics
(CFES) – without a priori assumption of the validity of FEP. The idea of
generating quantum effects in classical systems is investigated by
understanding the difference between the two limiting cases – i. completely
distinguishable MB statistics, ii. completely indistinguishable BE statistics.
It is shown that the FEP – much similar to the QFES models – emerges naturally
under certain constraints in classical systems obeying the CFES model.

The paper is summarized as follows – section II: the degree of
indistinguishability is formulated from BE microstate equation, section III:
the principal equations of CFES is derived, section IV: the origin of FEP and
other properties of CFES is described, section V: application of CFES in
classical thermodynamics is discussed.

##  II Degree of Indistinguishability

It was understood that quantum effects can be generated by controlling the
degree of indistinguishability (DI) of the particles in the system Kaniadakis
and Quarati (1994). The physical meaning of varying indistinguishability can
be that some particles of the same species (i.e. similar physical traits) can
be made distinguishable using some unknown traits which cannot be incorporated
in the model. Based on the idea of varying indistinguishability, Medvedev
Medvedev (1997); Meljanac  _et al._ (1999) developed another QFES model –
namely ambiguous statistics – which is intermediate between FD and BE
statistics.

Let’s consider a system of total $N$ non-interacting particles have a
population of $n_{i}\geq 0$ at $i^{\text{th}}$ energy level $\epsilon_{i}\geq
0$ with degeneracy $g_{i}\geq 1$ under NVE condition with total energy
$E=\sum_{i}n_{i}\epsilon_{i}$ and total degeneracy $G=\sum_{i}g_{i}$. The
reduced unit formalism is followed in all subsequent sections, i.e.,
$k_{B}=1$, where $k_{B}$ is the Boltzmann constant. For indistinguishable
particles, the number of microstates can be calculated from the BE statistics.
Using Stirling’s approximation $n_{i}!\sim n_{i}^{n_{i}}\exp(-n_{i})$ and
$n_{i}<<g_{i}$, one can simplify BE equation for total number microstates,
$W_{\text{BE}}$, as,

$W_{\text{BE}}\approx
W_{\text{MB}}\Bigg{[}\frac{1}{N!}\prod_{i}\Big{(}1+\frac{n_{i}}{g_{i}}\Big{)}^{n_{i}}\Bigg{]}$
(1)

For details of the derivation, see supporting information section SI.I. The
first product of the final Equation 1 is the traditional microstate counting
from MB statistics; $W_{\text{MB}}$ = $\prod_{i}N!(g_{i}^{n_{i}}/n_{i}!)$; for
distinguishable particles. Equation 1 is usually reduced to the traditional MB
statistics by considering $n_{i}/g_{i}\sim 0$ – i.e. dilute systems – and the
second product $(1+{n_{i}}/{g_{i}})^{n_{i}}~{}\sim 1$. However, by not
neglecting the second product, we get an interesting interpretation of the BE
statistics. If the entropy of a BE system is defined as
$S_{\text{BE}}=\ln(W_{\text{BE}})$, then using Stirling’s approximation and
logarithmic expansion on Equation 1 (see supporting information section SI.I),
one can write,

$S_{\text{BE}}=S_{\text{MB}}+\Bigg{[}\sum_{m=0}^{\infty}f_{m}S_{m}-(N\ln
N+G)\Bigg{]}$ (2)

where $f_{m}$’s are the coefficients from the logarithmic expansion. Equation
2 can be viewed as a relationship between the entropic contributions arising
from the distinguishable ($S_{\text{MB}}$) and indistinguishable
($S_{\text{BE}}$) properties of the system with unlimited occupancy at each
energy level. The residual entropic term,
$S^{*}=\sum_{m=0}^{\infty}f_{m}S_{m}-(N\ln N+G)$, can be viewed as a
superposition of multiple entropic terms, $S_{m}$’s, where $0\leq
m\leq\infty$. The $\sum_{m}f_{m}S_{m}$ term controls the transition between
the distinguishable and indistinguishable particles, hence, one can identify
this term as the DI. Maximizing $S_{\text{BE}}$ with respect to the population
at each energy state, $n_{i}$, Equation 2 becomes,

$\Bigg{(}\frac{\partial S_{\text{BE}}}{\partial
n_{i}}\Bigg{)}_{\text{max}}=\Bigg{(}\frac{\partial S_{\text{MB}}}{\partial
n_{i}}\Bigg{)}_{\text{max}}+\sum_{m=0}^{\infty}f_{m}\Bigg{(}\frac{\partial
S_{m}}{\partial n_{i}}\Bigg{)}_{\text{max}}=0$ (3)

Both $S_{\text{MB}}$ and $S_{m}$ terms and their derivatives are real positive
quantities. Hence, each individual $S_{m}$ have to be constant at the maximum
value of $S_{\text{BE}}$. This result can be stated in a different way as,
($S_{m}$ = constant) constraint acts as an additional restriction on top of
($S_{\text{MB}}$ = constant) to generate the maximum entropy in BE statistics
for indistinguishable particles. However, determination of the effects of
individual $S_{m}$ values are not required to get $S_{\text{BE}}$ from
$S_{\text{MB}}$, as their linear superposition with coefficients $f_{m}$ leads
to an $m$ independent logarithmic function in Equation 2. Question is, how
will the statistics of the system evolve if the superposition of each of these
restrictions does not lead to an $m$ independent function.

To answer such a question, one has to understand how a particle statistics
evolves under each of these restrictions – i.e. ($S_{m}$ = constant) – in
addition to ($S_{\text{MB}}$ = constant). A statistical model can be prepared
which maximize $S_{m}$ in addition to $S_{\text{MB}}$. MaxEnt method Jaynes
(1957a, b) is suitable for introducing any additional restriction in a
statistical model using Lagrange multipliers. One can expect that, properties
of such a model should lie in between BE and MB statistics.

##  III A Classical Fractional Exclusion Statistics

Taking inspiration from Equation 2, the following relation between $n_{i}$ and
$g_{i}$ is used to control DI in a system of distinguishable particles as,

$\sum_{i}\frac{n_{i}^{m}}{g_{i}^{(m-1)}}=S_{m}$ (4)

where, $m$ is defined as $m\in\mathbb{R}$, and $S_{m}$ is a constant. The aim
of the following analysis is to study the effect of different $m$ values on
$S_{\text{MB}}$ using MaxEnt method. For $m$=0 or 1, Equation 4 generates the
trivial conditions for NVE ensemble. For Equation 4 to be an non-trivial
constraint, $S_{m}$ cannot be linearly dependent on $N$, $E$, or $G$, and must
be an independent external parameter. Later it is shown that, $S_{m}$ is
related to an effective particle number. Therefore, starting from a purely MB
type statistics, CFES incorporates the individual effects of the constraints
($S_{m}$=constant) from Equation 2 on the maximum value of $S_{\text{MB}}$. It
can be shown that, one cannot use any other exponent combinations on $n_{i}$
and $g_{i}$ in Equation 4 (see supporting information section SI.II).

The derivation of CFES considers the absolute number of microstates to be same
as used in the modified MB statistics as, $\widetilde{W}=W_{\text{MB}}/N!$. At
equilibrium, one can write the optimal distribution of the particles
corresponding to the maximum entropy as (see supporting information section
SI.III),

$n_{i}=g_{i}e^{-(\alpha+\beta\epsilon_{i})}e^{-m\gamma(n_{i}/g_{i})^{(m-1)}}$
(5)

where $\alpha$, $\beta$, and $\gamma$ are the Lagrange multipliers. If one
uses the approximation that in dilute systems $g_{i}>>n_{i}$, then

$\frac{n_{i}}{g_{i}}=e^{-(\alpha+\beta\epsilon_{i})}\Bigg{[}1-m\gamma\Bigg{(}\frac{n_{i}}{g_{i}}\Bigg{)}^{(m-1)}\Bigg{]}$
(6)

Equation 6 is the final expression of the probability distribution function in
the CFES. $\exp(-\alpha)\equiv N/Z_{m}$ is the normalization factor and
$Z_{m}$ is the partition function corresponding to Equation 6. The average
population per individual degenerate levels with energy $\epsilon_{i}$ is
defined as $\bar{n_{i}}=(n_{i}/g_{i})$. Equation 6 can be reduced using
Ramanujan formula Berndt  _et al._ (1983); Murthy and Shankar (1999). Details
of the derivation is given in supporting information section SI.IV. The
reduced solution of Equation 6 for $m\geq 2$ can be written as,

$\bar{n_{i}}=\sum_{j=0}^{\infty}C_{j}(m)(-m\gamma)^{j}e^{-[(mj-j+1)(\alpha+\beta\epsilon_{i})]}$
(7)

where $C_{j}(m)$ is the first weight to the probability function. It is
defined as,

$C_{j}(m)=\frac{1}{j!}\prod_{k=1}^{j-1}[1+j(m-1)-k]=\prod_{l=2}^{j}\Bigg{(}\frac{(m-2)j}{l}+1\Bigg{)}$
(8)

From the definition of $C_{j}(m)$, it is clear that it is always a positive
number for $m\geq 2$. The second weight, $(-m\gamma)^{j}$, is always positive
for $\gamma<0$ and $m>0$. For $\gamma>0$ and $m>0$, this factor can either be
positive or negative, depending on whether $j$ is odd or even. If the total
weights are negative at a certain $(m,\gamma)$ combination, the total
probability function will become negative as well; which is a common problem
in HW statistics. Later it is shown that, within the permissible range of $m$
and $\gamma$, negative probabilities do not arise in CFES.

For $m=0$, it is straight forward to show that Equation 6 becomes traditional
MB distribution. If $m=1$, then Equation 6 becomes,

$\bar{n_{i}}=e^{-(\alpha+\beta\epsilon_{i})}(1-\gamma)$ (9)

Equation 9 is also equivalent to the traditional MB distribution. Via
normalizing, one can find $\exp(-\alpha)=N/[Z_{\text{MB}}(1-\gamma)]$, where
$Z_{\text{MB}}=\sum_{i}g_{i}\exp(-\beta\epsilon_{i})$ is the partition
function corresponding to the MB statistics. If $m=2$, then Equation 6
becomes,

$\bar{n_{i}}=\frac{1}{e^{(\alpha+\beta\epsilon_{i})}+2\gamma}$ (10)

Using $\gamma=\pm 1/2$, traditional FD or BE statistics can be retrieved from
Equation 10. The form of Equation 10 is essentially similar to the probability
distribution function derived from AP statistics Polychronakos (1996a, b). One
can generate many more probability distribution functions by solving the
polynomial in Equation 6 at different $m$ (see supporting information section
SI.V). However, the general solution given in Equation 7 is much too complex
to understand the properties of $m$, $\gamma$, and their underlying
thermodynamic connections. In the following section, it is shown how a
permissible range of $m$ and $\gamma$ factors can be determined for generating
a realistic probability distribution function using CFES, via logical and
algebraic reasoning.

##  IV Properties of CFES

To confirm that CFES generates a realistic probability distribution function,
one has to consider the dependence of $n_{i}$ on $\epsilon_{i}$ such that
$\bar{n_{i}}=\\{\bar{n_{i}}\in\mathbb{R}|\bar{n_{i}}\geq 0\\}$ throughout the
energy spectrum. Using this basic idea, many interesting features of CFES can
be deduced by studying and rearranging Equation 6 itself. Replacing
$a=\exp(\alpha+\beta\epsilon_{i})/m\gamma$ and $b=(1/m\gamma)$, Equation 6 can
be recognized as a polynomial of $\bar{n_{i}}$ as,

$\bar{n_{i}}^{(m-1)}+a\bar{n_{i}}-b=0$ (11)

For a particular $m\geq 2$, this equation has $(m-1)$ roots, including real
and imaginary. For CFES to apply to a system obeying thermodynamic principles,
at least one of the root of the Equation 11 must be real and positive. Looking
at Equation 11, it can be found that the two coefficients of the polynomial,
$(+a)$ and $(-b)$, have opposite signs at all values of $m$ and $\gamma$. This
indicates that at least one real root of $\bar{n_{i}}$ must exist for all
$m>0$. Using Descartes’ sign rule, it is found that there is only one positive
root is present at all permissible values of $m$ and $\gamma$. This rules out
any ambiguity in the general solution of CFES, and shows that $\bar{n_{i}}$
always has one, and only one, real positive value at all $\epsilon_{i}$ for
$m>0$.

###  IV.1 Limits of m, $\gamma$

Further rearranging Equation 11, one can show that, if
$\bar{n_{i}}=\\{\bar{n_{i}}\in\mathbb{R}|\bar{n_{i}}\geq 0\\}$, $m\neq 1$,
$\gamma\neq 0$, and $m\gamma>0$; a few possible conditions arise as,

$\displaystyle\begin{split}&\text{Condition 1: for }m\gamma>0\\\
&\bar{n_{i}}\leq e^{-(\alpha+\beta\epsilon_{i})}\end{split}$ (12a)
$\displaystyle\begin{split}&\text{Condition 2: for }0>m>1\text{ and
}\gamma>0\\\
&\bar{n_{i}}\geq\Big{(}\frac{1}{m\gamma}\Big{)}^{\Big{(}\frac{1}{m-1}\Big{)}}\end{split}$
(12b) $\displaystyle\begin{split}&\text{Condition 3: for }m>1\text{ and
}\gamma>0\\\
&\bar{n_{i}}\leq\Big{(}\frac{1}{m\gamma}\Big{)}^{\Big{(}\frac{1}{m-1}\Big{)}}\\\
\end{split}$ (12c) $\displaystyle\begin{split}&\text{Condition 4: for
}m<0\text{ and }\gamma<0\\\
&\bar{n_{i}}\geq\Big{(}\frac{1}{m\gamma}\Big{)}^{\Big{(}\frac{1}{-|m|-1}\Big{)}}\end{split}$
(12d)

The condition in Equation 12a is straightforward, which states that the number
of particles at an energy level $\epsilon_{i}$ will decrease with increasing
energy. However, in the rest of the sub-equations in Equation 12, no energy
terms are present. Hence, conditions in Equation 12b and 12d are in
contradiction with 12a, as they predict a non-zero lower bound for
$\bar{n_{i}}$, which is independent of energy. This is impossible at
$\epsilon_{i}\to\infty$, where Equation 12a predicts $\bar{n_{i}}\to 0$.
Hence, $m<1$ values are not allowed in Equation 12 within the given range of
parameters. The third condition in Equation 12c predicts a constant upper
bound, $n_{c}$, of the occupancy numbers at all available energy levels as,

$n_{c}=\Bigg{(}{\frac{1}{m\gamma}}\Bigg{)}^{\Big{(}\frac{1}{m-1}\Big{)}}$ (13)

which should be a valid condition at all $m$ and $\gamma$ values. However, by
definition, an available energy level means the allowed number of particles
cannot be less than one at any energy level, i.e. $n_{i}^{\text{min}}=1$. At
the limiting case, it means
$n_{c}^{\text{\text{min}}}=1/g^{\text{\text{min}}}$, where
$g^{\text{\text{min}}}$ is lowest available degeneracy at any of the available
energy level. Hence, Equation 13 can be written as,

$\gamma\leq\frac{(g^{\text{\text{min}}})^{(m-1)}}{m}$ (14)

In most classical systems, the degeneracy factor has a monotonous dependence
on energy. In fact, most realistic systems (classical or quantum) have a
ground state with single degeneracy, i.e. $g^{\text{\text{min}}}=1$. Hence, as
a special case of Equation 12c and 14 where $g^{\text{\text{min}}}=1$ and
$\gamma>0$ (labeled as ’$r$’ and ’$+$’, for ’real’ and ’positive’,
respectively), one can write,

$\gamma_{r}^{+}\leq\frac{1}{m}$ (15)

At $m=2$, Equation 15 yields the FD distribution at the upper limit of
$\gamma_{r}^{+}$. Hence, all systems which have positive $\gamma$ values and
obeys CFES model can be termed as fermion-like systems.

Similarly, one can write the possible conditions for
$\bar{n_{i}}=\\{\bar{n_{i}}\in\mathbb{R}|\bar{n_{i}}\geq 0\\}$, $m\neq 1$,
$\gamma\neq 0$, and $m\gamma<0$ from Equation 11 as,

$\displaystyle\begin{split}&\text{Condition 1: for }m\gamma<0\\\
&\bar{n_{i}}\geq e^{-(\alpha+\beta\epsilon_{i})}\\\ \end{split}$ (16a)
$\displaystyle\begin{split}&\text{Condition 2: for }m>0\text{ and }\gamma<0\\\
&\bar{n_{i}}\geq(-1)^{\Big{(}\frac{1}{m-1}\Big{)}}\Big{(}\frac{1}{m|\gamma|}\Big{)}^{\Big{(}\frac{1}{m-1}\Big{)}}\\\
\end{split}$ (16b) $\displaystyle\begin{split}&\text{Condition 3: for
}m<0\text{ and }\gamma>0\\\
&\bar{n_{i}}\geq(-1)^{\Big{(}\frac{1}{-|m|-1}\Big{)}}\Big{(}\frac{1}{|m|\gamma}\Big{)}^{\Big{(}\frac{1}{-|m|-1}\Big{)}}\\\
\end{split}$ (16c)

From Equation 16a, it is clear that $\bar{n_{i}}$ have a non-zero lower bound
at the ground state energy $\epsilon_{i}=0$ as
$\bar{n_{0}}^{\text{min}}=\exp(-\alpha)$. As $\bar{n_{i}}$ is a real number,
both sides of all sub-equations in 16 has to be real. Using this concept, one
can find that in the Equation 16b, negative values of $\gamma$ with $m>0$ are
allowed only for $m=2$ case. As a result, $n_{c}$ in Equation 16b becomes
negative, which indicates that there are no upper limit in the occupation
numbers and FEP does not apply here. From Equation 16c, one can find that
negative values of $m$ are not possible to find in a realistic probability
distribution function with $\gamma>0$; as they will produce imaginary numbers.
Combining all sub-equations from Equation 12 and 16, one can state that
negative or fractional ($<1$) values of $m$ are not allowed in CFES.

Performing the same analysis as in Equation 14 at $m=2$ and $\gamma<0$
(labeled as ’$r$’ and ’$-$’, for ’real’ and ’negative’, respectively) case,
one can write,

$\gamma_{r}^{-}\geq-\frac{1}{m}$ (17)

Figure 1: Population distribution function corresponding to CFES model at
various $m$. $\gamma_{r}^{+}=1/m$ and $\gamma_{r}^{-}=-1/m$. At m =1, system
follows MB statistics and at m=2 system follows FD and BE statistics. Data is
produced using Scilab-6.1.1 sci (v611) polynomial solver.

Equation 17 can be related to the BE statistics at $m=2$ at the lower bound of
$\gamma_{r}^{-}$. Therefore, any system that has a negative $\gamma$ value and
obeys CFES can be termed as boson-like systems. Boson-like systems are quite
different from fermion-like systems, as all boson-like systems have a single
$m$ value, whereas fermion-like systems have $\infty\geq m\geq 2$.

In this section, it is shown how the properties of both FD and BE statistics
can be generated from the CFES model. From Equation 15 and Equation 17, one
can state that for any real systems obeying CFES model with
$g^{\text{min}}=1$, FD and BE statistics are the two limits of the particle
distribution. This statement can be clearly understood by observing Figure 1.
The choice of $m$ determines how closely the system follows the MB statistics
at low energy. From Equation 15 and 13, it is evident that at large values of
$m\to\infty$, $\gamma_{r}^{+}\sim 0$ and $n_{c}\to 1$. As $n_{i}<<g_{i}$,
Equation 6 will be reduced to the traditional MB distribution at $m\to\infty$.

The interesting outcome of the CFES is the presence of the upper bound in
$\bar{n}_{i}$ for fermion-like systems, which was not an initial criterion for
the derivation in section III. In the CFES model, the use of the MaxEnt method
leads to a realistic probability function that follows FEP without considering
any arguments from the quantum mechanics, other than the concept of DI in the
form of a constraint in Equation 4. Drawing parallels with HW or AP statistics
Wu (1994); Polychronakos (1996a, b), one can compare $(1/n_{c})$ as the single
level statistical interaction parameter defined by Haldane Haldane (1991) in
HW statistics. It was shown earlier Bhaduri  _et al._ (2010) that at infinite
statistical correlation among the energy levels, HW statistics becomes
analogous to the classical systems. This is similar to the result we have
obtained in the CFES model as well. In this paper, however, no mutual
statistics are considered; i.e., no explicit statistical interactions between
the populations of two energy levels have been assumed a priori to derive
CFES. The exclusion at each energy level arises naturally to maintain the
constrain in Equation 4. Of course, unlike quantum systems, the applicability
of the FEP is valid in CFES only at very dilute conditions.

As the permissible ranges of (m,$\gamma$) are now determined, one can now
comment on the properties of the weights in Equation 7 and 8. Boson-like
systems are clearly free from any possibility of negative weights as
$\gamma<0$ and $m>0$. For fermion-like systems at $m>0$ and $\gamma>0$, the
positive-negative combinations of the weights cancel out to yield a net
positive probability because of the following reasons – $(a)$ As $m>0$, the
probability factor $\exp[-(\alpha+\beta\epsilon_{i})(jm-j+1)]$ is strictly
decreasing at $j\to\infty$, $(b)$ $C_{j}(m)$ is always positive, $(c)$
$\gamma_{r}^{+}\leq(1/m)$, which means the second weight in Equation 8,
$(-m\gamma)^{j}\to 0$, at $j\to\infty$. As a result, $\bar{n_{i}}$ in Equation
7 is always a converging positive function with ($j=even$) sums being larger
than ($j=odd$) sums within the permissible range of $m$, $\gamma$. $(d)$ The
convergence of the probability function is faster than the HW statistics
Murthy and Shankar (1999) due to the presence of the second weight
$(m\gamma)^{j}$, which is fractional within the permissible range.

###  IV.2 Scaled probability distribution

The interpretation of $S_{m}$ is an important aspect in the CFES model.
Although the constraint in Equation 4 may seem empirical, it’s use in CFES
model generates the probability distribution function corresponding to all
three fundamental statistical models. This raises the question, whether this
constraint has any physical meaning. In the earlier section, it is shown that
this constraint is related to the DI. To relate $S_{m}$ to thermodynamic
quantities, we first rewrite the final equation of CFES in terms of a scaled
particle distribution as,

$p_{i}^{\text{scl}}(\epsilon_{i},\beta)=\frac{(p_{i}^{2}/g_{i})e^{\beta\epsilon_{i}}}{\sum\limits_{i}(p_{i}^{2}/g_{i})e^{\beta\epsilon_{i}}}$
(18)

where $p_{i}^{\text{scl}}$ is the probability of finding a scaled number of
particles, $n_{i}^{\text{scl}}=n_{i}-\gamma m(n_{i}^{m}/g_{i}^{(m-1)})$, at a
certain energy level $\epsilon_{i}$; and $p_{i}=n_{i}/N$ is the actual
probability distribution of the CFES model. Details of the derivation is given
in supporting information section SI.VI. Equation 18 is an alternate version
of CFES equation which can be studied without the explicit knowledge of the
$m$, $\gamma$ factors. Using the idea of the scaled particle distribution, one
can derive the value of $S_{m}$ using Equation 4, Equation 6, and Equation 18
as (see supporting information section SI.VI),

$S_{m}=\frac{1}{m\gamma}(N-N^{\text{scl}})$ (19)

where $N^{\text{scl}}=\sum_{i}n_{i}^{\text{scl}}$ is the total scaled particle
number. $S_{m}$ thus represents the total effective particle number,
$N^{\text{eff}}=N-N^{\text{scl}}$, which is scaled by $m\gamma$ factor.
$S_{m}$, $m$, and $\gamma$ are independent quantities but together they
determine $N^{\text{eff}}$.

##  V Application of CFES to classical thermodynamics

The relationship between any statistical model and thermodynamics depends on
the equation of microstate counting. Using various forms microstate counting
formula, the application of QFES models in quantum thermodynamics has been
studied extensively in the literature Anghel (2013). However, in this paper,
the CFES model is derived from a classical framework. Naturally, the
application of CFES in classical thermodynamics is a main focus in this
section.

###  V.1 Free Energy equation

Ideally, the Gibbs equation of chemical potential for classical particles at
equilibrium ($\mu(\beta,m,\gamma)$) can be written as,

$N\mu=E+\frac{N}{\beta}-\frac{\ln(\widetilde{W})}{\beta}-\frac{\chi}{\beta}$
(20)

where $\widetilde{S}=\ln(\widetilde{W})$ is the total entropy of the system,
and $\chi/\beta$ – namely the exclusion potential – represents the excess
contribution to the free energy arising due to the dependence of $\mu$ on the
$m$ and $\gamma$ factors. Equation 20 is akin to the thermodynamic equation
used in the field of nanothermodynamics Hill (1962, 2001a, 2001b, 2001c);
Chamberlin (2015); Chamberlin  _et al._ (2021), where a subdivision potential
was used to describe the thermodynamics of a subsystem. However, origin of the
exclusion potential doesn’t lie on the segmentation of the system, rather on
the FEP maintained at each energy level. For the CFES model, it is
straightforward to show that, $\alpha=-\mu\beta$. However, definition of
$\beta$ is reserved for later. In equilibrium, $\chi(\beta,m,\gamma)$ has the
following form,

$\chi=\sum_{i}n_{i}\ln\Big{(}1-m\gamma\bar{n}_{i}^{(m-1)}\Big{)}$ (21)

The constant $\chi$ factor controls the occupation constraints discussed in
the previous section as $\\{\chi\in\mathbb{R}\\}$. For fermion-like systems,
$\chi$ is negative, and for boson-like systems, $\chi$ is positive. $\chi$ is
a function of both the distribution of the particles in the energy levels, as
well as the degeneracy factor $g_{i}$. Using the concept of the scaled
probability distribution, one can simplify $\chi$ (see supporting information
section SI.VII) as,

$\chi=\ln\widetilde{W}-\ln\widetilde{W}^{\text{scl}}$ (22)

where, $\widetilde{S}^{\text{scl}}=\ln\widetilde{W}^{\text{scl}}$ is the
cross-entropic term where $\widetilde{W}^{\text{scl}}$ is defined as,

$\widetilde{W}^{\text{scl}}=\prod_{i}\Bigg{(}\frac{g_{i}^{n_{i}}}{n_{i}!}\Bigg{)}\Bigg{(}\frac{n_{i}}{n_{i}^{\text{scl}}}\Bigg{)}^{n_{i}}$
(23)

Equation 22 shows that, $\chi$ is related to entropy and
$S^{\text{scl}}=\ln\widetilde{W}^{\text{scl}}$ is essentially a correction to
the total entropy of the system. Inserting Equation 22 in Equation 20, we get,

$N\mu=E+\frac{N}{\beta}-\frac{\widetilde{S}^{\text{eff}}}{\beta}$ (24)

where $\widetilde{S}^{\text{eff}}=2\widetilde{S}-\widetilde{S}^{\text{scl}}$
the effective entropy of the system. From Equation 23, it is evident that if
$\gamma$ is positive (i.e. fermion-like systems),
$\widetilde{S}^{\text{scl}}>\widetilde{S}$ and subsequently
$\widetilde{S}^{\text{eff}}$ will decrease as compared to $\widetilde{S}$, and
vice verse. At constant volume $v$, inverse temperature of the system is
defined as $\beta\equiv(\partial\widetilde{S}^{\text{eff}}/\partial E)_{v}$.
Interestingly, $\beta\neq\beta_{\text{MB}}$, where $\beta_{\text{MB}}$
corresponds to the inverse temperature derived form the definition in
traditional MB statistics, i.e., $\beta_{\text{MB}}\equiv(\partial S/\partial
E)_{v}$. Therefore, the definition of temperature in CFES model is distinct
from MB statistics. One can show that at a concentrated system ($n_{i}\sim
g_{i}$ or $n_{i}>>g_{i}$, where Equation 5 cannot be approximated to Equation
6) $\beta=\beta_{\text{MB}}$. The distinction arises because $\chi$ derived
from Equation 6 (dilute case) is directly related to the entropy. This results
in, $\chi\equiv f(E)$ and $(\partial\chi/\partial E)_{v}\neq 0$. Whereas,
$\chi$ derived from Equation 5 (concentrated case) will be independent of
entropy or energy as,

$\chi_{\text{conc.}}=m\gamma S_{m}=constant$ (25)

###  V.2 Effective temperature approximation

The equivalence between $\chi$ and system energy allows simplifying CFES to a
classical ideal gas scenario, which obeys slightly modified MB statistics. If
the particles are not interacting with one another, the entropy of the system
has a direct proportionality to the total energy of the system. Assuming that
$n_{i}\sim n_{i}^{\text{scl}}$ – i.e. $\chi$ is relatively small – one can
approximate the exclusion potential as,

$\frac{\chi}{\beta}\approx\lambda E$ (26)

where $\lambda\equiv f(\beta,m,\gamma)$ is a constant and independent of
energy. Substituting Equation 26 to Equation 20, an effective energy parameter
can be found as, $E^{\text{eff}}=E(1-\lambda)$. Using this effective energy in
place of the system energy, the usual Gibbs free energy equation is recovered,
where $\beta(1-\lambda)=(\partial S/\partial E)_{v}$. Here, an effective
inverse temperature, $\beta^{\text{eff}}=\beta(1-\lambda)$, replaces the
original inverse temperature ($\beta$) of the system. The advantage of
reformulating Equation 20 with effective energy is that, one can easily derive
the corresponding MB statistics in a retrospective way as,

$p_{i}\sim g_{i}e^{-\beta(1-\lambda)\epsilon_{i}}$ (27)

Equation 27 is valid only if $\lambda$ is not a function of energy. This
description of CFES is essentially the same as the ideal gas scenario in MB
statistics, albeit with an effective temperature. For fermion-like systems,
$n_{i}^{\text{scl}}<n_{i}$ from Equation 18; hence
$\widetilde{W}^{\text{scl}}>\widetilde{W}$ and $\chi,\lambda<0$. Hence, from
the effective inverse temperature relation,
$\beta^{\text{eff}}=\beta(1-\lambda)$, one can state that the effective
temperature in fermion-like systems will be lower than the system temperature.
This means, the presence of FEP in an ideal gas will decrease the effective
system temperature as compared to the original temperature, as part of the
entropy is lost to maintain the FEP. Similarly, for boson-like systems,
$\chi,\lambda>0$ and $\beta^{\text{eff}}<\beta$.

##  VI Conclusion and Outlook

In this paper, the classical analog of the FES model is investigated and
analyzed. We have shown that an additive DI parameter can be isolated from the
BE microstate formula, which linearly combine with $S_{\text{MB}}$ to generate
$S_{\text{BE}}$. Such a linear relationship between $S_{\text{BE}}$ and
$S_{\text{MB}}$ is interesting, as it suggests that BE statistics can be
generated from MB statistics via MaxEnt methods. The CFES model is derived
based on this idea. Moreover, it is shown that CFES model predicts presence of
FEP in fermion-like systems. However, it should be noted that such quantum
effects are only applicable in the CFES model when the system is sufficiently
dilute. Since we have considered a classical system, the microstate counting
formula in CFES is not changed from the MB statistics. Given the similarity of
CFES with AP statistics, it is perhaps possible to derive a microstate
counting equation that can generate similar parental equations as CFES for
hypothetical quantum systems. In this way, the current version of the CFES
model may have a quantum interpretation as well.

The constraint in Equation 4 is a semi-empirical one, as it is related to the
DI. The presence of DI in classical systems can be explained if some of the
degenerate particles of the same species can be distinguished by additional
degrees of freedom, which are not incorporated in the statistics model
Medvedev (1997). However, presence of these additional degrees of freedom
allows the degeneracy of the energy levels to vary which, in turn, affect the
entropy of the system. If such degrees of freedom are measurable and
incorporated in the model, then DI becomes the traditional binary function.
However, if such quantities are not measurable, then one has to consider that
DI is a variable quantity. More theoretical work is needed to understand the
concept of DI in classical systems.

The microcanonical CFES model shown in this paper should be easily extended to
Canonical systems and grand-Canonical systems as well. Derivation of several
thermodynamic quantities, such as specific heat capacities, virial
coefficients, thermodynamic potential etc., and their relation to the
partition function $Z_{m}$ is the next logical step to understand the relation
between thermodynamics and different parameters in CFES. The current version
of the CFES model assumes particles in an ideal gas scenario with no
interactions among each other. Further mathematical analysis is needed to
understand the effect of pairwise interactions Anghel (2008, 2012) in a CFES
model. Of course, the present form of CFES is valid for a uniform maximum
occupancy at all energy levels. One needs to investigate the properties of
CFES in case different energy levels have different $\bar{n_{c}}$.

It will be interesting to find realistic classical systems where the CFES
model is applicable. One possible candidate is the two-dimensional silica
(2D-silica) Lichtenstein  _et al._ (2012). Previously, in the equilibrium
simulation of two-dimensional silica melt using molecular dynamics Roy  _et
al._ (2018), the presence of effective temperature Roy and Heuer (2019a, b)
was observed in the probability distribution of various ring-sizes. It was
found that the probability distributions of the individual rings also show the
presence of an effective temperature. Here, the deviation in the temperature
factor was attributed to the fact that the rings are spatially correlated in
such a way, that there are certain preferences among the neighbours Roy and
Heuer (2019a). Due to these correlations, the density of states of the
individual rings are reduced, and the statistics of the system deviates from
the supposed MB statistics. It will be interesting to apply the CFES technique
in 2D-silica to understand the presence of effective temperature in terms of
FEP, and quantify the extent of correlations in terms of $m$, $\gamma$
parameters.

## Acknowledgement

The author would like to thank Prof. Pinaki Chaudhuri, Prof. M.V.N. Murthy,
Prof. R. Shankar, and Umang Dattani at IMSc., Chennai for useful discussions.

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# A Class of $(n,k,r,t)$ IS-LRCs Via Parity Check Matrix

Deep Mukhopadhyay1, Sanjit Bhowmick1, Kalyan Hansda2, Satya Bagchi1 The first
author of the paper would like to thank the Ministry of Education (MoE),
Government of India, for financial support to carry out this research. This
work is also supported by the National Board of Higher Mathematics (NBHM),
Government of India (Grant No. 02011/2/2019 NBHM (R.P) /R& D
II/1092).1Department of Mathematics, National Institute of Technology
Durgapur, Durgapur-713209, West Bengal, India. 2Department of Mathematics,
Visva-Bharati University, Santiniketan, Bolpur-731235, West Bengal, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A code is called $(n,k,r,t)$ information symbol locally repairable code (IS-
LRC) if each information coordinate can be achieved by at least $t$ disjoint
repair sets containing at most $r$ other coordinates. This letter considers a
class of $(n,k,r,t)$ IS-LRCs, where each repair set contains exactly one
parity coordinate. We explore the systematic code in terms of the standard
parity check matrix. First, we propose some structural features of the parity
check matrix by showing a connection with the membership matrix. After that,
we place parity check matrix based proof of several bounds associated with the
code. In addition, we provide two constructions of optimal parameters of
$(n,k,r,t)$ IS-LRCs with the help of two Cayley tables of a finite field.
Finally, we present a generalized result on optimal $q$-ary $(n,k,r,t)$ IS-
LRCs related to MDS codes.

###### Index Terms:

Locally repairable codes, optimal $(n,k,r,t)$ IS-LRCs, parity check matrix.

##  I Introduction

To increase data reliability, while storing data in distributed storage
systems, the method that usually employed is replication. However, in this
case, the storage cost is much higher than the data reliability. Thus, instead
of replication, coding theory (erasure coding) is then used, where at first,
the data is divided into $k$ parts and stored into $n$ different data nodes.
It is efficient to overcome in terms of both node failures and storage
overhead problems, but it suffers from substantial repair costs. For this
sense, some coding theory techniques are designed. Locally repairable code is
one of them.

A code symbol has locality $r$, if it can be fetched by at most $r$ other code
symbols. An $[n,k,d]$ linear code with $r$ locality is said to be $(n,k,r)$
locally repairable code (LRC). LRCs with $r$ locality for information symbols
is called an $(n,k,r)$ information symbol locally repairable codes (IS-LRCs),
and if all symbols have locality $r$, then the formatted code is $(n,k,r)$ all
symbol locally repairable codes (AS-LRCs).

Gopalan et al. [3] firstly introduced $(n,k,r)$ IS-LRCs and gave lucidly a
Singleton like upper bound on minimum distance which is given by

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

Later on, it is apparently seen that the bound (I.1) is true for $(n,k,r)$ AS-
LRCs also. Following the same approach of Gopalan et al. [3], Forbes and
Yekhanin proposed locality and proved the bound (I.1) for nonlinear codes in
[2]. A family of optimal $(n,k,r)$ LRCs against the bound (I.1) was
constructed by Tamo and Barg [11]. Hao et al. [5] overviewed $(n,k,r)$ LRCs
via parity check matrix and gave different possible classes of optimal binary
$(n,k,r)$ LRCs.

In an LRC, the set of other symbols that help to reconstruct a particular
symbol is called the repair set or recovery set of that particular symbol. For
an $(n,k,r)$ LRC, the number of repair sets for a code symbol is one.
Subsequently, this notion is generalized by choosing $t$ repair sets for a
code symbol, which is the availability of an LRC. The notion of availability
was initiated by Rawat et al. [9].

######  Definition I.1.

[4, 9] A linear code is said to be $(n,k,r,t)$ LRC if it holds the following
properties:

  1. 1.

For $i$th code symbol, there exist $t$ codewords $c_{1},\dots,c_{t}$ of
$\mathcal{C}^{\perp}$ such that each $c_{j}$ covers the $i$th symbol.

  2. 2.

$|Supp(c_{j})|\leq r+1$, for all $j\in[t]:=\\{1,2,\dots,t\\}$.

  3. 3.

$Supp(c_{j})\cap Supp(c_{l})=\\{i\\}$, for all $j\neq l$ and $j,l\in[t]$.

Rawat et al. [9] also originated a restricted class of $(n,k,r,t)$ LRCs, where
there is exactly one parity symbol in each repair set and placed an upper
bound on the minimum distance, which applies to both AS and IS, is given by

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

Design of the membership matrix [9] is an important tool which helps in the
construction of the above class of $(n,k,r,t)$ LRCs. There are several
techniques for the construction of membership matrix for some locality and
availability parameters in [10]. Due to the achievability of the bound (I.2),
this special class of $(n,k,r,t)$ LRCs is mostly studied. Several
constructions on optimal $(n,k,r,t)$-LRCs against the bound (I.2) can be found
in [4], [7], [9], [13], [14] for different classes of parameters.

In support of $(n,k,r,t)$ IS-LRCs without the restriction of the parity bits
on the repair set, Wang and Zhang [15] derived

$d\leq n-k-\left\lceil\frac{t(k-1)+1}{t(r-1)+1}\right\rceil+2$ (I.3)

and established that, if $n\geq k(rt+1)$ then there are certain codes which
satisfy (I.3) with equality.

In [12], for $(n,k,r,t)$ AS-LRCs, an upper bound on the code rate and minimum
distance are developed, which are

$\frac{k}{n}\leq\frac{1}{\prod_{j=1}^{t}(1+\frac{1}{jr})},$ (I.4) $d\leq
n-\sum_{i=0}^{t}\left\lfloor\frac{k-1}{r^{i}}\right\rfloor.$ (I.5)

Codes satisfying bounds (I.3) and (I.5) with equality can be found in [15],
[16]. Several new bounds are developed for codes with strict availability in
[1], which perform tighter than (I.4). Prakash et al. [8] presented an upper
bound on the code rate of $(n,k,r,t)$ LRCs with $t$ = 2, which is

$\frac{k}{n}\leq\frac{r}{r+2}.$ (I.6)

Codes that attain the bound (I.6) was constructed in [6].

In this letter, we focus only on $(n,k,r,t)$ IS-LRCs, where each recovery set
has precisely one parity. The rest of the letter is arranged in the following
manner. In Section II, we analyze some bounds and properties of the code
through the standard parity check matrix. In Section III, two constructions of
distance optimal codes are given, which are also rate optimal. Section IV
shows a property for a distance optimal $q$-ary $(n,k,r,t)$ IS-LRCs. The
letter concludes in Section V.

##  II Parity check matrix viewpoint of $(n,k,r,t)$ IS-LRCs

This section is divided into two subsections. First, we set up some
characteristics on the parity check matrix for the code. Next, by using the
properties of a parity check matrix, some new proofs for different upper and
lower bounds are given.

###  II-A Structural features of parity check matrix

######  Theorem II.1.

Let $\mathcal{C}$ be a systematic $[n,k,d]$ linear code over a finite field
$\mathbb{F}_{q}$ with parity check matrix $H:=[P|I_{n-k}]$, then $\mathcal{C}$
forms an $(n,k,r,t)$ IS-LRC if and only if there exists a submatrix $P_{1}$ of
order $l\times k$ of $H\equiv\left[\dfrac{P_{1}}{P_{2}}\vline I_{n-k}\right]$
such that,

  * (i)

the weight of each row of $P_{1}$ is at most $r$,

  * (ii)

all the columns of $P_{1}$ has weight at least $t$, and

  * (iii)

$|\Omega_{i}\cap\Omega_{j}|\leq 1$, where $\Omega_{i}$ and $\Omega_{j}$ are
the supports of any two rows of $P_{1}$.

###### Proof.

Consider $\mathcal{C}$ be a systematic $(n,k,r,t)$ IS-LRC with the parity
check matrix $H$. Suppose the encoding function of the code $\mathcal{C}$ is

$(m_{1},\dots,m_{k})\mapsto(m_{1},\dots,m_{k},p_{1},\dots,p_{n-k})$

where $m_{1},\dots,m_{k}$ are information bits and $p_{1},\dots,p_{n-k}$ are
parities, each of which is linear combination of some $m_{j}$’s. Let
$\textbf{x}=(m_{1},\dots,m_{k},p_{2},\dots,p_{n-k})$ be a codeword in
$\mathcal{C}$. Now for every information symbol $m_{i}$ of x, there is at
least $t$ repair sets $\Gamma_{1}(i),\dots,\Gamma_{t}(i)$ (where
$\Gamma_{j}(i)$ is the $j$th repair set of the $i$th information coordinate)
of cardinality at most $r$ and each of them contains exactly one parity. Thus,
to ensure $t$ availability of $m_{i}$, $m_{i}$ involves in at least $t$
parities. Among these $t$ parities, each $p_{i}$ is a linear combination of at
most $r$ information bits including $m_{i}$ as $m_{i}$ has locality $r$.
Similarly, by considering all information symbols, let $l$ be the number of
such distinct parities. Consequently, there are $l$ rows of $P$ in $H$ those
who forms $P_{1}$ and due to the above combination of information bits in the
parities, each row of $P_{1}$ has weight at most $r$ and weight of each column
of $P_{1}$ is at least $t$. Let $\Omega_{i}$ and $\Omega_{j}$ are the supports
of any two rows of $P_{1}$ and also consider $\Omega_{i}\cap\Omega_{j}$ has at
least two points. Choose any one of them, now it must corresponds to an
information symbol $m_{i}$ and parallelly $\Omega_{i}$ and $\Omega_{j}$
resemble the positions of the information symbols of two recovery sets of
$m_{i}$. As $\Omega_{i}\cap\Omega_{j}$ has at least two points, the
corresponding two recovery sets of $m_{i}$ intersect in at least one element.
Hence it violates the fact that $m_{i}$ has disjoint recovery sets. So we get
our result.

Conversely, a parity check matrix $H$ is given for a linear code, with a
submatrix $P_{1}$ that satisfies the given three properties. Now the rows of
$P_{1}$, including the $I_{n-k}$ part in $H$ produces $l$ codewords of
$\mathcal{C}^{\perp}$ and hence it can be seen that the three properties of
$P_{1}$ help the code $\mathcal{C}$ to match those three properties of
$(n,k,r,t)$ IS-LRC, which is given in Definition I.1. ∎

######  Theorem II.2.

For each of the $m$ distinct local groups (subsets) $S_{1},S_{2},\dots,S_{m}$
of the form $\Gamma_{j}(i)\cup\\{i\\}$, there is a row in $P_{1}$ whose
support set is equal to $S_{i}\cap[k]$ for some $i\in[m]$ and vice-versa.

###### Proof.

$\Gamma_{j}(i)$ contains the positions of at most $r-1$ information symbols
(excluding the single parity bit) that help to construct the $i$th symbol.
Thus, the corresponding parity must be a combination of those $r-1$
information symbols as well as the $i$th information symbol. Hence we can say
that the rows of $P_{1}$ are non zero in those $r$ positions indexed by the
set $\left\\{\Gamma_{j}(i)\cup\\{i\\}\right\\}\cap[k]$. Similarly, when a
codeword is multiplied with $H$, each row of $P_{1}$ produces a locality
relation which helps to recover a code symbol and hence there is a local group
consisting the support of the rows of $P_{1}$. ∎

######  Corollary II.1.

The number of rows in $P_{1}$ is equal to the number of local groups
(subsets), i.e., $l=m$. Again, we know
$m\geq\left\lceil\frac{kt}{r}\right\rceil$ from [9, Lemma 1], so
$l\geq\left\lceil\frac{kt}{r}\right\rceil$.

######  Corollary II.2.

By choosing a suitable order of the local groups (subsets), for a binary
$(n,k,r,t)$ IS-LRC, the transpose of $P_{1}$ is equal to the membership matrix
$R$.

###  II-B Review of several bounds using parity check matrix

######  Theorem II.3.

For an $(n,k,r,t)$ IS-LRC, the minimum distance (Hamming) $d$ satisfies

$t+1\leq d\leq n-k-\left\lceil\frac{kt}{r}\right\rceil+t+1.$

###### Proof.

For $t+1\leq d$, we show that any $t$ columns of
$H\equiv\left[\dfrac{P_{1}}{P_{2}}\vline I_{n-k}\right]$ are linearly
independent. Since supports of any two rows of $P_{1}$ intersect in at most
one position, so it implies that the supports of any two columns of $P_{1}$
also intersect in at most one position. Again weight of any column of $P_{1}$
is at least $t$. Now if $w$ is the weight of any nonzero linear combination of
any $t$ columns of $P_{1}$, then $w\geq t\cdot t-2\cdot{t\choose 2}=t\neq 0.$
Hence any $t$ columns of $\dfrac{P_{1}}{P_{2}}$ are linearly independent as
any $t$ columns of $P_{1}$ are linearly independent. Likewise, if we choose
any $h~{}(<t)$ columns from $\dfrac{P_{1}}{P_{2}}$ and rest any $t-h$ columns
from $I_{n-k}$, then by applying the same approach we can say these $t$
columns are linearly independent. Hence we have our result.

Now for the second part, as the availability of the code is $t$, thus there is
an information bit which has exactly $t$ repair sets and hence there is a
column of $P_{1}$ whose weight is exactly $t$. Let it be the $j$th column.
Therefore, in $P_{1}$, the $j$th column has exactly $l-t$ zeros. Accordingly
the $j$th column has at most $n-k-l+t$ nonzeros in $\dfrac{P_{1}}{P_{2}}$.
Henceforth, due to $I_{n-k}$, there always exist $n-k-l+t+1$ linearly
dependent columns in $H$. Consequently,

$d\leq n-k-l+t+1\leq n-k-\left\lceil\frac{kt}{r}\right\rceil+t+1.$

The second inequality holds due to Corollary II.1. This completes the proof. ∎

######  Theorem II.4.

For any $(n,k,r,t)$ IS-LRC,

$n\geq k+\left\lceil\frac{kt}{r}\right\rceil.$

###### Proof.

By using the fact that the rows of $P_{1}$ must lie in $H$ and since
$l\geq\left\lceil\frac{kt}{r}\right\rceil$, so accordingly it follows. ∎

######  Corollary II.3.

The code rate of $(n,k,r,t)$ IS-LRC is upper bounded by

$\frac{k}{n}\leq\left\\{\begin{array}[]{ll}\dfrac{k}{k+\left\lceil\frac{kt}{r}\right\rceil},&\text{
if~{}~{}}r\nmid kt;\\\ \dfrac{r}{r+t},&\text{ if~{}~{}}r\mid
kt.\end{array}\right.$

######  Remark II.1.

It is noted that comparatively, the code rate is higher when $r\mid kt$.
Furthermore, an $(k+\frac{kt}{r},k,r,t)$ IS-LRC, where $r\mid kt$, is always
distance optimal and has the highest code rate among all possible values of
the parameters of $(n,k,r,t)$ IS-LRCs.

##  III Construction of binary optimal $(n,k,r,t)$ IS-LRC

This section aims to present a construction of two optimal parameters of
$(n,k,r,t)$ IS-LRC. The construction depends on the formation of the $P_{1}$
matrix of Theorem II.1 with the help of two different Cayley tables of a
finite field.

###  III-A Construction 1

Let $r=t=p^{m}$, where $p$ is prime and $m\in\mathbb{N}$ and
${\mathbb{F}_{p^{m}}}$ is a field of order $p^{m}$. Consider the Cayley table
of $({\mathbb{F}_{p^{m}}},+)$. We can assume it to be a square matrix
$T^{+}=(a_{ij})$ of order $p^{m}$. Using the matrix $T^{+}=(a_{ij})$, we built
a square matrix $M_{c}$ of order $p^{m}$ for each $c\in{\mathbb{F}_{p^{m}}}$
in the following way

$M_{c}=(b_{ij})=\left\\{\begin{array}[]{ll}b_{ij}=1,&\text{if}~{}~{}a_{ij}=c~{}~{}\text{in}~{}~{}T^{+},\\\
b_{ij}=0,&\text{otherwise}.\end{array}\right.$

Likewise by considering the Cayley table of $({\mathbb{F}_{p^{m}}},\star)$, we
can again imagine it as a square matrix $T^{\star}$ of order $p^{m}$, as we
include $0$ in the Cayley table. Now directly place the matrix $M_{c}$ where
there is a $c$ in $T^{\star}$. It forms a square matrix of order $p^{2m}$
which is the desired matrix $P_{1}$.

######  Remark III.1.

If $H=[P_{1}|I_{p^{2m}}]$, where $P_{1}$ is the matrix constructed above, then
$H$ forms a parity check matrix of an optimal $(2p^{2m},p^{2m},p^{m},p^{m})$
IS-LRC with distance $p^{m}+1$ and rate $\frac{1}{2}$.

######  Example III.1.

Let $r=t=5$, we construct $(50,25,5,5)$ IS-LRC using Construction 1. First of
all, consider the field $\mathbb{F}_{5}$ with $5$ elements and its two Cayley
tables are the following,

+ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
0 | 0 | 1 | 2 | 3 | 4
1 | 1 | 2 | 3 | 4 | 0
2 | 2 | 3 | 4 | 0 | 1
3 | 3 | 4 | 0 | 1 | 2
4 | 4 | 0 | 1 | 2 | 3
TABLE I:

$\star$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | 0
1 | 0 | 1 | 2 | 3 | 4
2 | 0 | 2 | 4 | 1 | 3
3 | 0 | 3 | 1 | 4 | 2
4 | 0 | 4 | 3 | 2 | 1
TABLE II:

From Table I, we can obtain

$\displaystyle M_{0}=\begin{bmatrix}1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&1&0\\\
0&0&1&0&0\\\ 0&1&0&0&0\\\ \end{bmatrix},\quad
M_{1}=\begin{bmatrix}0&1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&1&0\\\
0&0&1&0&0\\\ \end{bmatrix},$ $\displaystyle M_{2}=\begin{bmatrix}0&0&1&0&0\\\
0&1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&1&0\\\ \end{bmatrix},\quad
M_{3}=\begin{bmatrix}0&0&0&1&0\\\ 0&0&1&0&0\\\ 0&1&0&0&0\\\ 1&0&0&0&0\\\
0&0&0&0&1\\\ \end{bmatrix},$ $\displaystyle M_{4}=\begin{bmatrix}0&0&0&0&1\\\
0&0&0&1&0\\\ 0&0&1&0&0\\\ 0&1&0&0&0\\\ 1&0&0&0&0\\\ \end{bmatrix},\quad
P_{1}=\begin{bmatrix}M_{0}&M_{0}&M_{0}&M_{0}&M_{0}\\\
M_{0}&M_{1}&M_{2}&M_{3}&M_{4}\\\ M_{0}&M_{2}&M_{4}&M_{1}&M_{3}\\\
M_{0}&M_{3}&M_{1}&M_{4}&M_{2}\\\ M_{0}&M_{4}&M_{3}&M_{2}&M_{1}\\\
\end{bmatrix}.$

We then place $M_{0}$, $M_{1}$, $\dots$, $M_{4}$ accordingly using Table II.
This gives our wished $P_{1}$ matrix of order $25\times 25$. Now, If we
arrange the parity check matrix as $H=[P_{1}|I_{25}]$, then $H$ forms an
optimal $(50,25,5,5)$ IS-LRC.

###  III-B Construction 2

Suppose $r=t=p^{m}+1$, where $p$ is prime and $m\in\mathbb{N}.$ First, we take
the matrix $P_{1}$ from Construction 1. Then we build $P^{\prime}_{1}$ using
$P_{1}$ in the following manner, which gives a square matrix of order
$(p^{2m}+p^{m}+1)$.

$P^{\prime}_{1}=\left[\begin{array}[]{cc}\lx@intercol\hfil\textbf{1}_{p^{m}+1}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\textbf{0}_{p^{2m}}\hfil\lx@intercol\\\
\hline\cr&\vrule\lx@intercol\hfil\hfil\lx@intercol\\\
I_{p^{m}+1}\otimes\textbf{1}_{p^{m}}^{T}&\vrule\lx@intercol\hfil
I_{p^{m}}\otimes\textbf{1}_{p^{m}}\hfil\lx@intercol\\\
\cline{2-2}\cr&\vrule\lx@intercol\hfil
P_{1}\hfil\lx@intercol\end{array}\right],$

where $X\otimes Y$ means the Kronecker product of two matrices $X$ and $Y$;
$\boldsymbol{v}_{g}$ denotes $g$-tuple row vector whose all entries are
$\boldsymbol{v}$ and $\boldsymbol{v}^{T}_{g}$ is the transpose of
$\boldsymbol{v}_{g}$.

######  Remark III.2.

If $H=[P^{\prime}_{1}|I_{p^{2m}+p^{m}+1}]$, where $P^{\prime}_{1}$ is the
above made matrix, then $H$ becomes a parity check matrix of an optimal
$\left(2(p^{2m}+p^{m}+1),p^{2m}+p^{m}+1,p^{m}+1,p^{m}+1\right)$ IS-LRC with
distance $p^{m}+2$ and rate $\frac{1}{2}$.

######  Example III.2.

Let $r=t=6$, we construct $(62,31,6,6)$ IS-LRC using construction 2. First
consider the square matrix $P_{1}$ of order 25 of Example III.1. Accordingly
the required matrix $P^{\prime}_{1}$ is

$P^{\prime}_{1}=\left[\begin{array}[]{cc}\lx@intercol\hfil\textbf{1}_{6}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\textbf{0}_{25}\hfil\lx@intercol\\\
\hline\cr&\vrule\lx@intercol\hfil\hfil\lx@intercol\\\
I_{6}\otimes\textbf{1}_{5}^{T}&\vrule\lx@intercol\hfil
I_{5}\otimes\textbf{1}_{5}\hfil\lx@intercol\\\
\cline{2-2}\cr&\vrule\lx@intercol\hfil
<EMAIL_ADDRESS>
Now if we consider $H=[P^{\prime}_{1}|I_{31}]$, then $H$ forms a parity check
matrix of an optimal $(62,31,6,6)$ IS-LRC.

######  Remark III.3.

The above two optimal parameters of $(n,k,r,t)$ IS-LRC can be constructed for
any finite field $\mathbb{F}_{q}$, in that case, we just need to replace the
$1$’s of $P_{1}$ and $P^{\prime}_{1}$ by any nonzero element of
$\mathbb{F}_{q}$.

##  IV property of $q$-ary optimal $(n,k,r,t)$ IS-LRC

This section proposes a generalised property of a $q$-ary optimal $(n,k,r,t)$
IS-LRC. Furthermore, the result reduces to [5, Theorem 1] for $t=1$.

######  Theorem IV.1.

Suppose $\mathcal{C}$ is an $(n,~{}k,~{}r,~{}t)$ IS-LRC having distance
$d=n-k-\left\lceil\frac{kt}{r}\right\rceil+t+1$ and
$H\equiv\left[\dfrac{P_{1}}{P_{2}}\vline I_{n-k}\right]$ is its parity check
matrix. Let $H_{1}$ and $H_{2}$ be two $m_{1}\times n_{1}$ and $m_{2}\times
n_{2}$ submatrices acquired from $H$ by deleting any fixed
$\left\lceil\frac{kt}{r}\right\rceil-t$ and
$\left\lceil\frac{kt}{r}\right\rceil-t-1$ rows among the first $l$ rows and
all the columns whose coordinates are covered by the supports of those
$\left\lceil\frac{kt}{r}\right\rceil-t$ and
$\left\lceil\frac{kt}{r}\right\rceil-t-1$ rows respectively. Then $H_{1}$ and
$H_{2}$ have full row rank with $m_{1}=d-1$ and $m_{2}=d$. Moreover, the
linear codes $\mathcal{C}_{1}\equiv[n_{1},k_{1},d_{1}]$ and
$\mathcal{C}_{2}\equiv[n_{2},k_{2},d_{2}]$ with $H_{1}$ and $H_{2}$ as parity
check matrices, accordingly become $q$-ary MDS code and almost MDS code.

###### Proof.

First of all, it is easy to verify that

$m_{1}=n-k-\left\lceil\frac{kt}{r}\right\rceil+t=d-1,~{}\text{and}$
$m_{2}=n-k-\left\lceil\frac{kt}{r}\right\rceil+t+1=d.$

Let $\gamma_{1}$ and $\gamma_{2}$ be the number of columns covered by the
omitted $\left\lceil\frac{kt}{r}\right\rceil-t$ and
$\left\lceil\frac{kt}{r}\right\rceil-t-1$ rows successively. Then
$\gamma_{1}\leq k+\left\lceil\frac{kt}{r}\right\rceil-t$ and $\gamma_{2}\leq
k+\left\lceil\frac{kt}{r}\right\rceil-t-1$. Thus, we have

$n_{1}\geq
n-(k+\left\lceil\frac{kt}{r}\right\rceil-t)=n-k-\left\lceil\frac{kt}{r}\right\rceil+t=m_{1},$
$n_{2}\geq
n-(k+\left\lceil\frac{kt}{r}\right\rceil-t-1)=n-k-\left\lceil\frac{kt}{r}\right\rceil+t+1=m_{2}.$

Now it follows that, $H_{1}\equiv\left[\dfrac{X_{1}}{Y_{1}}\vline
I_{m_{1}}\right]$ and $H_{2}\equiv\left[\dfrac{X_{2}}{Y_{2}}\vline
I_{m_{2}}\right]$. Hence, we can say that $H_{1}$ and $H_{2}$ have full row
rank with $m_{1}=d-1$ and $m_{2}=d$.
For the later part, at first we notice that
$\left\lceil\frac{kt}{r}\right\rceil-t-1$ is non-negative, as
$\left\lceil\frac{kt}{r}\right\rceil>t$ and if
$\left\lceil\frac{kt}{r}\right\rceil=t+1$, then $H_{2}=H$. Here we use the
fact that $d\leq d_{2}\leq d_{1}$, since the above entries of those $n_{2}$ or
$n_{1}$ columns of $H$, which resemble to the columns of $H_{2}$ or $H_{1}$,
are all zeros. Again from the Singleton bound for the code $\mathcal{C}_{1}$,
$d_{1}\leq m_{1}+1=d$ holds.
Combining, we get $d_{2}=d_{1}=d$. Therefore, it is easy to observe that
$\mathcal{C}_{1}$ forms a $q$-ary MDS code and $\mathcal{C}_{2}$ becomes a
$q$-ary almost MDS code, as the Singleton defect of $\mathcal{C}_{2}$ is $1$.
∎

##  V Conclusion

We outlook $(n,k,r,t)$ IS-LRC through the standard parity check matrix point
of view. We provide the structural form of the parity check matrix for an
$(n,k,r,t)$ IS-LRC. We disclose how the membership matrix or the local groups
(subsets) and the parity check matrix are interlinked. Besides that, by
utilizing the parity check matrix tools, we prove some bounds. Additionally,
we place precise construction of two optimal parameters of the code using the
$P_{1}$ matrix which performs well for non-binary cases also. Finally, we give
a generalized result on optimal $(n,k,r,t)$ IS-LRC.

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