# Stochastic approach to study the properties of the complex patterns observed
in cytokine and T-cells interaction process

Moirangthem Shubhakanta Singh1, Mairembam Kelvin Singh1 and R.K. Brojen Singh2
<EMAIL_ADDRESS>1Department of Physics, Manipur University, Canchipur-795003,
Imphal, Manipur, India.
2School of Computational $\&$ Integrative Sciences, Jawaharlal Nehru
University, New Delhi-110067, India.

###### Abstract

Patterns in complex systems store hidden information of the system which is
needed to be explored. We present a simple model of cytokine and T-cells
interaction and studied the model within stochastic framework by constructing
Master equation of the system and solving it. The solved probability
distribution function of the model show classical Poisson pattern in the large
population limit $M,Z\rightarrow large$ indicating the system has the tendency
to attract a large number small-scale random processes of the cytokine
population towards the basin of attraction of the system by segregating from
nonrandom processes. Further, in the large $\langle Z\rangle$ limit, the
pattern transform to classical Normal pattern, where, uncorrelated small-scale
fluctuations are wiped out to form a regular but memoryless spatiotemporal
aggregated pattern. The estimated noise using Fano factor shows clearly that
the cytokine dynamics is noise induced process driving the system far away
from equilibrium.

Keywords: Stochastic process; Master equation; Cytokine, T-cells; Poisson
Pattern; Normal pattern.

##  I INTRODUCTION

Patterns are inherent fundamental building blocks of complex systems. Each
pattern is created by a self-organized certain population of interacting
components of the system resulting aggregation having certain form of
regularity, symmetry and structure which is the source of certain form of
information and can participate in the system’s collective decision making
process Julia . However, the whole process of each pattern may not be the
exact sum of the individual activities of the constituting components which
are nonlinear in characters, sensitive to the surrounding internal and
external fluctuations and initial conditions, dissipative in nature, and the
dynamics are generally strongly coupled Newman . Each pattern is generally
open in nature which allow exchange of energy, accessible states are far away
from equilibrium, and the information flow is permitted across the boundary
Werner . The interaction among the components in a pattern could be of diverse
form which drives the dynamics of individual components quite complicated in
nature which could be the origin of complicated properties of each pattern
Karaca . Technically, the degree of pattern is generally characterized by
complexity Cosma . The complexity of the pattern can be measured by different
techniques, some of which are information theoretical approach Hale ,
statistical complexity measures Crutchfield , Kolmogorov complexity Kolmogorov
, Lempel-Ziv complexity Aboy  etc. The study of the properties of these
patterns may provide the information of how complex systems work at
fundamental level.

The observation of the emergent properties of the patterns in the systems can
be done mathematically using traditional central limit theorem by calculating
large number of outcomes perceived from hidden processes of the distribution
of the interacting random variables $\\{x_{i}\\};i=1,2,...,N$ corresponding to
the aggregated $N$ components of the pattern Jaynes ; Feller1 ; Feller2 .
Depending on the observed properties of the probability distribution, the
pattern can be in one of the classical patterns, namely, Norma, Poisson, Power
law patterns etc Frank  and these patterns, in most of the biological systems,
are dynamic in nature depending on the fluctuations driving them and limiting
distribution conditions Wootton . If the spatiotemporal events are scattered
randomly throughout the system, the probability distribution function becomes
Poisson distribution and the pattern corresponds to classical Poisson pattern
Feller1 ; Feller2 ; Frank . Since Poisson pattern is generally generated from
the random processes smaller-scales, the basin of attraction of such patterns
have aggregation of large scale small-scale random processes by segregating
from nonrandom processes Kleiber . On the other hand, classical Normal pattern
stores the information of mean $\langle x\rangle$ and variance $\sigma^{2}$ in
the aggregation process of the components in the pattern $\mathcal{P}[\langle
x\rangle,\sigma^{2}]$, where, sum of the uncorrelated small-scaled
fluctuations of the components during the aggregation process cancels out to
attract towards a certain form of spatiotemporal regular aggregated pattern
Jaynes ; Frank . Hence, the dynamics of the aggregated components in such
classical Normal patterns follow Brownian motion Uhlenbeck . Further, if the
aggregated pattern keeps only the information of mean, the pattern falls in
exponential pattern $\displaystyle\mathcal{P}[x]=\frac{1}{\langle
x\rangle}e^{-\frac{x}{\langle x\rangle}}$, where, the pattern is contributed
from the waiting time characterization of random events and the distribution
has memoryless property Frank . If the aggregation pattern preserves the
information of geometric mean, the pattern follows classical Power law pattern
which can have fractal patterns which have self-similar process
$\displaystyle\frac{\mathcal{P}[\alpha
x]}{\mathcal{P}[x]}=\alpha^{h}=F(\alpha)$, where, $h$ is self-similarity
exponent Mandelbrot ; Barabasi . From these probability distribution
functions, one can measure hidden information stored in the respective
patterns by calculating Shannon entropy,
$\displaystyle\mathcal{H}=-\sum_{i}\mathcal{P}_{i}[x]ln\mathcal{P}_{i}[x]$
Shannon .

The time evolution of the components in the patterns/systems is in general
fluctuation driven stochastic process which can be well described in a
probabilistic manner Kolmogorov . The inherent fluctuations in the dynamics of
the constituting variables arise from the internal (due to inter-molecular
interaction) and external (due to fluctuations driven by surrounding the
system i.e. temperature fluctuation, environmental fluctuations etc)
fluctuations and have variety of roles in regulating the system Kampen ; Swain
. The trajectory of the variables of the pattern/system and their realizations
can be obtained using the Master equation, which is the discrete form of the
Chapman-Kolmogorov equation Kolmogorov ; Gardiner , constructed from the
information of the transitions of the states due to inter-variables
interaction Kampen ; Papoulis ; McQuarrie . On the other hand,
computationally, the trajectory of each variables in the system can be traced
using Monte carlo like stochastic simulation algorithm (SSA) due to Gillespie
Gillespie . The stochastic approach has wide variety of applications, ranging
from microscopic biomolecules Bressloff  to macroscopic ecology Russell  to
cosmology Kodama , and then to finance Wolfgang ; Michael , cryptography
Jonathan  etc.

Cytokines are certain category of diverse family of small proteins or
glycoproteins that are secreted by specific cells of immune system and are
produced throughout the body by cells of diverse embryological origin Jodie .
Cytokines provide the mechanism by which lymphocytes, inflammatory cells and
haematopoietic cells can communicate with each other to mount and coordinate
an effective immune response Dimitriou . Interleukins (ILs) are the Cytokines
secreted by leukocytes and affect the cellular responses of leucocytes Delgado
. Cytokines are pleiotropic which means that they have multiple effects and
can increase both proliferation and death of a certain cell type Hart . On the
other hand, a pleiotropic cytokine that helps in the growth of T-cells is
Interleukin-2 (IL-2) and intervenes in the activation-induced cell death
(AICD) and causes differentiation of regulatory T-cells Wei . Hence,
interleukin-2 (IL-2) plays a major role in the proliferation of activated T
cells Crawley . These T-cells can identify and eliminate invading pathogens,
and also attack human tissues Booki . Further, a successful activation and
differentiation of T-cells is aided by the presence of cytokines and specify
the type of response that the T-cells give to a foreign antigen Anaya . Now,
it has been shown that, a particular type of T-cells known as the CD4+ T-cell
produces cytokine IL-2, which is responsible for both the proliferation and
death of T-cells Refaeli ; Li . This cytokine can also be produced by T-cells,
the T-cells proliferate to the cytokine again and are removed, they can also
uptake and secrete the cytokine and the cytokine can then degrade itself Hart
. However, in the complicated regulating mechanism of cytokine and T-cells
interaction what are dynamical properties of the system, how does the dynamics
affect the patterns in the interacting system and how can we interrelate the
emergent patterns to phenotyical behavior are still not answered fully.
Further, what are the roles of the fluctuations in system’s dynamics are still
open questions.

In this work, we present a simple model of cytokine and T-cells interaction
model. We then analyze the model using stochastic approach to understand how
the probability distribution function of the cytokines aggregate to form
different patterns at various situations. The results also explain the role of
noise in regulating the system. Then some conclusions are made based on the
results obtained.

##  II Methodology

###  II.1 Master Equation Formalism

The dynamics of stochastic particles can generally be described by The
Chapman-Kolmogorov equation which is based on the probabilistic description of
state transitions during the course of the particle’s motion Kolmogorov ;
Gardiner . This Chapman-Kolmogorov equation in integral form Gardiner  can
relate the joint probability distributions of different sets of variables
involved in a stochastic process where the transition of states of the system
obey Markov process. If one consider a ${}^{\prime}n^{\prime}$-step transition
probability of the stochastic process, then the jump probability
$P(x_{n},t_{n}|x_{1},t_{1})$ from the state $(x_{1},t_{1})$ to the state
$(x_{n},t_{n})$ is given by,

$\displaystyle P(x_{n},t_{n}|x_{1},t_{1})=\int...\int
dx_{n-1}...dx_{3}dx_{2}P(x_{n},t_{n}|x_{n-1},t_{n-1}|x_{n-2},t_{n-2})...P(x_{2},t_{2}|x_{1},t_{1})$
(1)

The standard way of deriving Chapman-Kolmogorov equation in differential form
is to calculate the time evolution of the expectation value of a function
$f(x)$ defined in the $n-$dimensional system given by,

$\displaystyle\frac{\partial}{\partial t}\langle f(x)\rangle=\int
dxf(x)\frac{\partial}{\partial t}P(x,t|x^{\prime},t^{\prime})$ (2)

Now, using the definition of differentiation to the partial derivative of the
probability distribution function $P(x,t|x^{\prime},t^{\prime})$, expanding
$f(x)$ with Taylor series expansion and then rearranging the terms by keeping
terms upto second order derivative, one can get the following Chapman-
Kolmogorov equation Kolmogorov ; Gardiner ,

$\displaystyle\frac{\partial}{\partial t}P(x,t|x^{\prime},t^{\prime})$
$\displaystyle=$ $\displaystyle-\sum_{i}\frac{\partial}{\partial
u_{i}}\left[A_{i}(u,t)P(u,t|x^{\prime},t^{\prime})\right]+\frac{1}{2}\sum_{i,j}\frac{\partial^{2}}{\partial
u_{i}\partial u_{j}}\left[B_{ij}(u,t)P(u,t|x^{\prime},t^{\prime})\right]$ (3)
$\displaystyle+\int
dx\left[W(u|x,t)P(x,t|x^{\prime},t^{\prime})-W(x|u,t)P(u,t|x^{\prime},t^{\prime})\right]$

where, $\left\\{W\right\\}$ are Wiener functions which are transition
probabilities of the states and $A_{i}$ and $B_{ij}$ are the functions which
depend the properties of the system. The continuum approximation of the
equation (3), which is contributed from the first two terms in the right hand
side, provides Fokker-Planck-Kolmogorov equation Gardiner . The Master
equation is generally the discrete form of the Chapman-Kolmogorov equation
Kolmogorov ; Gardiner ; Kampen  which is contributed from the jump process of
the particle dynamics (keeping the last two terms in the right hand side),
which is given by,

$\displaystyle\frac{\partial}{\partial t}P(x,t|x^{\prime},t^{\prime})=\int
dx\left[W(u|x,t)P(x,t|x^{\prime},t^{\prime})-W(x|u,t)P(u,t|x^{\prime},t^{\prime})\right]$
(4)

This equation can also be expressed by replacing integration by summation over
configurational states indicating jump process Gardiner ; Kampen ; Bressloff
which is easier to handle and more directly related to physical systems Kampen
. Master equation can also be seen as the time evolution of the
configurational probability for Markov processes of systems that jump from one
to another state which is due to birth and death processes of the particles in
the system. There are different techniques to solve Master equation, namely,
using generating function technique McQuarrie , Fourier transform technique
Kodama , Laplace transform technique McNeil , Laplace-Fourier transform
technique Scalas  etc. However, solving Master equation of complex systems
involving multi-variable is quite difficult except for simple systems.

###  II.2 Generating function technique to solve Master equation

The generating function (GF) technique is generally used to solve Master
equations for simple low dimensional systems McQuarrie . Brief procedure to
solve Master equation using GF is given as follows. The $n-$dimensional GF
$G(s_{1},s_{2},...,s_{n})$ is the transformation map of the probability
distribution function $P(x_{1},x_{2},...,x_{n};t)$ as defined by,

$\displaystyle
G(s_{1},s_{2},...,s_{n})=\sum_{x_{1},x_{2},...,x_{n}}\prod_{i=1}^{n}s_{i}^{x_{i}}P(x_{1},x_{2},...,x_{n};t)$
(5)

Now, multiplying the above Master equation (4) by
$\sum_{x_{1},x_{2},...,x_{n}}\prod_{i=1}^{n}s_{i}^{x_{i}}$ and substituting
equation (5), one can transform the Master equation to spatiotemporal partial
differential equation in the GF. The partial differential equation in $G$ can
be solved using the boundary condition at $t=0$,

$\displaystyle G(s_{1},s_{2},...,s_{n};0)=\prod_{i=1}^{n}s_{i}^{N_{i}^{[0]}}$
(6)

where, $\\{N_{i}^{[0]}\\};i=1,2,...,n$ is the set of initial populations of
the respective variables $x_{1},x_{2},...x_{n}$. Now, the solution $G$ is put
back to the GF (5) and equating the coefficients of the terms containing
$\prod_{i=1}^{n}s_{i}^{x_{i}}$ from both sides, one can arrive at the
functional form of $P(x_{1},x_{2},...,x_{n})$ which is the solution of Master
equation (4). However, solving Master equation for any high dimensional system
is quite difficult and is still an open question.

The generating function also allows us to determine the observables which will
describe the properties of the system, such as, the role of fluctuations in
the system and many others. The analysis of GF and probability distribution
function could reveal various other hidden patterns and their properties of
the system.

The observables of the system namely, mean, variance and other measurable
parameters can be determined using the generating function. The mean of any
variable $x_{i}$ can be obtained from the following relation,

$\displaystyle\langle x_{i}\rangle$ $\displaystyle=$
$\displaystyle\left.\frac{\partial G(s_{i},t)}{\partial
s_{i}}\right|_{s_{i}=1;i=1,2,...,n}$ (7) $\displaystyle\langle
x_{i}^{2}\rangle$ $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}G(s_{i},t)}{\partial
s_{i}^{2}}\right|_{s_{i}=1;i=1,2,...,n}+\langle x_{i}\rangle$ (8)
$\displaystyle\sigma_{i}^{2}$ $\displaystyle=$ $\displaystyle\langle
x_{i}^{2}\rangle-\langle x_{i}\rangle^{2}$ (9)

The fluctuations involved in the dynamics of the particles in the system can
be well quantified by measuring Fano factor $F$ of the variable corresponding
to the particle which is given by Fano ,

$\displaystyle F=\frac{\sigma_{i}^{2}}{\langle x_{i}\rangle}$ (10)

The Fano Factor is an important parameter which can determine the role of
fluctuations or noise in the dynamics of the particles of the system Kamil .
Depending on the values of $F$, one can determine the behavior of the particle
which undergo various processes Kamil ; Chanu .

  * •

If $F=1$, then, the process is a Poisson process and the particles in the
system undergo Brownian motion. In such situation, there is no specific role
of the fluctuations in the dynamics of the system.

  * •

If $F>1$, the process is a noise-enhanced process and is also known as super-
Poisson process. Here, the fluctuations has significant role in regulating the
dynamics of the system.

  * •

If $F<1$, the process is called a sub-Poisson process. Here, the noise has no
significance but still has a tendency to regulate the dynamics of the system.

###  II.3 Cytokines and T-cell interaction model

The proliferation of T-cells (T) involves the production of a cytokine $IL-2$
(Z), and this cytokine plays a significant role in the T-cell proliferation
and death Hart . In this simple model, if we denote the rate of proliferation
as $X$ and rate of death as $Y$, and when these two rates are functions of the
produced cytokine $Z$, we can write the concentration of the cytokine $Z$ at
any instant of time by a simple first order differential equation Hart ,

$\displaystyle\frac{dZ}{dt}=X(Z)T-Y(Z)Z$ (11)

The maximum value of $Z$ during the process can be obtained by calculating the
fixed point of the equation (11), $\frac{dZ}{dt}=0$ giving the value,
$Z^{*}=\frac{X^{*}(Z^{*})}{Y^{*}(Z^{*})}T$. This steady state population of
the cytokine $Z$ is generally independent of its initial population at time
$t=0$. However, the initial population of cytokine trigger the T-cell
proliferation by producing another type of cytokine in response Hart .
Moreover, the cytokine IL-2 can be produced from other external sources other
than T-cell proliferation. Further, T-cells can uptake and secrete the
cytokine IL-2 which are needed to be incorporated to the model.

Considering various processes involved in the proliferation of T-cells and its
death with the production of cytokine IL-2, the above model (11) can be
represented by following form of the equation Hart ,

$\displaystyle\frac{dZ}{dt}=k_{1}+k_{2}T-k_{3}TZ-k_{4}$ (12)

where $k_{1}$ is the secretion rate by an external source, $k_{2}$ is the
secretion rate by T-cells, $k_{3}$ is the rate of uptake of Z by T-cells and
$k_{4}$ is the rate of degradation of Z by itself. This equation can
translated into the following four reactions,

$\displaystyle\alpha\stackrel{{\scriptstyle\rm
k_{1}}}{{\longrightarrow}}Z;~{}~{}T\stackrel{{\scriptstyle\rm
k_{2}}}{{\longrightarrow}}Z;~{}~{}T+Z\stackrel{{\scriptstyle\rm
k_{3}}}{{\longrightarrow}}T;~{}~{}Z\stackrel{{\scriptstyle\rm
k_{4}}}{{\longrightarrow}}\phi$ (13)

Figure 1: Schematic diagram representing T-cell and cytokine IL-2 (Z)
interaction mechanism.

The first reaction represents the secretion of cytokine $Z$ by an external
source $\alpha$, the second reaction shows the secretion of cytokine $Z$ by
T-cells, the third reaction indicates uptake of cytokine $Z$ by T-cells and
the fourth reaction represents the degradation of cytokine $Z$. The schematic
diagram of these reactions is given in Figure 1.

##  III Results

###  III.1 Stochastic approach to study cytokine and T-cells interaction model

The study of the stochastic dynamics of the system can lead us to explore the
role of fluctuations and other factors which regulate the behavior of the
system. Since the patterns exhibited in the interaction of the cytokine and
T-cells at any instant of time is random and probabilistic in nature,
understanding the properties of the patterns within the stochastic framework
could be important to correlate with the exhibited phenotypic behavior. We
consider the equivalent reaction set (13) of the model equation (12) and
assume that the total population of cytokine $Z$ and T-cells is a constant,
i.e, $T+Z=constant=M$. Further, as the T-cells recruits cytokine in a quite
fast rate Rogers  and the T cells population in our immune system always tends
to an equilibrium/steady state Almeida , the population of T-cells is also
taken to be a constant as compared to cytokine population in our calculation.

Consider the transition of states given by the model reaction set (12) obey
the Markov property Jodie , which means that the future state of a system
depends only on the present state and not on any of the past states, we can
determine the transition probabilities from one state to the next for each
reaction using the Law of mass action. From the model (12), the reaction
$\alpha\stackrel{{\scriptstyle\rm k_{1}}}{{\longrightarrow}}Z$ indicates the
production of cytokine Z by T-cells, where, $\alpha$ is a constant source.
Since the firing of the reaction follow Markov process, the transition rates
of the process can be calculated using mass action law in the state change
processes and are given by, $\omega_{1}=\alpha k_{1}\Delta t+O(\Delta t^{2})$
and $\omega_{1}^{\prime}=\alpha k_{1}\Delta t+O(\Delta t^{2})$ respectively.
Next, the reaction $T\stackrel{{\scriptstyle\rm k_{2}}}{{\longrightarrow}}Z$
reveals the proliferation of T-cells to recruit cytokine Z and removal of
itself with a rate $k_{2}$. The corresponding transition rates are given by,
$\omega_{2}=k_{2}\beta\Delta t+O(\Delta t^{2})$ and
$\omega_{2}^{\prime}=k_{2}\beta\Delta t+O(\Delta t^{2})$ respectively, where,
$\beta$ is the mean value of population of T-cells as its fluctuation is
approximated to be negligible. Further, the reaction
$T+Z\stackrel{{\scriptstyle\rm k_{3}}}{{\longrightarrow}}T$ shows that the
T-cells can also uptake a cytokine Z, secrete another cytokine Z and then
remove itself. This means that the state of cytokine Z remains constant at
state Z. The transition rates of this reaction are given by,
$\omega_{3}=k_{3}\beta(Z+1)\Delta t+O(\Delta t^{2}$ and
$\omega_{3}^{\prime}=k_{3}\beta Z\Delta t+O(\Delta t^{2}$ respectively.
Lastly, the degradation of cytokine Z is indicated by the reaction
$Z\stackrel{{\scriptstyle\rm k_{4}}}{{\longrightarrow}}\phi$. The transition
rates are given by, $\omega_{4}=k_{4}(Z+1)\Delta t+O(\Delta t^{2})$ and
$\omega_{4}^{\prime}=k_{4}Z\Delta t+O(\Delta t^{2})$ respectively.

Now, according to the principle of detailed balance, the process of transition
from one state to another should be in equilibrium with the corresponding
reverse process of transition. Using this principle, we can determine the
probability, $P(T,Z;t+\Delta t)$, that the system is in the state $(T,Z)$ at
any instant of time $t+\Delta t$ provided that the system was in some other
state at time $t$. Now, using the transition rates for each reaction, we can
arrive at the following detailed balance condition,

$\displaystyle P(Z;t+\Delta t)$ $\displaystyle=$
$\displaystyle\omega_{1}P(Z-1,t)+\omega_{2}P(Z-1,t)+\omega_{3}P(Z+1,t)$ (14)
$\displaystyle+\omega_{4}P(Z+1,t)+[1-(\omega_{1}^{\prime}+\omega_{2}^{\prime}+\omega_{3}^{\prime}+\omega_{4}^{\prime})]P(Z,t)$
$\displaystyle=$ $\displaystyle\alpha k_{1}P(Z-1;t)\Delta t+\beta
k_{2}P(Z-1,t)\Delta t$ $\displaystyle+\beta k_{3}(Z+1)P(Z+1,t)\Delta
t+k_{4}(Z+1)P(Z+1,t)\Delta t$ $\displaystyle+[1-(\alpha k_{1}+\beta
k_{2}+\beta k_{3}Z+k_{4}Z)\Delta t]P(Z,t)$

Now, rearranging the above equation and using the definition of
differentiation in the limit $\Delta t\rightarrow 0$, we arrived at the
following Master equation,

$\displaystyle\frac{\partial P(Z;t)}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha k_{1}P(Z-1,t)+\beta k_{2}P(Z-1,t)+\beta
k_{3}(Z+1)P(Z+1,t)$ (15) $\displaystyle+k_{4}(Z+1)P(Z+1,t)-[\alpha k_{1}+\beta
k_{2}+\beta k_{3}Z+k_{4}Z]P(Z,t)$

This Master equation (15) can be solved using generating function technique
which we have explained in the methodology section. The generating function in
one dimension can be defined as.

$\displaystyle G(s,t)=\sum_{Z}s^{Z}P(Z,t)$ (16)

with the initial condition, $G(s,0)=S^{N}$ at $t=0$, where $N$ is the original
number of cytokine population in time $t=0$. Now, multiply equation (15) by
$s^{Z}$ on both sides, summing over Z, and after rearranging the terms, we
have,

$\displaystyle\frac{\partial G(s,t)}{\partial t}=-(\beta
k_{3}+k_{4})(s-1)\frac{\partial G(s,t)}{\partial s}+(\alpha k_{1}+\beta
k_{2})(s-1)G(s,t)$ (17)

This spatiotemporal partial differential equation (17) can be solved using
standard Lagrange’s Charpit characteristic method for partial differential
equations Delgado . Now, applying this method, the solution can be obtained
from the following set of equations,

$\displaystyle\frac{dt}{1}=\frac{ds}{(\beta k_{3}+k_{4})(s-1)}=\frac{\partial
G(S,t)}{(\alpha k_{1}+\beta k_{2})(s-1)G(s,t)}$ (18)

Now, let us consider the first equation constructed from the above set of
equations (18), $dt=\frac{ds}{(\beta k_{3}+k_{4})(s-1)}$. Integrating both
sides and rearranging the terms, we get,

$\displaystyle\lambda_{1}=e^{C_{1}}=(s-1)e^{-(\beta k_{3}+k_{4})t}$ (19)

where, $\lambda_{1}$ is a constant and $C_{1}$ is the constant of integration.
Next, we consider second equation from the set of the equations eqreflce,
given by, $\frac{\partial G(S,t)}{(\alpha k_{1}+\beta
k_{2})(s-1)G(s,t)}=\frac{ds}{(\beta k_{3}+k_{4})(s-1)}$. Now integrating and
rearranging the terms, we have,

$\displaystyle G(s,t)=\lambda_{2}e^{\left[\frac{\alpha k_{1}+\beta
k_{2}}{\beta k_{3}+k_{4}}\right]s}$ (20)

where, $\lambda_{2}=e^{C_{2}}$ is a constant and $C_{2}$ is another constant
of integration. Now, combining the equations (19) and (20), we can write $G$
in the following form,

$\displaystyle G(s,t)=F\left[(s-1)e^{-(\beta
k_{3}+k_{4})t}\right]e^{\left[\frac{\alpha k_{1}+\beta k_{2}}{\beta
k_{3}+k_{4}}\right]s}$ (21)

Now, to find out the functional form of the function $F$ in equation (21), we
take the initial boundary condition given by,

$\displaystyle G(s,0)=s^{M}$ (22)

where, $M$is the initial cytokine population at $t=0$. Putting this condition
to equation (21), we have the $F$ given by,

$\displaystyle F(s)=(s+1)^{M}e^{\left[-\frac{\alpha k_{1}+\beta k_{2}}{\beta
k_{3}+k_{4}}\right]s}$ (23)

Then, using this equation to the equation (21), we have the GF given by,

$\displaystyle G(s,t)=e^{ab}\left[(s-1)a+1\right]^{M}e^{bs(1-a)}$ (24)

where, $a=e^{-(\beta k_{3}+k_{4})t}$ and $b=\frac{\alpha k_{1}+\beta
k_{2}}{\beta k_{3}+k_{4}}$. Now, rearranging the first factor of the equation
(24) in the form $F_{1}=\left[(s-1)e^{-(\beta
k_{3}+k_{4})t}+1\right]^{M}=a^{M}\left[s+\frac{1-a}{a}\right]^{M}$, and using
Binomial expansion, we have,
$F_{1}=\displaystyle\sum_{i=0}^{M}~{}^{M}C_{i}s^{i}\left(\frac{1-a}{a}\right)^{M-i}$.
The second factor in the equation (24) can be rearranged and expand in the
form,
$e^{bs(1-a)}=\displaystyle\sum_{j=0}^{\infty}\frac{[b(1-a)]^{j}}{j!}s^{j}$.
Now putting the factors back to the equation (24), and taking $Z=i+j$, we
have,

$\displaystyle G(s,t)$ $\displaystyle=$ $\displaystyle
e^{ab}\sum_{i=0}^{M}\sum_{j=0}^{\infty}~{}^{M}C_{i}\frac{\left[b(1-a)\right]^{j}}{j!}s^{i+j}\left[\frac{1-a}{a}\right]^{M-i}a^{M}$
(25) $\displaystyle=$ $\displaystyle
e^{ab}\sum_{Z}s^{Z}\sum_{i=0}^{M}~{}^{M}C_{i}\frac{b^{Z-i}}{(Z-i)!}\left(1-a\right)^{M-2i+Z}a^{i}$
$\displaystyle=$ $\displaystyle\sum_{Z}s^{Z}P(Z,t)$

Equating the coefficient of $s^{Z}$, we arrived at the expression for
$P(Z,t)$, given by,

$\displaystyle
P(Z,t)=e^{ab}\sum_{i=0}^{M}~{}^{M}C_{i}\frac{b^{Z-i}}{(Z-i)!}\left(1-a\right)^{M-2i+Z}a^{i}$
(26)

Now, let us study the role of noise in cytokine dynamics triggered by T-cells
which can be analyzed by calculating Fano factor Fano . To calculate the Fano
factor $\mathcal{F}$ of the cytokine dynamics we need to calculate
observables, mean $\langle Z\rangle$ and variance $\sigma^{2}=\langle
Z^{2}\rangle-\langle Z\rangle^{2}$. Using equation eqrefgf, the mean of the
cytokine dynamics is given by,

$\displaystyle\langle Z\rangle$ $\displaystyle=$
$\displaystyle\left.\frac{\partial G(s,t)}{\partial s}\right|_{s=1}$ (27)
$\displaystyle=$ $\displaystyle e^{b}\left[a(M-b)+b\right]$

The asymptotic behaviors of the mean can be calculated as follows. Taking
$\displaystyle\lim_{t\rightarrow 0}\langle Z\rangle=e^{b}M$, which indicates
that the initial population of cytokine $Z$ depends on various interaction
processes involved in the model i.e. various rate constants and population of
T-cells $\beta$. Since the total population has to be $M$, we should have
$b=0$, which means that $\frac{\alpha k_{1}+\beta k_{2}}{\beta k_{3}+k_{4}}=0$
i.e. $\alpha k_{1}+\beta k_{2}=0$ revealing that the overall production of
cytokine both from external sources and secretion by T-cell should be zero. On
the other hand, since, $\displaystyle\lim_{t\rightarrow\infty}\langle
Z\rangle\rightarrow be^{b}=constant$ indicating the population of cytokine
reach a stable constant value at significantly large time. Next, the variance
is given by,

$\displaystyle\sigma^{2}$ $\displaystyle=$ $\displaystyle\langle
Z^{2}\rangle-\langle Z\rangle^{2}=\left.\frac{\partial^{2}G(s,t)}{\partial
s^{2}}\right|_{s=1}+\langle Z\rangle-\langle Z\rangle^{2}$ (28)
$\displaystyle=$ $\displaystyle e^{b}\left[(Ma-ab+b)^{2}-Ma^{2}\right]+\langle
Z\rangle-\langle Z\rangle^{2}$

Now, the Fano factor is calculated as follows,

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{\langle Z\rangle}$ (29) $\displaystyle=$
$\displaystyle 1+\langle
Z\rangle\left(e^{-b}-1\right)-e^{b}\frac{Ma^{2}}{\langle Z\rangle}$

Now, we analyze the Fano factor to understand the behavior of the cytokine
dynamics.

Proposition 1 The asymptotic behavior of the cytokine dynamics allows only
sub-Poissonian process.

Proof: Let us calculate the asymptotic values of Fano factor $\mathcal{F}$
from equation (29).

$\displaystyle\lim_{t\rightarrow 0}\mathcal{F}=M\left(1-e^{b}\right)$ (30)

From the expression of $b$, $b\neq 0$ and $b\rangle 0$, such that,
$e^{b}\rangle 1$. Hence, $\displaystyle\lim_{t\rightarrow 0}F\rightarrow-ve$
which can not be physically reliable and can not say anything about the
cytokine dynamics. On the other hand,

$\displaystyle\lim_{t\rightarrow\infty}\mathcal{F}=1-b\left(e^{b}-1\right)<1$
(31)

This indicates that the cytokine process follows sub-Poissonian nature. The
involved noise in the cytokine dynamics tries to stabilize its dynamical
behavior.

Proposition 2 At any instant of time $t~{}0\langle t\langle\infty$, the
cytokine dynamics noise-enhanced process.

Proof: The cytokine dynamics becomes noise enhanced process or super-
Poissonian process when $\mathcal{F}$ satisfies $\mathcal{F}>1$. Applying this
condition in equation (29), the condition becomes,

$\displaystyle\langle Z\rangle^{2}>\frac{e^{2b}Ma^{2}}{1-e^{b}}$ (32)

Since, $1\leq e^{b}<\infty$, $1-e^{b}<0$. This indicates that left hand side
term of equation (32) is negative value. Since, $\langle Z\rangle^{2}>0$, the
condition (32) is always satisfied. Further, since $\langle Z\rangle^{2}$ can
not be negative the process can not be Poissonian and sub-Poissonian process.
Hence, only possible role of noise for $0\langle t\langle\infty$ is noise
enhanced process.

###  III.2 Patterns observed in cytokines distribution dynamics

The simple cytokine and T-cells interaction model exhibits complicated
system’s dynamics where the role of noise is quite significant in regulating
the dynamics. Interesting question is even though the dynamics of the cytokine
molecules are complicated they should follow some traditional equation of
motion. Since the typical size of the single cytokine molecule ($IL2$) is
around $450nm$ in diameter Bakker  which is within mesoscopic scale size Salje
, they are very sensitive and significantly affected by internal and external
fluctuations Ritort . We intend to study the patterns of the distribution
functions these particles by analyzing the probability distribution function
given by equation (26).

Theorem 1 Near thermodynamic limit, $M,Z\rightarrow\infty$ and $\langle
Z\rangle\rightarrow finite$,

  * •

The cytokine distribution function $P(Z,t)$ follows modified Poisson
distribution:

$\displaystyle P(Z,t)\sim e^{-a(M-b)+be^{-a\left[1-\frac{M}{b}\right]}}\times
Pois\left[be^{-a\left(1-\frac{M}{b}\right)}\right]$ (33)

which is independent of cytokine population variable $Z$.

  * •

The information stored in classical Poisson pattern is given by,

$\displaystyle\mathcal{I}\sim e^{-a(M-b)+\Gamma}\times
ln\left[\frac{M}{\sqrt{2\pi e\Gamma}}\right]$ (34)

Proof: At large $M$ limit, the equation (26) can be written as,

$\displaystyle P(Z,t)$ $\displaystyle=$ $\displaystyle
e^{ab}\sum_{i=0}^{M}\frac{\left[b(1-a)\right]^{Z-i}}{(Z-i)!}\times\left[~{}^{M}C_{i}a^{i}(1-a)^{M-i}\right]$
(35) $\displaystyle\sim$ $\displaystyle
e^{ab}\sum_{i=0}^{M}\frac{\left[b(1-a)\right]^{Z-i}}{(Z-i)!}\times\left[\frac{\mu^{i}e^{-\mu}}{i!}\right];~{}~{}~{}\mu=Ma;~{}~{}M\rightarrow
large$ $\displaystyle\sim$ $\displaystyle
e^{ab}\sum_{i=0}^{M}\frac{M^{i}e^{-\mu}b^{Z-i}}{Z!}\times~{}^{Z}C_{i}a^{i}(1-a)^{Z-i}$
$\displaystyle\sim$ $\displaystyle
e^{ab}\sum_{i=0}^{M}\frac{M^{i}e^{-\mu}b^{Z-i}}{Z!}\times\frac{\epsilon^{i}e^{-\epsilon}}{i!}~{}~{}~{}\epsilon=Za;~{}~{}Z\rightarrow
large$ $\displaystyle\sim$ $\displaystyle
e^{ab}\frac{e^{-(\mu+\epsilon)}b^{Z}}{Z!}\sum_{i=0}^{M}\frac{\left[\frac{MZa}{b}\right]^{i}}{i!}$
$\displaystyle\sim$ $\displaystyle
e^{ab}\frac{e^{-(\mu+\epsilon)}b^{Z}}{Z!}\times e^{\frac{MZa}{b}}$
$\displaystyle\sim$ $\displaystyle
e^{-a(M-b)+be^{-a\left(1-\frac{M}{b}\right)}}\times
Pois\left[be^{a\left(\frac{M}{b}-1\right)}\right]$

The Poisson distribution is independent of $Z$ indicating the cytokines’
distribution becomes homogeneous approximately everywhere in the system when
the population of $Z$ is significantly large.

Now, the complexity in the distribution of the cytokine molecules whose
dynamics undergo stochastic process can be calculated using Shannon entropy
$\mathcal{H}$ using the modified Poisson probability distribution function
$\mathcal{P}(\Gamma,t)=Pois(\Gamma,t)$ given by equation (35), where,
$\Gamma=be^{a\left(\frac{M}{b}-1\right)}$. We follow the procedure of
calculation given in the work Evans . Then the terms are rearranged to get the
expression for $\mathcal{H}$ as in the following,

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-\sum_{u=0}^{M}\mathcal{P}(\Gamma\rightarrow
u)ln[\mathcal{P}(\Gamma\rightarrow u)]$ (36) $\displaystyle=$ $\displaystyle
e^{-a(M-b)+\Gamma}\times\left[\sum_{u=0}^{\infty}\mathcal{P}(u)ln\left(\frac{\Gamma^{u}e^{-\Gamma}}{u!}\right)\right]$
$\displaystyle=$ $\displaystyle e^{-a(M-b)+\Gamma}\times\left[\Gamma-\Gamma
ln\Gamma+\sum_{u=0}^{\infty}\mathcal{P}(u)ln\left(u!\right)\right]$

Then substituting the series expression for $ln(i!)$ given by the Malmsten’s
representation Erdilyi  as in the following,

$\displaystyle
ln(u!)=\int_{0}^{\infty}dx\left[u-\frac{1-e^{-ux}}{1-e^{-u}}\right]\frac{e^{-x}}{x}$
(37)

The equation (36) becomes,

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
e^{-a(M-b)+\Gamma}\times\left[\Gamma-\Gamma
ln\Gamma+\int_{0}^{\infty}dx\frac{e^{-x}}{x}\left(\Gamma-\frac{1-\langle
e^{-ux}\rangle}{1-e^{-x}}\right)\right]$ (38) $\displaystyle=$ $\displaystyle
e^{-a(M-b)+\Gamma}\times\left[\Gamma-\Gamma
ln\Gamma+\int_{0}^{\infty}dx\frac{e^{-x}}{x}\left(\Gamma-\frac{1-e^{-x(1-e^{-x})}}{1-e^{-x}}\right)\right]$

The integration in equation (38) can be done using series of formulae listed
Erdilyi , doing integration by parts and rearranging the terms, we get the
following expression for entropy Evans ,

$\displaystyle\mathcal{H}$ $\displaystyle\sim$ $\displaystyle
e^{-a(M-b)+\Gamma}\times\left[\frac{1}{2}ln(2\pi
e\Gamma)-\sum_{i=0}^{\infty}c_{i}(i-1)!\Gamma^{-i}\right]$ (39)
$\displaystyle\sim$ $\displaystyle\frac{1}{2}e^{-a(M-b)+\Gamma}\times ln(2\pi
e\Gamma)~{}~{}(\Gamma\rightarrow\infty)$

Now, the information stored in the Poisson pattern of the cytokine population
can be calculated using the following expression for information $\mathcal{I}$
due to the surprises given by the Shannon entropy given by equation (36),

$\displaystyle\mathcal{I}=\mathcal{H}_{max}-\mathcal{H}$ (40)

where, $\mathcal{H}_{max}$ is the maximum entropy which can be calculated
using maximum entropy principle Guiasu . The main procedure of this principle
is to define a function $\Omega=\mathcal{H}-\kappa\Phi$, where, $\kappa$ is
the Lagrange’s multiplier, and $\Phi$ is a function constructed from the
constraint $\displaystyle\sum_{i=0}^{M}\mathcal{P}_{i}=1$, such that,
$\Phi=\displaystyle\sum_{i=0}^{M}\mathcal{P}_{i}-1$. The Shannon entropy given
in equation (36) can also be written as
$\mathcal{H}=-\displaystyle\sum_{i=0}^{M}\mathcal{P}_{i}ln\mathcal{P}_{i}$.
Now the function $\Omega$ becomes,

$\displaystyle\Omega=-\sum_{i=0}^{M}\mathcal{P}_{i}ln\mathcal{P}_{i}-\kappa\left[\sum_{i=0}^{M}\mathcal{P}_{i}-1\right]$
(41)

Taking the derivative of the function $\Omega$ with respect to
$\mathcal{P}_{i}$, and putting
$\displaystyle\frac{\partial\Omega}{\partial\mathcal{P}_{i}}=0$, and solving
for $\mathcal{P}_{i}$ from the resulting equation, we have,
$\displaystyle\mathcal{P}_{i}=e^{-[1+\kappa]}$ The Lagrange’s multiplier
$\kappa$ can be calculated by putting this expression to the condition,
$\displaystyle\sum_{i=0}^{M}\mathcal{P}_{i}=1$, and it is found that
$\displaystyle\mathcal{P}_{i}=\frac{1}{M}$ for all values of $i$. Now, putting
this expression to equation (36), we get,
$\displaystyle\mathcal{H}_{max}=e^{-a(M-b)+\Gamma}\times ln[M]$. Now, putting
this expression to the equation (40), and using the equation (39), we get,

$\displaystyle\mathcal{I}\sim e^{-a(M-b)+\Gamma}\times
ln\left[\frac{M}{\sqrt{2\pi e\Gamma}}\right]$ (42)

This equation (42) indicates that $\mathcal{I}=\mathcal{I}(t)$. From the
asymptotic values of $\mathcal{I}$, we get that, (i) at large time limit,
($t\rightarrow\infty$),
$\displaystyle\mathcal{I}_{\infty}=\lim_{t\rightarrow\infty}\mathcal{I}\sim
e^{b}\times ln\left[\frac{M}{\sqrt{2\pi eb}}\right]$, indicating if the
overall production rate of the cytokine ($\alpha k_{1}+\beta k_{2}$) is faster
than overall decay ($\beta k_{3}+k_{4}$) then, $b\gg 1$, and hence,
$\mathcal{I}_{0}$ will become small. This condition indicate that the
probability of finding of cytokines will be increased due to the increase in
the population of cytokines, and hence, information stored will be decreased
as the system goes to more ordered state. However, if overall production rate
of the cytokines is smaller than the overall decay rate then $b\ll 1$
indicating $\mathcal{I}_{0}$ increases significantly indicating the system
goes more disorder state. On the other hand, (ii) at initial time $t=0$,
$\mathcal{I}_{0}=\displaystyle\lim_{t\rightarrow
0}\mathcal{I}=e^{-(M-b)+be^{\frac{M}{b}-1}}\times ln\left[\frac{M}{\sqrt{2\pi
be^{\frac{M}{b}}}}\right]$.

Theorem 2 Near thermodynamic limit, and at the large mean value
$\Gamma\rightarrow\infty$ (overall mean cytokine production rate is much
higher than overall mean decay rate: $b\gg 1$),

  * •

The distribution approaches to classical Normal pattern:

$\displaystyle P(Z,t)\sim
e^{-a(M-b)+\Gamma}\times\mathcal{N}\left[\Gamma,\Gamma\right]$ (43)

  * •

The information stored in Normal pattern is given by,

$\displaystyle\mathcal{I}=e^{-a(M-b)+\Gamma}\left[ln(M)+a(M+b)-\Gamma-\frac{d_{1}ln(\Gamma\sqrt{2\pi})+d_{2}}{\Gamma\sqrt{2\pi}}\right]$
(44)

Proof: The proof is straight forward. Near thermodynamic limit,
$M,Z\rightarrow large$ $P(Z,t)$ becomes Poisson distribution given by (35).
Further, taking $\Gamma\rightarrow large$, where,
$\Gamma=be^{a\left(\frac{M}{b}-1\right)}$ and using Stirling’s formula,
$Z!\approx Z^{Z}e^{-Z}\sqrt{2\pi Z}$ to the Poisson distribution,
$\displaystyle Pois(\Gamma)=\frac{\Gamma^{Z}e^{-\Gamma}}{Z!}$, we expand it
around the mean $\Gamma$ and get Normal distribution, $\displaystyle
Pois(\Gamma)\approx\mathcal{N}(\Gamma,\Gamma)$. This proves equation (43).

Now, let us calculate the information stored in the classical normal pattern
which can be calculated using the equation (40). The Shannon entropy, which
can be used to calculate the complexity in complex system, can be obtained by
using the equation (43) and rearranging the terms as given below,

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-\sum_{u=0}^{M}\mathcal{P}(\Gamma\rightarrow
u)ln\mathcal{P}(\Gamma\rightarrow u)$ (45) $\displaystyle=$
$\displaystyle-e^{-a(M-b)+\Gamma}\times\left[\sum_{u=0}^{M}\mathcal{N}(\Gamma\rightarrow
u)ln\mathcal{N}(\Gamma\rightarrow u)+\Gamma-a(M+b)\right]$

We, then, substitute the expression for Normal distribution given by
$\mathcal{N}(\Gamma\rightarrow
u)=\frac{1}{\Gamma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{u-\Gamma}{\Gamma}\right)^{2}}$
to the equation (45) to obtain the expression for $\mathcal{H}$. At the
$M\rightarrow\infty$ limit,
$\displaystyle\sum_{u=0}^{M}\rightarrow\int_{0}^{\infty}$. Then, after
integration and rearranging the terms, we have,

$\displaystyle\mathcal{H}=e^{-a(M-b)+\Gamma}\left[\Gamma+\frac{d_{1}ln(\Gamma\sqrt{2\pi})+d_{2}}{\Gamma\sqrt{2\pi}}-a(M+b)\right]$
(46)

where, $d_{1}=\sqrt{\frac{\pi}{2}}+\sqrt{2}erf\left(\frac{1}{\sqrt{2}}\right)$
and
$d_{2}=\frac{1}{2}erf\left(\frac{1}{\sqrt{2}}\right)-\frac{e^{-\frac{1}{2}}}{\sqrt{\pi}}$
are constants respectively. $erf(x)$ is the error function of $x$. From
equations (40) and (46) and putting the value of
$\mathcal{H}_{max}=e^{-a(M+b)+\Gamma}\times ln(M)$, we have the information
stored in classical Normal pattern, $\mathcal{I}$ as in the following,

$\displaystyle\mathcal{I}=e^{-a(M-b)+\Gamma}\left[ln(M)+a(M+b)-\Gamma-\frac{d_{1}ln(\Gamma\sqrt{2\pi})+d_{2}}{\Gamma\sqrt{2\pi}}\right]$
(47)

In order to have non-negative information, one has to have the condition
$ln(M)+a(M-b)>\Gamma+\frac{d_{1}ln(\Gamma\sqrt{2\pi})+d_{2}}{\Gamma\sqrt{2\pi}}$.
We further get that at the sufficiently large time,
$\displaystyle\mathcal{I}_{\infty}=\lim_{t\rightarrow\infty}\mathcal{I}=e^{b}\displaystyle\left[ln(M)-b-\frac{d_{1}ln(b\sqrt{2\pi})+d_{2}}{b\sqrt{2\pi}}\right]$.
This indicates that at large time limit, the overall production rate of
cytokine should be controlled such that $ln(M)\gg b$ as the third term is less
than one, otherwise, $\mathcal{I}_{\infty}<0$ which is physically have no
meaning.

##  IV Conclusion and discussion

Patterns are manifestation of regular self-organised aggregated random
components. They have significant roles as they keep diverse information
stored in them to take part in collective decision making. Hence, it is
important to study these patterns and information associated with them
specially in biological systems.

The interaction of cytokine and T-cells in our body keeps the regulation of
immune system and helps in various biological functions Wu . Since the
cytokine production by T-cells is quite essential for developing immune
system, we considered a simple of cytokines and T cells interaction, where,
all possible cytokine production mechanisms are associated. We used stochastic
formalism to study the model by constructing Master equation of the model
system specifically considering the dynamics of cytokines. The constructed
Master equation is solved analytically using generating function technique and
the solved probability distribution function, which describes the pattern of
distribution of dynamical cytokines in the system, is found to be quite
complex. However, under certain limiting conditions, we explored few
interesting patterns which carry the information of the system. Near
thermodynamics limit, the probability distribution function becomes modified
Poisson distribution which indicates that the distribution of the cytokines in
the system exhibit classical Poisson pattern. This property indicates that the
cytokine events occurring in the system is purely random and the pattern is
exhibited mostly by the self-organised aggregation of spatiotemporal random
cytokine events scattered over the entire system. In such situation, Poisson
pattern behaves as a basin of attraction which attracts cytokine random
processes towards it and nonrandom processes are away from it Frank . The
uncertainty or surprise of random cytokine events is measured by complexity
measurement parameter Shannon entropy which we found to be time dependent. The
information stored in the Poisson pattern is then calculated as the difference
of maximum entropy and entropy at any instant of time, and found that the
competition between overall production rate of cytokines and overall decay
rate of it need to be balanced such that information stored in the Poisson
pattern should be optimised and non-zero.

On the other hand, near the thermodynamics limit, if the mean of the cytokine
distribution is taken to be significantly large, then the distribution of the
cytokines exhibits classical Normal pattern. In such situation, the phenomenon
indicates that the sum of the observations of the small-scale random and
independent cytokine events is also a random variable which is a larger-scale
random process in the system, no matter how small the distributed random
variables are Lyon . The distribution of such larger-scale random processes in
the system approaches to classical Normal pattern. It also reveals that not
only the small-scale random processes, but also larger-scale random processes
of the cytokine distribution exhibit Normal pattern. This could be an
indication of hierarchical organization of the cytokine random processes which
approaches to similar behaviour at various scales. The information contained
in the Normal pattern is calculated using Shannon entropy of the distribution.

The role of the noise is important and significant specially in microscopic
systems. In the cytokine and T-cells interaction model, the strength of the
noise is measured using Fano factor which can be calculated from the
information of mean and variance of the distribution. The calculated Fano
factor shows noise enhanced or super-Poissonian process which indicates that
the noise associated with the distribution can significantly drive the system
at various possible accessible states. In such situation, the state of the
cytokine distribution and dynamics is driven far away from the equilibrium and
are generally in active state.

The study in this simple model could able to understand few important features
and properties of cytokines distribution which is dynamic in principle in
cytokine and T-cells interaction. This system is multi-variable system which
is needed to incorporate various other interacting molecular species. In such
situation, analytical work could have limitation but definitely large scale
simulation can be implemented to the system to study dynamics and distribution
of the cytokine and related other variables.

Acknowledgements
RKBS is financially supported DST-Matrics scheme.

Author Contributions:
RKBS conceptualized the work. MSS, MKS and RKBS did the analytical work. All
authors read, analyzed the results, and approved the final manuscript.

Additional Information
Competing interests:
The authors declare no competing interests.

## References

  * (1) Julia K. Parrish and Leah Edelstein-Keshet. Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science, 284:99–101 (1999).
  * (2) Newman MEJ. Complex systems: A survey. Am J Phys. 2011;79:800–810.
  * (3) B. T. Werner. Complexity in natural landform patterns. Science, 284:102–104, April 1999.
  * (4) Y. Karaca. Theory of complexity, origin and complex systems. Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, Academic Press (2022).
  * (5) Cosma R. Shalizi and James P. Crutchfield. Computational mechanics: Pattern and prediction, structure and simplicity. Journal of Statistical Physics, 104(3):817–879, 2001.
  * (6) J. P. Crutchfield and N. H. Packard. Inferring statistical complexity. Physica D 7, 201 (1983).
  * (7) Hale, J. T. (2016). Information-theoretical complexity metrics. Language and Lin7 guistics Compass, 10(9):397–412.
  * (8) Kolmogorov, A.: ‘Three approaches to the definition of the concept “quantity of information”’, Probl. Inform. Transmission1 (1965), 1–7.
  * (9) M. Aboy, R. Hornero, D. Abasolo, and D. Alvarez, ”Interpretation of the Lempel-Ziv Complexity Measure in the Context of Biomedical Signal Analysis,” Biomedical Engineering, IEEE Transactions on, vol. 53, 2282-2288, (2006).
  * (10) Frank, S. A. (2009). The common patterns of nature. Journal of evolutionary biology, 22(8), 1563–1585.
  * (11) Jaynes, E.T. 2003. Probability Theory: The Logic of Science. Cambridge University Press, New York.
  * (12) Feller, W. 1968. An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.
  * (13) Feller, W. 1971. An Introduction to Probability Theory and its Applications Vol. II, 2nd edn. Wiley, New York.
  * (14) Wootton JT, Pfister CA, Forester JD. 2008. Dynamic patterns and ecological impacts of declining ocean pH in a high-resolution multi-year dataset. Proc. Natl. Acad. Sci. USA 105: 18848–53.
  * (15) Kleiber, C. and Kotz, S. 2003. Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, New York.
  * (16) Mandelbrot, B.B. 1983. The Fractal Geometry of Nature. WH Freeman, New York.
  * (17) Barabasi, A.L. and Albert, R. 1999. Emergence of scaling in random networks. Science 286: 509–512.
  * (18) Uhlenbeck GE, Ornstein LS. On the theory of the Brownian motion. Phys Rev. 1930;36:823–841.
  * (19) Shannon, C.E., 1948. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656.
  * (20) Kolmogorov A.N., On analytic methods in probability, Uspehi Mat. Nauk, 1938, V.5, 5–41.
  * (21) Swain, Peter S., Michael B. Elowitz, and Eric D. Siggia. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proceedings of the National Academy of Sciences 99.20 (2002): 12795-12800.
  * (22) N.G. Van Kampen - Stochastic Processes in Physics and Chemistry \- Elsevier Science Ltd (1981).
  * (23) Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984.
  * (24) McQuarrie D.A., Gillespie D.T. Stochastic Theory and Simulations of Chemical Kinetics. J. Appl. Prob. 4, 413-478, 1967.
  * (25) Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340-2361 (1977).
  * (26) Gardiner C.W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Second Edition, Fourth Printing. New Zealand. 1997.
  * (27) Paul C. Bressloff (2014). Stochastic Processes in Cell Biology. Springer.
  * (28) Russell Lande, Steiner Engen, Bernt-Erik Sther (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press.
  * (29) Kodama, H., and Ishibashi, A., A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions. Prog. Theor. Phys., 110, 701, (2003).
  * (30) Wolfgang Paul; Jrg Baschnagel (11 July 2013). Stochastic Processes: From Physics to Finance. Springer Science Business Media.
  * (31) J. Michael Steele (2001). Stochastic Calculus and Financial Applications. Springer Science Business Media.
  * (32) Jonathan Katz, Yehuda Lindell (2007-08-31). Introduction to Modern Cryptography: Principles and Protocols. CRC Press. p. 26.
  * (33) Jodie Testar, Imperial College London, UK. Cytokines: Introduction. Category: Receptors and Molecules. Bitesized Immunology. British Society for Immunology.
  * (34) Dimitriou, I. D., L. Clemenza, A. J. Scotter, G. Chen, F. M. Guerra, and R. Rottapel. 2008. Putting out the fire: coordinated suppression of the innate and adaptive immune systems by SOCS1 and SOCS3 proteins. Immunol. Rev. 224: 265–283.
  * (35) Delgado AV, McManus AT, Chambers JP. Production of tumor necrosis factor-alpha, interleukin 1-beta, interleukin 2, and interleukin 6 by rat leukocyte subpopulations after exposure to substance P. Neuropeptides. 2003;37:355–61.
  * (36) Hart Y, Antebi YE, Mayo AE, Friedman N, Alon U. Design principles of cell circuits with paradoxical components. Proc Natl Acad Sci U S A. 109, 83468351 (2012).
  * (37) Wei Liao, Jian-Xin Lin, and Warren J. Leonard. IL-2 Family Cytokines: New Insights into the Complex Roles of IL-2 as a Broad Regulator of T helper Cell Differentiation. Curr Opin Immunol. 2011 Oct; 23(5): 598–604.
  * (38) James B. Crawley, Lesley Rawlinson, Ferdinand V. Lali, Theresa H. Page, Jeremy Saklatvala, Brian M.J. Foxwell. T Cell Proliferation in Response to Interleukins 2 and 7 Requires p38MAP Kinase Activation. Cell Biology And Metabolism, Volume 272, P15023-15027, June 1997.
  * (39) Booki Min. Spontaneous T Cell Proliferation: A Physiologic Process to Create and Maintain Homeostatic Balance and Diversity of the Immune System. Front Immunol. 2018; 9: 547.

  * (40) Anaya JM, Shoenfeld Y, Rojas-Villarraga A, et al. Autoimmunity: From Bench to Bedside. Editors. Bogota (Colombia): El Rosario University Press; 2013 Jul 18.
  * (41) Refaeli Y, Van Parijs L, London CA, Tschopp J, Abbas AK (1998) Biochemical mechanisms of IL-2-regulated Fas-mediated T cell apoptosis. Immunity 8:615–623.
  * (42) Li XC, et al. (2001) IL-15 and IL-2: A matter of life and death for T cells in vivo. Nat Med 7:114–118.
  * (43) R. J. Evans and J. Boersma. The Entropy of a Poisson Distribution. SIAM Review Vol. 30, 314. (1988).
  * (44) A. Erdilyi, W. Magnus, F. Oberhettinger and F.G. Tricomi. Higher Transcendental Functions. Vols. I, II, McGraw-Hill, New York, 1953.
  * (45) Guiasu, S. and Shenitzer, A. (1985). The Principle of Maximum Entropy. The Mathematical Intelligencer, 7(1).
  * (46) Wu, J.; Xie, A.; Chen, W. Cytokine regulation of immune tolerance. Burns Trauma 2014, 2, 11–17.
  * (47) Lyon A, Why are Normal Distributions Normal? The British Journal for the Philosophy of Science, 65 (3) 621-649, (2014).
  * (48) McNeil K J and Walls D F. A master equation approach to nonlinear optics. J. Phys. A: Math. Gen. 7 617 (1974).
  * (49) Scalas, E., R. Gorenflo and F. Mainardi (2004). Uncoupled continuous-time random walks: Solution and limiting behaviour of the master equation. Phys. Rev. E 69, 011107.
  * (50) U. Fano. Ionization yield of radiations. 2. The fluctuations of the number of ions. Phys. Rev. 72, 26 (1947).
  * (51) Kamil Rajdl, Petr Lansky, Lubomir Kostal. Fano Factor: A Potentially Useful Information. Front. Comput. Neurosci., 20 November 2020.
  * (52) Chanu, A.L., Bhadana, J., Singh, R.B.: Stochastic fluctuations as a driving force to dissipative non-equilibrium states. J. Phys. A Math. Theor. 53(42), 425002 (2020).
  * (53) Rogers, P. R., C. Dubey, and S. L. Swain. 2000. Qualitative changes accompany memory T cell generation: faster, more effective responses at lower doses of antigen. J. Immunol. 164:2338.
  * (54) Almeida AR, Rocha B, Freitas AA, Tanchot C. Homeostasis of T cell numbers: from thymus production to peripheral compartmentalization and the indexation of regulatory T cells. Semin. Immunol. 2005;17:239–249.
  * (55) Delgado, M. The Lagrange-Charpit Method. SIAM Review. 39, 298–304 (1997).
  * (56) de Bakker, B. I. et al. Nanometer-scale organization of the $\alpha$ subunits of the receptors for IL2 and IL15 in human T lymphoma cells. J. Cell Sci. 121, 627–633 (2008).
  * (57) Salje, E. (2010). ”Multi-scaling and mesoscopic structures.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368(1914): 1163-1174.
  * (58) Ritort F 2007 Nonequilibrium fluctuations in small systems: from physics to biology Advances in Chemical Physics ed S A Rice, vol 137.
  * (59) Raúl Toral, Pere Colet. Stochastic Numerical Methods: An Introduction for Students and Scientists. ISBN: 978-3-527-68312-3, June 2014.


# Fractional decoding of algebraic geometry codes over extension fields

Eduardo Camps-Moreno  Department of Mathematics
Virginia Tech
Blacksburg, VA USA<EMAIL_ADDRESS>,  Gretchen L. Matthews  Department of
Mathematics
Virginia Tech
Blacksburg, VA USA<EMAIL_ADDRESS>and  Welington Santos  Department of
Mathematics, Statistics and Computer Science University of Wisconsin-Stout
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study algebraic geometry codes from curves over
$\mathbb{F}_{q^{\ell}}$ through their virtual projections which are algebraic
geometric codes over $\mathbb{F}_{q}$. We use the virtual projections to
provide fractional decoding algorithms for the codes over
$\mathbb{F}_{q^{\ell}}$. Fractional decoding seeks to perform error correction
using a smaller fraction of $\mathbb{F}_{q}$-symbols than a typical decoding
algorithm. In one instance, the bound on the number of correctable errors
differs from the usual lower bound by the degree of a pole divisor of an
annihilator function. In another, we view the virtual projections as
interleaved codes to, with high probability, correct more errors than
anticipated.

##  1\. Introduction

The fractional decoding problem is motivated by distributed systems in which
there are limitations on the disk operation and on the amount of information
transmitted for the purpose of decoding. Sometimes thought of as error
correction with partial information, fractional decoding considers codes
defined over an extension field and algorithms for error correction that use
fewer symbols from the base field than is typical, thus operating using a
restricted amount of information in the decoding process. Tamo, Ye, and Barg
[13] introduced the concept, inspired by regenerating codes and codes for
erasure recovery which focus on the repair bandwidth which measures the
proportion of a received word used to recovery an erasure (see, for instance,
[2, 4, 14]).

In [11], Santos provided a connection between fractional decoding of Reed-
Solomon codes, which can be considered as codes from the projective line, a
curve of genus $0$, and collaborative decoding of interleaved Reed-Solomon
codes. Later in [7, 8], the authors present the first fractional decoding
algorithm for codes from the Hermitian curve. The $r$-Hermitian codes are
constant field extensions of Hermitian codes with a bound on the $y$-degree of
the rational functions in the Riemann-Roch space used to define codewords.

In this paper, we present a fractional decoding approach for a large family of
algebraic geometry codes. The codes may be considered as evaluation codes over
the field $\mathbb{F}_{q^{l}}$ of cardinality $q^{l}$ with evaluation points
whose coordinates that lie in a subfield. Taking evaluation points to have
coordinates in a proper subfield (such as $\mathbb{F}_{q}$ where $l>1$) to
define a family of codes for particular purpose is an idea utilized by
Guruswami and Xing [5] and Gao, Yue, Huang, and Zhang [3] among others
(including [6]). To be more precise, we focus on algebraic geometry codes
defined as follows. Consider $n$ distinct affine points $P_{1},\dots,P_{n}$ on
a smooth, projective curve $\mathcal{X}$ with coordinates in $\mathbb{F}_{q}$
and a divisor $G$ of degree less than $n$ on $X$ whose support consists of
only $\mathbb{F}_{q}$-rational points and contains none of the points
$P_{1},\dots,P_{n}$. We are interested in the $q^{l}$-ary algebraic geometry
code
$C_{l}(G,D):=\textnormal{ev}(\mathcal{L}_{l}(G))\subseteq\mathbb{F}_{q^{l}}^{n}$
where

$\begin{array}[]{lccc}\textnormal{ev}:&\mathbb{F}_{q^{l}}(\mathcal{X})&\rightarrow&\mathbb{F}_{q^{l}}^{n}\\\
&f&\mapsto&(f(P_{1}),\dots,f(P_{n}))\end{array}$

and $D=P_{1}+\dots+P_{n}$. Noting that $C_{l}(G,D)$ has length $n$, dimension
$\ell(G)$, which is the dimension of the Riemann-Roch space $\mathcal{L}(G)$,
and minimum distance $d$ at least $n-\deg G$, we know that up to $e$ errors
may be corrected in a received word $w\in\mathbb{F}_{q^{l}}^{n}$, where
$e\leq\frac{d-1}{2}$. Hence, viewing $\mathbb{F}_{q^{l}}$ as a degree $l$
extension of $\mathbb{F}_{q}$, we may see this as correcting $e$ errors using
$ln$ symbols of $\mathbb{F}_{q}$.

In this work, we use partitions of the set of evaluation points
$P_{1},\dots,P_{n}$ to demonstrate that up to $\frac{d^{\prime}-1}{2}$ errors
may be corrected using only $mn$ symbols of the base field $\mathbb{F}_{q}$,
where $m<l$ is number of parts of a partition and $d^{\prime}$ is specified by
pole divisors of certain rational functions on $\mathcal{X}$. This allows for
error correction using algebraic geometry codes in applications where limiting
network traffic is beneficial, such as distributed storage. We emphasize two
key advantages to this approach. First, while quite general, these techniques
apply particularly nicely to codes constructed from Kummer extensions or
Castle curves and a constant field extension of the Riemann-Roch space of a
divisor on the curve. Second, the perspective may be combined with advances in
traditional decoding algorithms for algebraic geometry codes, providing
immediate adapation to the distributed context considered in this work.

This paper is organized as follows. Section 2 presents the main tools to be
used in fractional decoding of codes from curves. The codes we consider are
described in Subection 2.2 where we study their virtual projections. These
virtual projections may be conveniently captured as an interleaved code, as
seen in Subsection 3.2. Particular instances for important families of codes
are given in Subsection 2.3. Fractional decoding algorithms are presented in
Section 3. The first, in Subsection 3.1, makes use of the virtual projections
directly. The second is via interleaved codes found in Subsections 3.2 and
3.3. These frameworks can make use of advances in decoding of the component or
interleaved codes.

##  2\. Virtual projections of algebraic geometry codes

In this section, we will consider convenient ways to represent algebraic
geometry codes defined over extention fields. In [5], Guruswami and Xing
considered algebraic-geometric codes over a field $\mathbb{F}_{q^{l}}$ whose
evaluation points belong to a subfield $\mathbb{F}_{q}$ and provided a list
decoding algorithm for those codes. Here, we consider properties of a similar
family of codes, in preparation for demonstrating that fractional decoding can
be applied to them.

###  2.1. Notation

We use the standard notation from coding theory and the algebraic curves over
finite fields; see [12] for instance. Given a genus $g$ curve $\mathcal{X}$
over a finite field $\mathbb{F}$, the associated field of rational functions
is denoted $\mathbb{F}(\mathcal{X})$. For a rational function
$f\in\mathbb{F}(\mathcal{X})$ and $\mathbb{F}$-rational point $P$ on
$\mathcal{X}$, the valuation of $f$ at $P$ is denoted $v_{P}(f)$. The divisor
of a nonzero rational function $f$ is $(f)=(f)_{0}-(f)_{\infty}$ where
$(f)_{0}=\sum_{{\footnotesize{\\{P\in\mathcal{X}(\overline{\mathbb{F}}):v_{P}(f)>0\\}}}}v_{P}(f)P$
and
$(f)_{\infty}=\sum_{{\footnotesize{\\{P\in\mathcal{X}(\overline{\mathbb{F}}):v_{P}(f)<0\\}}}}v_{P}(f)P$
are the zero and pole divisors of $f$. For a divisor $D$ on $\mathcal{X}$, the
Riemann-Roch space of $G$ is vector space of functions
$\mathcal{L}(G):=\left\\{f\in\mathbb{F}(\mathcal{X}):(f)+G\geq 0\right\\}$
whose dimension over $\mathbb{F}$ is denoted $\ell(G)$. If $\deg G\geq 2g-1$,
then $\ell(G)=\deg G+1-g$, and $\Omega(G)$ is the vector space of
differentials with divisors at least $G$, together with the zero differential.
We use $\mathbb{N}$ to denote the set of nonnegative integers. Given a
positive integer $n$, $[n]:=\left\\{1,\dots,n\right\\}$. For integers $i,j$,
$\delta_{i,j}=1$ if and only if $i=j$ and $\delta_{i,j}=0$ otherwise. The set
of all $m\times n$ matrices over $\mathbb{F}$ is denoted $\mathbb{F}^{m\times
n}$. Given a matrix $A=(a_{ij})\in\mathbb{F}^{m\times n}$, we let
$D_{Row_{i}A}$ denote the diagonal matrix whose diagonal is given by the
$i^{th}$ row of $A$ and $A\mid_{I}\in\mathbb{F}^{m\times\mid I\mid}$ denote
the submatrix of $A$ whose columns are the columns of $A$ indexed by
$I\subseteq[n]$.

An $[n,k,d]$ code $C$ over a finite field $\mathbb{F}$ is a $k$-dimensional
$\mathbb{F}$-subspace of $\mathbb{F}^{n}$ in which the minimum weight of a
nonzero vector is $d$. Here, $n$ is the length of $C$, $k$ is the dimension of
$C$, $d$ is the minimum distance of $C$, and the weight of a word
$w\in\mathbb{F}^{n}$, denoted $wt(w)$ is its number of nonzero coordinates.
Such a code can be described by a generator matrix, which is a matrix
$G\in\mathbb{F}^{k\times n}$ whose rows form a basis for $C$. We say that a
subset of $k$ coordinate positions $I\subseteq[n]$ is an information set if
and only if the $k\times k$ submatrix of $G$ formed by columns indexed by $I$
is nonsingular.

We will consider algebraic geometry codes $C_{l}(G,D)$ as described in Section
1. The points in the support of $D$ are referred to as the evaluation points
of the code. We typically take them to be affine points on the curve, in which
case an evaluation point may be expressed as $(a_{1},\dots,a_{M})$ where
$\mathcal{X}$ is given by equations involving variables $x_{1},\dots,x_{M}$.
In the case where $D$ is the sum of all $\mathbb{F}_{q}$-rational points not
in the support of $G$ or is clear from the context, we write $C_{l}(G)$ rather
than $C_{l}(G,D)$. If the support of $G$ consists of a single point, the codes
$C_{l}(G,D)$ and $C(G,D)$ are called one-point codes. We say that a collection
of points
$\left\\{P_{i_{1}},\dots,P_{i_{k}}\right\\}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
is an information set to mean that the positions
$\left\\{{i_{1}},\dots,{i_{k}}\right\\}$ corresponding to these points is an
information set.

###  2.2. Codes from constant field extensions

Let $\left\\{\zeta_{1},\ldots,\zeta_{l}\right\\}$ be a basis of
$F:=\mathbb{F}_{q^{l}}$ over $B:=\mathbb{F}_{q}$, and let
$\left\\{\nu_{1},\ldots,\nu_{l}\right\\}$ be its dual basis, meaning
$tr(\zeta_{s}\nu_{j})=\delta_{s,j}$ for all $s,j\in[l]$. Then for all
$\beta\in F$,

$\beta=\sum_{s=1}^{l}tr(\zeta_{s}\beta)\nu_{s}.$

In other words, any element $\beta$ in $F$ can be calculated from its $l$
projections $tr(\zeta_{s}\beta)$, $s\in[l]$, on $B$.

Consider a smooth projective curve $\mathcal{X}$ over a field $\mathbb{F}_{q}$
and a divisor $G$ whose support contains only $\mathbb{F}_{q}$-rational
points. At times, we may wish consider the Riemann-Roch space of $G$ on
$\mathcal{X}$ considered over the extension $\mathbb{F}_{q^{l}}$, denoted
$\mathcal{L}_{l}(G)$. It is well-known that

$\mathcal{L}_{l}(G)=\mathcal{L}(G)\otimes\mathbb{F}_{q^{l}}.$

######  Definition 1.

Keep the notation above and assume that
$\mathcal{B}=\left\\{h_{1},\dots,h_{k}\right\\}$ is a basis for
$\mathcal{L}(G)$. For $s\in[l]$, the $s$-projection of the function
$f(x_{1},\ldots,x_{M})=\sum_{i=1}^{k}a_{i}h_{i}(x_{1},\ldots,x_{M})\in\mathcal{L}_{l}(G)$
to $\mathcal{L}(G)$ with respect to $\mathcal{B}$ is defined to be

$f_{s}(x_{1},\ldots,x_{M})=\sum_{i=1}^{k}tr(\zeta_{s}a_{i})h_{i}(x_{1},\ldots,x_{M}).$

Typically, a basis for $\mathcal{L}(G)$ is fixed, and we refer to $f_{s}$ as
the $s$-projection of $f$ to $\mathcal{L}(G)$, omitting mention of the
particular choice of basis.

Note that $f\in\mathcal{L}_{l}(G)\subseteq\mathbb{F}_{q^{l}}(\mathcal{X})$
while $f_{s}\in\mathcal{L}(G)\subseteq\mathbb{F}_{q}(\mathcal{X})$.
Furthermore, $f(x_{1},\ldots,x_{M})$ is fully determined by
$\left\\{f_{s}(x_{1},\ldots,x_{M}):s\in[l]\right\\}$, since

$\begin{array}[]{lll}f(x_{1},\ldots,x_{M})&=\sum_{i=1}^{k}a_{i}h_{i}(x_{1},\ldots,x_{M})\\\
\\\
&=\sum_{i=1}^{k}\left[\sum_{s=1}^{l}tr(\zeta_{s}a_{i})\nu_{s}\right]h_{i}(x_{1},\ldots,x_{M})\\\
\\\
&=\sum_{s=1}^{l}\left[\sum_{i=1}^{k}tr(\zeta_{s}a_{i})h_{i}(x_{1},\ldots,x_{M})\right]\nu_{s}\\\
\\\ &=\sum_{s=1}^{l}f_{s}(x_{1},\ldots,x_{M})\nu_{s}.\end{array}$

This observation will be useful in the fractional decoding of certain
algebraic geometry codes. It is also straightforward to verify that the
$s$-projection of a $\mathbb{F}_{q}$-linear combination of functions behaves
nicely, as mentioned in the next observation.

######  Fact 2.

For all $f,h\in\mathcal{L}_{l}(G)$ and $a,b\in\mathbb{F}_{q}$,

$(af+bh)_{s}=af_{s}+bh_{s}.$

To set the stage for fractional decoding, fix a partition
$A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
where $m<l$. Consider an annihilator polynomial
$p_{t}(x_{1},\dots,x_{M})\in\mathbb{F}_{q}[x_{1},\dots,x_{M}]$ of the set
$A_{t}$, $t\in[m]$, meaning $p(x_{1},\dots,x_{M})=0$ for all
$(x_{1},\dots,x_{M})\in A_{t}$.

######  Definition 3.

Given $f\in\mathcal{L}_{l}(G)$,
$A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$,
and $t\in[m]$, define the function

$\begin{array}[]{ll}T_{t}(f)(x_{1},\dots,x_{M})=&f_{l-m+t}(x_{1},\dots,x_{M})(p_{t}(x_{1},\dots,x_{M}))^{l-m}\\\
\\\
&+\sum_{s=1}^{l-m}f_{s}(x_{1},\dots,x_{M})(p_{t}(x_{1},\dots,x_{M}))^{s-1}\end{array}$

and the $t$-virtual projection of $\mathcal{C}_{l}(G,D)$ to be

$\mathcal{VP}_{t}(G,D)=\left\\{(T_{t}(f)(P_{1}),\ldots,T_{t}(f)(P_{n})):f\in\mathcal{L}_{l}(G)\right\\}.$

If $D$ is the sum of all $\mathbb{F}_{q}$-rational points not in the support
of $G$ or is clear from the context, we write $\mathcal{VP}_{t}(G)$ rather
than $\mathcal{VP}_{t}(G,D)$.

The next result demonstrates how the virtual projection of an algebraic
geometry code may be seen as an algebraic geometry code itself.

######  Proposition 4.

Given divisors $G$ and $D$ whose supports consist only of
$\mathbb{F}_{q}$-rational points, the $t$-virtual projection
$\mathcal{VP}_{t}(G,D)$ of $\mathcal{C}_{l}(G,D)$ is a subcode of
$\mathbb{F}_{q}^{n}$. In particular,

$\mathcal{VP}_{t}(G,D)=\textnormal{ev}\left(T_{t}\left(\mathcal{L}_{l}(G)\right)\right)\subseteq
C\left(D,G-(l-m)\left(p_{t}\right)_{\infty}\right).$

###### Proof.

According to Fact 2, for all $f,h\in\mathcal{L}_{l}(G)$ and
$a,b\in\mathbb{F}_{q}$,

$T_{t}\circ af+T_{t}\circ bh=T_{t}\circ\left(af+bh\right)$

and

$a\cdot\textnormal{ev}\left(T_{t}(f)\right)+b\cdot\textnormal{ev}\left(T_{t}(h)\right)=\textnormal{ev}\left(T_{t}(af+bh)\right).$

Now the result follows from the fact that

$\mathcal{VP}_{t}(G)=\textnormal{ev}(T_{t}(\mathcal{L}_{l}(G)))\subseteq\mathbb{F}_{q}^{n}.$

Indeed, note that for any function $f\in\mathcal{L}_{l}(G)$ and $t\in[m]$,

(2.1)
$\begin{array}[]{lcl}v_{P}(T_{t}(f))&\geq&\min\left\\{v_{P}\left(f_{i}p_{t}^{i}\right),v_{P}\left(f_{l-m+t}p_{t}^{l-m}\right):i\in[l-m]\right\\}\\\
\\\
&\geq&\min\left\\{\begin{array}[]{l}v_{P}\left(f_{i}\right)+iv_{P}\left(p_{t}\right),\\\
v_{P}\left(f_{l-m+t}\right)+(l-m)v_{P}\left(p_{t}\right)\end{array}:i\in[l-m]\right\\}\\\
\\\
&\geq&-v_{P}\left(G\right)+\min\left\\{iv_{P}\left(p_{t}\right):i\in[l-m]\right\\}\\\
\\\ &\geq&\begin{cases}-v_{P}(G)&\textnormal{if }v_{P}\left(p_{t}\right)\geq
0\\\ -v_{P}(G)+(l-m)v_{P}\left(p_{t}\right)&\textnormal{if
}v_{P}\left(p_{t}\right)<0.\end{cases}\end{array}$

Hence,

$T_{t}(f)\in\mathcal{L}\left(G-(l-m)\sum_{P\in\textnormal{supp}(p_{t})_{\infty}}v_{P}\left(p_{t}\right)P\right).$

∎∎

Proposition 4 demonstrates the influence of the partition and the annihilator
functions on the virtual projection. More information about the annihilator
functions will allow us to better bound $v_{P}(T_{t}(f))$, resulting in a
better understanding of the space that contains
$\left\\{T_{t}(f):f\in\mathcal{L}_{l}(G)\right\\}$ for each $t\in[m]$. We will
see that a key idea to achieving fractional decoding of $C_{l}(G)$ is decoding
$\mathcal{VP}_{t}(G)$, $t\in[m]$.

######  Theorem 5.

Suppose $I\subseteq
A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$,
for some $m\in\mathbb{N}$ and some information set $I$ for the code $C_{l}(G)$
with generator matrix $A=\left(h_{i}(P_{j})\right)$. Then the function
$f\in\mathcal{L}_{l}(G)$ depends only on $\left\\{T_{t}(f)(P):t\in[m],P\in
I\right\\}$ and $A\mid_{I}$.

###### Proof.

Given
$A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
and $\left\\{T_{t}(f)(x_{1},\ldots,x_{M}):t\in[m]\right\\}$, we aim to
determine $f$. For $i\in[l-m]$ and $t\in[m]$, let

$T_{t}^{(i)}(f)(x_{1},\dots,x_{M})=\frac{T_{t}(f)(x_{1},\dots,x_{M})-\sum_{s=1}^{i-1}f_{s}(x_{1},\dots,x_{M})p_{t}(x_{1},\dots,x_{M})^{s-1}}{p_{t}(x_{1},\dots,x_{M})^{i-1}}.$

First, we will determine $f_{1}$. Notice that $T_{t}^{(1)}(f)=T_{t}(f)$ and

$f_{1}(P)=T_{t}(f)(P)$

for all $P\in A_{t}$. Since $f_{1}=\sum_{j=1}^{k}a_{1j}h_{j}$ for some
$a_{1j}\in\mathbb{F}_{q}$ and $I\subseteq A_{1}\cup\dots\cup A_{m}$, we may
use the values $h_{j}(P)$ for $j\in[k]$ and $P\in I$ to set up a system of
equations

$\sum_{j=1}^{k}a_{1j}h_{j}(P)=f_{1}(P).$

Because $I$ is an information set, we can determine the $a_{1j}$ and hence
$f_{1}$.

Next, induct on $i\in[m]$, assuming that $f_{s}$, $s\in[i-1]$, is known.
Notice that $T_{t}^{(i)}(f)$ can be determined from $T_{t}(f)$ and
$\left\\{f_{s}:s\in[i-1]\right\\}.$ Because

$T_{t}^{(i)}(f)=f_{l-m+t}(p_{t})^{l-m-i+1}+\sum_{s=i}^{l-m}f_{s}(p_{t})^{s-i},$

evaluating at $P\in A_{t}$ gives

$f_{i}(P)=T_{t}^{(i)}(P).$

The rest of the argument is similar to the $s=1$ case. In particular, recall
that $f_{i}=\sum_{j=1}^{k}a_{ij}h_{j}$ for some $a_{ij}\in\mathbb{F}_{q}$.
Then using the values $h_{j}(P)$, $P\in I$ gives a system of $k$ equations
$f_{i}(P)=\sum_{j=1}^{k}a_{ij}h_{j}(P)$ in $k$ unknowns $a_{ij}$, $j\in[k]$.
Because $I$ is an information set, the $a_{ij}$ can be found, hence revealing
$f_{i}$. In this way, $\left\\{f_{s}(x_{1},\dots,x_{M}):s\in[m]\right\\}$ may
be determined, and
$f(x_{1},\ldots,x_{M})=\sum_{s=1}^{l}f_{s}(x_{1},\ldots,x_{M})\nu_{s}.$ ∎∎

To obtain a partition of the evaluation points for which Theorem 5 applies, it
is most helpful if an information set is known, especially one of cardinality
$k$. In the absence of this information, one may take $I$ to be the set of all
evaluation points. The downside to doing so is that then one needs
$\left\\{T_{t}(f)(P):t\in[m],P\in I\right\\}$ and the full generator matrix
$A$.

The next result provides a hypothesis on the partition which is easier to
check than whether a set $I$ is an information set.

######  Corollary 6.

Suppose
$I:=A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
with $\mid I\mid\geq\deg G$ and $\sum_{P\in I}P-G$ is not principal. Then
$f\in\mathcal{L}_{l}(G)$ depends only on $\left\\{T_{t}(f)(P):t\in[m],P\in
I\right\\}$ and $A\mid_{I}$.

###### Proof.

Consider the divisor $D^{\prime}=\sum_{P\in A_{1}\cup\cdots\cup A_{m}}P$. The
evaluation map restricted to the points of $D^{\prime}$ has kernel
$\mathcal{L}(G-D^{\prime})$. If $\deg D^{\prime}\geq\deg G$ and $G-D^{\prime}$
is not principal, then $\mathcal{L}(G-D^{\prime})=\\{0\\}$. Thus,
$A_{1}\cup\cdots\cup A_{m}$ contains an information set. The proof follows
from Theorem 5.∎∎

The requirement of the non-principality of $D^{\prime}-G$ is unavoidable, as
the following example demonstrates.

######  Example 7.

Let $q=2$ and $\mathcal{X}$ be the Hermitian curve given by $x^{3}=y^{2}+y$
over $\mathbb{F}_{4}$. Set $G=2P_{\infty}$, where $P_{\infty}$ denotes the
unique point at infinity on $\mathcal{X}$. Take $A_{1}\cup
A_{2}=\\{(0,0),(0,1)\\}$. Notice that $\mathcal{L}(G)=\left<1,x\right>$. The
evaluation of $\mathcal{L}(G)$ in the points of $A_{1}\cup A_{2}$ are

$\begin{pmatrix}0&0\\\ 1&1\end{pmatrix}$

and thus $A_{1}\cup A_{2}$ does not form an information set of
$C((0,0)+(0,1),G)$.

One may note that Theorem 5 and Corollary 6 recover $f\in\mathcal{L}_{l}(G)$
by utilizing an array that can be expressed as

(2.2)
$\left[\begin{array}[]{cccc}T_{1}(f)(P_{1})&T_{1}(f)(P_{2})&\cdots&T_{1}(f)(P_{n^{\prime}})\\\
T_{2}(f)(P_{1})&T_{2}(f)(P_{2})&\cdots&T_{2}(f)(P_{n^{\prime}})\\\
\vdots&\vdots&&\vdots\\\
T_{m}(f)(P_{1})&T_{m}(f)(P_{2})&\cdots&T_{m}(f)(P_{n^{\prime}})\end{array}\right]\in\mathbb{F}_{q}^{m\times
n^{\prime}}$

where $\left\\{P_{1},\dots,P_{n^{\prime}}\right\\}=A_{1}\cup\dots\cup A_{m}$.
Since $m<l$, the array (2.2) consists of fewer elements of $\mathbb{F}_{q}$
than the $ln$ elements required to express the coordinates of the codeword
$\textnormal{ev}(f)$.

We will see in the next subsection how information about the annihilator
functions provides a better description of the virtual projection. Then, in
Section 3, we will see how these ideas support fractional decoding.

###  2.3. Virtual projections from particular partitions

We consider virtual projections by specifying the partitions of evaluation
points and associated annihilator functions. This will allow us to better
understand the virtual projection in Proposition 4 and better control it for
fractional decoding in Section 3.

Let $\mathcal{X}$ be a curve over $\mathbb{F}_{q}^{\ell}$ with a single point
at infinity $P_{\infty}$. Let
$\\{P_{1},\ldots,P_{n}\\}\subseteq\mathcal{X}(\mathbb{F}_{q})$ and assume
there is a polynomial $z\in\mathbb{F}_{q}(\mathcal{X})$ such that there exists
$\alpha_{1},\ldots,\alpha_{r}\in\mathbb{F}_{q}$ such that

$(z-\alpha_{i})=\sum_{j=1}^{s}P_{ij}-rP_{\infty}$

and $\\#\\{P_{ij}\ :\ i\in[r],\ j\in[s]\\}=rs\leq n$. Let
$A^{\prime}_{1}\dot{\cup}\cdots\dot{\cup}A^{\prime}_{m}$ be a partition of
$\\{\alpha_{1},\ldots,\alpha_{r}\\}$ and let

$A_{t}=\cup_{a\in A^{\prime}_{t}}P_{a}.$

This is equivalent to taking $A_{t}=\cup_{a\in
A^{\prime}_{t}}\mathrm{supp}(z-a)_{0}$ and we may take the annihilator
polynomial of $A_{t}$ to be $p_{t}=\prod_{a\in A^{\prime}_{t}}(z-a)$.

With this, we can understand where the virtual projection lies.

######  Theorem 8.

For any $A^{\prime}_{1}\dot{\cup}\cdots\dot{\cup}A^{\prime}_{m}$ as above and
$t\in[m]$, the $t$-virtual projection of $\mathcal{C}_{l}(\beta P_{\infty})$
is a subcode of $\mathcal{C}(\beta_{t}P_{\infty})$ where
$\beta_{t}=\beta+s\mid A^{\prime}_{t}\mid(l-m)$.

###### Proof.

Let $f\in\mathcal{L}_{l}(\beta P_{\infty})$. Using the fact that
$v_{P_{\infty}}(p_{t})=s\mid A^{\prime}_{t}\mid$ and $v_{P}(p_{t})\geq 0$ for
all other $P\neq P_{\infty}$, with Equation (2.1) we can see that

$v_{P}(T_{t}(f))\geq\begin{cases}-v_{P}(\beta P_{\infty})&\text{if}\
v_{P}(p_{t})\geq 0\\\ -v_{P}(\beta P_{\infty})+(l-m)v_{P}(p_{t})&\text{if}\
v_{P}(p_{t})<0\end{cases}.$

Then $v_{P}(T_{t}(f))\geq 0$ for all $P\neq P_{\infty}$ and
$v_{P_{\infty}}(T_{t}(f))\geq-\beta-(l-m)s\mid A^{\prime}_{t}\mid$ and we have
the conclusion. ∎

As an application of Theorem 8, we consider codes from a family of Kummer
extensions and two natural partitions of evaluation points that give rise to
virtual projections.

######  Corollary 9.

Consider the curve $\mathcal{X}$ given by $L(y)=x^{u}$ over
$\mathbb{F}_{q^{l}}$ where
$L(y)=\sum_{i=0}^{r}a_{i}y^{q^{i}}\in\mathbb{F}_{q^{l}}[y]$ is a separable,
linearized polynomial with $a_{0},a_{r}\neq 0$, and $u\mid\frac{q^{l}-1}{q-1}$
and the associated one-point code $\mathcal{C}_{l}(D,\beta P_{\infty})$ where
$D$ is supported by $n<\beta$ $\mathbb{F}_{q}$-rational points on
$\mathcal{X}$.

  1. (1)

Suppose
$A^{\prime}_{1}\dot{\cup}\cdots\dot{\cup}A^{\prime}_{m}\subseteq\mathbb{F}_{q^{r}}$,
and let $A_{i}=\\{P_{ab}\in\mathcal{X}(\mathbb{F}_{q^{r}})\ :\ a\in
A^{\prime}_{i}\\}$ for $i\in[m]$. Then

$\mathcal{V}_{t}(\beta P_{\infty})\subseteq
C\left((\beta+(l-m)|A^{\prime}_{i}|q^{r-1}\right)P_{\infty}).$

  2. (2)

Suppose
$A^{\prime}_{1}\dot{\cup}\cdots\dot{\cup}A^{\prime}_{m}\subseteq\mathbb{F}_{q^{r}}^{\ast}$.
Let $D$ be the sum of all $\mathbb{F}_{q^{r}}$-rational points $(a,b)$ of
$\mathcal{X}_{q,r}$ with $a\neq 0$, and consider
$A_{i}=\\{P_{ab}\in\mathcal{X}(\mathbb{F}_{q^{r}})\ :\ b\in
A^{\prime}_{i}\\}$, $i\in[m]$. Then the virtual projection of the code with
evaluation points in the support of $D$ is

$\mathcal{V}_{t}(\beta P_{\infty})\subseteq
C\left(\left(\beta+(l-m)|A^{\prime}_{i}|u\right)P_{\infty}\right).$

Several well-known one-point algebraic geometry codes satisfy the hypotheses
given in Theorem 8, allowing them to be considered as collections of codes
over $\mathbb{F}_{q}$ rather than over $\mathbb{F}_{q^{l}}$.

######  Example 10.

Let $\mathbb{F}_{4}$ with $\alpha$ a primitive element such that
$\alpha^{2}+\alpha+1=0$ and $\mathcal{X}$ be the Hermitian curve defined by
$x^{3}=y^{2}+y$. We can define two partitions using the polynomials $x$ and
$y$.

For any $a\in\mathbb{F}_{q}^{\ast}$ we have

$(x-a)_{0}=P_{a\alpha}+P_{a\alpha^{2}}$

and

$(x)_{0}=P_{00}+P_{01}.$

Thus, we can use any partition of $\mathbb{F}_{q}$ to define a virtual
projection as in Theorem 8 for any code $\mathcal{C}(\beta P_{\infty})$.

We can do the same with a partition of $\\{\alpha,\alpha+1\\}$ since for
$a=\alpha,\alpha+1$, we have $(y-a)_{0}=P_{1a}+P_{\alpha a}+P_{\alpha+1a}$. We
can use this to make a virtual projection satisfying the conditions of Theorem
5 for any $G=\beta P_{\infty}$ with $\beta<8$.

##  3\. Decoding via virtual projections

In this section, we detail how virtual projections give rise to fractional
decoding algorithms for algebraic geometry codes over extension fields. First,
we consider fractional decoding directly from virtual projections in
Subsection 3.1. Next, in Subsection 3.2, we use virtual projections to define
related interleaved codes, which gives rise to another approach to fractional
decoding in Subsection 3.3.

Throughout, we consider a curve $\mathcal{X}$ over $\mathbb{F}_{q^{l}}$ with
divisors $G$ and $D$ whose supports consist only of $\mathbb{F}_{q}$-rational
points and the algebraic geometry code
$\mathcal{C}_{l}(G,D)\subseteq\mathbb{F}_{q^{l}}^{n}$. As before, we also take
$\left\\{\zeta_{1},\ldots,\zeta_{l}\right\\}$ to be a basis of the extension
$\mathbb{F}_{q^{l}}/\mathbb{F}_{q}$ with
$\left\\{\nu_{1},\ldots,\nu_{l}\right\\}$ its dual basis.

###  3.1. Fractional decoding directly from virtual projections

Given a received word $w=\textnormal{ev}(f)+e\in\mathbb{F}_{q^{ln}}$ where
$f\in\mathcal{L}_{l}(G)$, for each $i\in[n]$ and $t\in[m]$, download

(3.1)
$w_{i}^{t}:=tr\left(\zeta_{l-m+t}w_{i}\right)p_{t}(P_{i})^{l-m}+\sum_{s=1}^{l-m}tr\left(\zeta_{s}w_{i}\right)p_{t}(P_{i})^{s-1};$

We aim to show that $\textnormal{ev}(f)$ can be recovered from

(3.2)
$\pi(w):=\left[\begin{array}[]{cccc}w_{1}^{1}&w_{2}^{1}&\cdots&w_{n^{\prime}}^{1}\\\
w_{1}^{2}&w_{2}^{2}&\cdots&w_{n^{\prime}}^{2}\\\ \vdots&\vdots&&\vdots\\\
w_{1}^{m}&w_{2}^{m}&\cdots&w_{n^{\prime}}^{m}\end{array}\right]\in\mathbb{F}_{q}^{m\times
n^{\prime}}$

if $wt(e)$ is not too large. Notice that if $w=\textnormal{ev}(f)$, meaning
$wt(e)=0$, then

$\begin{array}[]{lcl}w_{i}^{t}&=&tr\left(\zeta_{l-m+t}f(P_{i})\right)p_{t}(P_{i})^{l-m}+\sum_{s=1}^{l-m}tr\left(\zeta_{s}f(P_{i})\right)p_{t}(P_{i})^{s-1}\\\
\\\
&=&f_{l-m+t}(P_{i})p_{t}(P_{i})^{l-m}+\sum_{s=1}^{l-m}f_{s}(P_{i})p_{t}(P_{i})^{s-1}\\\
\\\ &=&T_{t}(f)(P_{i}).\end{array}$

Indeed, if
$f(x_{1},\ldots,x_{M})=\sum_{j=1}^{k}a_{j}h_{j}(x_{1},\ldots,x_{M})$ where
$h_{j}\in\mathbb{F}_{q}[x]$, then

$tr\left(\zeta_{s}f(x_{1},\ldots,x_{M})\right)=tr\left(\zeta_{s}\sum_{j=1}^{k}a_{j}h_{j}(x_{1},\ldots,x_{M})\right)=tr\left(\zeta_{s}\sum_{j=1}^{k}a_{j}\right)h_{j}(x_{1},\ldots,x_{M}).$

We have already seen in Theorem 5 that $\textnormal{ev}(f)$ can be recovered
from the array

$\left[\begin{array}[]{cccc}T_{1}(f)(P_{1})&T_{1}(f)(P_{2})&\cdots&T_{1}(f)(P_{n^{\prime}})\\\
T_{2}(f)(P_{1})&T_{2}(f)(P_{2})&\cdots&T_{2}(f)(P_{n^{\prime}})\\\
\vdots&\vdots&&\vdots\\\
T_{m}(f)(P_{1})&T_{m}(f)(P_{2})&\cdots&T_{m}(f)(P_{n^{\prime}})\end{array}\right]\in\mathbb{F}_{q}^{m\times
n^{\prime}}.$

The challenge now is to confirm that this is the case if
$w=\textnormal{ev}(f)+e$ where $e\in\mathbb{F}_{q^{l}}$ has $0<wt(e)<B$,
meaning some (but fewer than $B$) positions are in error, for some specified
value $B$. Given that $mn<ln$, it is reasonable to suspect that
$B<\frac{n-\deg G}{2}$, the number of correctable errors for the code. This
challenge is addressed in the next result where we see the value of $B$
depends on the degree of the pole divisor of an annihilator function.

######  Theorem 11.

Suppose $I\subseteq
A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$,
for some $m\in\mathbb{N}$ and some information set $I$ for the code
$C_{l}(G)$. The code $C_{l}(D,G)$ can correct any

(3.3) $\left\lfloor\frac{n-\left(\deg
G+(l-m)\max\left\\{\deg(p_{t})_{\infty}:t\in[m]\right\\}\right)}{2}\right\rfloor$

errors using $mn^{\prime}$ entries of $\mathbb{F}_{q}$.

###### Proof.

Consider a received word $w$ in which at most $d^{\prime}$ errors have
occurred, where $d^{\prime}$ denotes the value in (3.3), is the meaning
$w=ev(f)+e$ where $wt(e)\leq d^{\prime}$. Let
$\mathcal{E}=\left\\{i\in[n]:e_{i}\neq 0\right\\}$ denote the set of positions
in error.

Download the $w_{i}^{t}$, $i\in[n^{\prime}]$, $t\in[m]$ as in Equation (3.2).
Notice that $w_{i}^{t}=T_{t}(f)(P_{i})+e_{ti}$ for some
$e_{ti}\in\mathbb{F}_{q}$. Moreover, $e_{ti}\neq 0$ only if $i\in\mathcal{E}$.
Thus, we may consider

$Row_{t}\left(\pi(w)\right)=ev\left(T_{t}(f)\right)+\left(e_{t1},\dots,e_{tn^{\prime}}\right)$

as a received word in $C(G_{t})\subseteq\mathbb{F}_{q}^{n^{\prime}}$ where

$G_{t}=G+(l-m)\left(p_{t}\right)_{\infty}.$

Since the minimum distance of $C(G_{t})$ is at least $n-\deg G_{t}$, at least
$\frac{n-\deg G_{t}}{2}$ errors may be corrected in each row
$Row_{t}\left(\pi(w)\right)$, giving rise to $ev\left(T_{t}(f)\right)$ if
there are no more than $\frac{n-\deg G_{t}}{2}$ errors. Hence, the up to
$d^{\prime}$ errors in each $Row_{t}\left(\pi(w)\right)$ are corrected to give
$ev\left(T_{t}(f)\right)$. This gives an array as shown in (2.2).

Now, we apply Theorem 5 to this array to obtain
$\textnormal{ev}(f)\in\mathbb{F}_{q^{l}}^{n}$. ∎

Observe that this fractional decoding consists of the following steps for a
received word $w\in\mathbb{F}_{q^{l}}^{n}$:

  1. (1)

Download $\pi(w)\in\mathbb{F}_{q}^{m\times n^{\prime}}$ as in Equation (3.2).

  2. (2)

Apply a decoding algorithm for
$C(G+G_{t})\subseteq\mathbb{F}_{q}^{n^{\prime}}$ to $w^{t}$ to obtain
$T_{t}(f)$ for all $t\in[m]$.

  3. (3)

Apply Theorem 5 to obtain
$\textnormal{ev}(f)\in\mathbb{F}_{q^{l}}^{n^{\prime}}$.

The performance will be dictated by the partition of the evaluation set
$A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
and associated choice of annihilator functions. Moreover, one may select among
the various decoding algorithms for algebraic geometry codes for Step (1) and
evolve with related advances such as [10].

While Theorem 11 is stated in terms of an information set, the hypothesis of
Corollary 6 allows us to give the next, similar result which does not rely on
knowledge of an information set.

######  Corollary 12.

Suppose
$I:=A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}\subseteq\left\\{P_{1},\dots,P_{n}\right\\}$
with $\mid I\mid\geq\deg G$ and $\sum_{P\in I}P-G$ is not principal. Then the
code $C_{l}(D,G)$ can correct any

$\left\lfloor\frac{n-\left(\deg
G+(l-m)\max\left\\{\deg(p_{t})_{\infty}:t\in[m]\right\\}\right)}{2}\right\rfloor$

errors using $mn$ entries of $\mathbb{F}_{q}$.

###### Proof.

According to the proof of Corollary 6, $A_{1}\dot{\cup}\cdots\dot{\cup}A_{m}$
contains an information set. The result now follows as in the proof of Theorem
11. ∎

###  3.2. Interleaved codes from virtual projections

The use of interleaved codes in fractional decoding can, with a certain
probability, provide a larger decoding radius than non-interleaved codes. In
this subsection, we recast the virtual projections of algebraic geometry codes
as interleaved codes. We begin with recalling useful terminology.

Given a collection of codes $C_{1},\dots,C_{m}\subseteq\mathbb{F}_{q}^{n}$,
one may define an interleaved code

$\mathcal{IC}(C_{1},\dots,C_{m}):=\left\\{\left[\begin{array}[]{c}c_{1}\\\
c_{2}\\\ \vdots\\\ c_{m}\end{array}\right]:c_{i}\in C_{i},i\in[m]\right\\}.$

If $C_{i}=C$ for all $i\in[m]$, then $\mathcal{IC}(C_{1},\dots,C_{m})$ is said
to be homogeneous; otherwise, $\mathcal{IC}(C_{1},\dots,C_{m})$ is called
heterogeneous.

Given a curve $X$ over a finite field $\mathbb{F}$, an interleaved algebraic
geometry code on $\mathcal{X}$ is of the form

$\mathcal{IC}(D,G_{1},\dots,G_{m}):=\left\\{\left[\begin{array}[]{c}c_{1}\\\
c_{2}\\\ \vdots\\\ c_{m}\end{array}\right]:c_{i}\in
C(D,G_{i}),i\in[m]\right\\}.$

If $G_{i}=G$ for all $i\in[m]$, then we write $\mathcal{IC}(D,G,m)$ for this
homogeneous code.

According to Proposition 4, we may consider the set of arrays in (2.2) as
subcodes of interleaved codes. In particular, it may be considered as a
subcode of the heterogeneous interleaved code
$\mathcal{IC}(D,G_{1},\dots,G_{m})$ where

$G_{t}:=G+(l-m)\left(p_{t}(x_{1},\ldots,x_{M})\right)_{\infty}P$

for all $t\in[m]$:

$\left\\{\left[\begin{array}[]{c}\textnormal{ev}(T_{1}(f))\\\
\textnormal{ev}(T_{2}(f))\\\ \vdots\\\
\textnormal{ev}(T_{m}(f))\end{array}\right]:f\in\mathcal{L}_{l}(G)\right\\}\subseteq\mathcal{IC}(D,G_{1},\dots,G_{m}).$

Alternatively, it may be viewed as a subcode of the homogeneous interleaved
code $\mathcal{IC}(D,G,m)$ where

$G=\sum_{P\in S}\left(v_{P}(G)-(l-m)v_{P}(p_{t})\right)$

and $S=\cup_{t\in[m]}\textnormal{supp}(p_{t})_{\infty}$. :

$\left\\{\left[\begin{array}[]{c}\textnormal{ev}(T_{1}(f))\\\
\textnormal{ev}(T_{2}(f))\\\ \vdots\\\
\textnormal{ev}(T_{m}(f))\end{array}\right]:f\in\mathcal{L}_{l}(G)\right\\}\subseteq\mathcal{IC}(D,G,m).$

We will now see that viewing the collection of virtual projections of certain
algebraic geometry codes as interleaved codes facilitates fractional decoding.

###  3.3. Fractional decoding via interleaved algebraic geometry codes

In this subsection, we provide fractional decoding algorithms for certain one-
point algebraic geometry codes from constant field extension by viewing their
virtual projections as interleaved codes.

In [1], it is shown that interleaved one-point algebraic geometry codes $C(G)$
of length $n$ constructed from a curve of genus $g$ can be decoded up to $e$
errors where $e\leq(n-\deg G-g)b-\frac{c}{al+1}$ for some $c>0$,
$a:=\frac{\ln{(q^{l}-1)}}{\ln{q^{l}}}$, and $b:=\frac{la}{la+1}$. The same
decoder may be used in the case of $e$ errors where $e\leq g-1$ and succeeds
with probability at least $1-\frac{q}{q^{c}(q-1)}$. We will see how this
decoder, and those tailored to codes from particular codes, aid in fractional
decoding.

Consider the code $C_{l}(\beta P)$ where

$T_{t}\left(\mathcal{L}\left(\beta
P\right)\right)\subseteq\mathcal{L}\left(\alpha P\right)\ \forall t\in[m].$

Fix a parameter $t^{\prime}\in\mathbb{N}$. Then
$\mathcal{L}\left(\left(g+t^{\prime}\right)P\right)\subseteq\mathcal{L}\left(\left(g+t^{\prime}+\alpha\right)P\right)$.
Take a basis $\left\\{\varphi_{1},\dots,\varphi_{s}\right\\}$ for
$\mathcal{L}\left(\left(g+t^{\prime}+\alpha\right)P\right)$ so that
$\left\\{\varphi_{1},\dots,\varphi_{s^{\prime}}\right\\}$ is a basis for
$\mathcal{L}\left(\left(g+t^{\prime}\right)P\right)$. Then the transposes of

$V:=\left[\begin{array}[]{ccc}\varphi_{1}(P_{1})&\cdots&\varphi_{s}(P_{1})\\\
\varphi_{1}(P_{2})&\cdots&\varphi_{s}(P_{2})\\\ \vdots&&\vdots\\\
\varphi_{1}(P_{n})&\cdots&\varphi_{s}(P_{n})\end{array}\right]\mathrm{\ and\
}W:=\left[\begin{array}[]{ccc}\varphi_{1}(P_{1})&\cdots&\varphi_{s^{\prime}}(P_{1})\\\
\varphi_{1}(P_{2})&\cdots&\varphi_{s^{\prime}}(P_{2})\\\ \vdots&&\vdots\\\
\varphi_{1}(P_{n})&\cdots&\varphi_{s^{\prime}}(P_{n})\end{array}\right]$

are generator matrices of $C\left(\left(g+t^{\prime}+\alpha\right)P\right)$
and $C\left(\left(g+t^{\prime}\right)P\right)$ Determine a partition as in
Theorem 5 along with the associated annihilator polynomials $p_{t}$,
$t\in[m]$.

Now suppose $w=ev(f)+e$ is received for some $f\in\mathcal{L}_{l}(\beta P)$.
From the projection of $w$ given in Equation 3.2, for each $i\in[m]$, form the
matrix

(3.4)
$W_{i}:=-\left[\begin{array}[]{cccc}w_{1}^{i}\varphi_{1}(P_{1})&w_{1}^{i}\varphi_{2}(P_{1})&\cdots&w_{1}^{i}\varphi_{s^{\prime}}(P_{1})\\\
w_{2}^{i}\varphi_{1}(P_{1})&w_{2}^{i}\varphi_{2}(P_{1})&\cdots&w_{2}^{i}\varphi_{s^{\prime}}(P_{1})\\\
\vdots&\vdots&&\vdots\\\
w_{n}^{i}\varphi_{1}(P_{1})&w_{n}^{i}\varphi_{2}(P_{1})&\cdots&w_{n}^{i}\varphi_{s^{\prime}}(P_{1})\end{array}\right]\in\mathbb{F}_{q}^{m\times
n^{\prime}}.$

and for $i\in[l]\setminus[m]$, let $W_{i}\in\mathbb{F}_{q}^{n\times n}$ be the
zero matrix. Set

$A:=\left[\begin{array}[]{cccc|c}V&&&&W_{1}\\\ &V&&&W_{2}\\\
&&\ddots&&\vdots\\\ &&&V&W_{l}\end{array}\right].$

Given $v=(v_{1};\dots;v_{l};v_{l+1})\in
NS(A)\cap\mathbb{F}_{q}^{ls+s^{\prime}}$, set

$T^{t}(f):=\frac{\sum_{j=1}^{s}v_{tj}\varphi_{j}}{\sum_{j=1}^{s^{\prime}}v_{tj}\varphi_{j}}$

for all $t\in[m]$. Then apply Theorem 5 with the $T^{t}(f)$ in place of the
$T_{t}(f)$ to recover $ev(f).$ According to [1, Theorem 1], this proves the
following.

######  Proposition 13.

Consider the code $C_{l}(\beta P)$ where $T_{t}\left(\mathcal{L}\left(\beta
P\right)\right)\subseteq\mathcal{L}\left(\alpha P\right)\ \forall t\in[m]$. A
received word $w=\textnormal{ev}(f)+e\in\mathbb{F}_{q^{ln}}$, where
$f\in\mathcal{L}_{l}(\alpha P)$ for some $\mathbb{F}_{q}$-rational point $P$,
can be decoded from $mn<ln$ symbols of $\mathbb{F}_{q}$ provided
$wt(e)\leq(n-\alpha-g)b-\frac{c}{al+1}$ for some $c>0$. If
$(n-\alpha-g)b-\frac{c}{al+1}<e\leq g-1$, then decoder failure is possible but
the decoder succeeds with probability at least $1-\frac{q}{q^{c}(q-1)}$.

While Proposition 13 applies to a large family of algebraic geometry codes,
advances have been made for codes from particular curves.

Recall that the algorithm downloads symbols which are $m$ corrupted codewords
of $\mathcal{C}(\beta_{r}P_{\infty})$, where
$\beta_{r}=\beta+q\left\lceil\frac{\beta+1}{q}\right\rceil\left(\frac{1-\alpha}{\alpha}\right)$.
Those $m$ corrupted codewords can be arranged to form the following matrix

(3.5)
$\pi(h)=\left[\begin{array}[]{llll}T_{0}(h)(P_{1})&T_{0}(h)(P_{2})&\ldots&T_{0}(h)(P_{n})\\\
T_{1}(h)(P_{1})&T_{1}(h)(P_{2})&\ldots&T_{1}(h)(P_{n})\\\
\vdots&\vdots&\ddots&\vdots\\\
T_{m-1}(h)(P_{1})&T_{m-1}(h)(P_{2})&\ldots&T_{m-1}(h)(P_{n})\end{array}\right]\in
M_{m,n}(\mathbb{F}_{q^{r}}).$

The matrix $\pi(h)$ is called the homogeneous virtual projection of $h$.
Moreover, the downloaded symbols may be viewed as corrupted a codeword of a
homogeneous interleaved norm-trace code

(3.6)
$\mathcal{C}_{\mathcal{X}}(n,m,\beta_{r})=\left\\{\left[\begin{array}[]{c}c^{(0)}\\\
\vdots\\\ c^{(m-1)}\end{array}\right]\in
M_{m,n}(\mathbb{F}_{q^{r}}):\begin{array}[]{l}c^{(i)}\in\mathcal{C}(\beta_{r}P_{\infty})\\\
\beta_{r}=\beta+q\left\lceil\frac{\beta+1}{q}\right\rceil\left(\frac{1-\alpha}{\alpha}\right)\end{array}\right\\}.$

So, we point out that it is possible to apply the power decoding algorithm
present in [9] to improve the $\alpha$-decoding correction capability (with
some failure probability) of our decoding algorithm for certain families of
codes.

In the same way, when we are considering a small constant extension it is
possible to use the downloaded symbols to create the following matrix

(3.7)
$\pi^{\star}(h)=\left[\begin{array}[]{llll}R_{0}(h)(P_{1})&R_{0}(h)(P_{2})&\ldots&R_{0}(h)(P_{n})\\\
R_{1}(h)(P_{1})&R_{1}(h)(P_{2})&\ldots&R_{1}(h)(P_{n})\\\
\vdots&\vdots&\ddots&\vdots\\\
R_{m-1}(h)(P_{1})&R_{m-1}(h)(P_{2})&\ldots&R_{m-1}(h)(P_{n})\end{array}\right]\in
M_{m,n}(\mathbb{F}_{q^{r}}).$

The matrix $\pi(h)$ is called the homogeneous virtual projection of $h$.
Moreover, the downloaded symbols may be viewed as corrupted codeword of a
homogeneous interleaved norm-trace code

(3.8)
$\mathcal{C}_{\mathcal{X}}(n,m,\beta^{\star})=\left\\{\left[\begin{array}[]{c}c^{(0)}\\\
\vdots\\\ c^{(m-1)}\end{array}\right]\in
M_{m,n}(\mathbb{F}_{q^{r}}):\begin{array}[]{l}c^{(i)}\in\mathcal{C}(\beta_{r}P_{\infty})\\\
\beta^{\star}=\beta+q(q+1)(l-m)\end{array}\right\\}.$

So, we can apply the power decoding algorithm present in [9] to improve the
$\alpha$-decoding correction capability (with some failure probability) of our
decoding algorithm.

Analogous to fractional decoding directly from virtual projections and
advances in decoding algebraic geometry codes, fractional decoding of
algebraic geometry codes via projections represented as interleaved codes can
make use of emerging advances in the decoding of interleaved algebraic
geometry codes.

## Acknowledgments

The first and second author were partially supported by NSF DMS-2201075 and
the Commonwealth Cyber Initiative. The second and third author were partially
supported by DMS-1802345.

## References

  * [1] Andrew Brown, Lorenz Minder, and Amin Shokrollahi.  Improved decoding of interleaved AG codes.  In Nigel P. Smart, editor, Cryptography and Coding, pages 37–46, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.
  * [2] Alexandros G. Dimakis, P. Brighten Godfrey, Yunnan Wu, Martin J. Wainwright, and Kannan Ramchandran.  Network coding for distributed storage systems.  IEEE Transactions on Information Theory, 56(9):4539–4551, 2010.
  * [3] Yanyan Gao, Qin Yue, Xinmei Huang, and Jun Zhang.  Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes.  IEEE Transactions on Information Theory, 67(10):6619–6626, 2021.
  * [4] Venkatesan Guruswami and Mary Wootters.  Repairing Reed-Solomon codes.  IEEE Transactions on Information Theory, 63(9):5684–5698, 2017.
  * [5] Venkatesan Guruswami and Chaoping Xing.  List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the Singleton bound.  In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC 13, pages 843–852, New York, NY, USA, 2013. Association for Computing Machinery.
  * [6] Hiram H. López and Gretchen L. Matthews.  Multivariate Goppa codes.  IEEE Transactions on Information Theory, 69(1):126–137, 2023.
  * [7] Gretchen L. Matthews, Aidan W. Murphy, and Welington Santos.  Fractional decoding of codes from Hermitian curves.  In 2021 IEEE International Symposium on Information Theory (ISIT), pages 515–520, 2021.
  * [8] Gretchen L. Matthews, Aidan W. Murphy, and Welington Santos.  Fractional decoding of r-Hermitian codes.  Finite Fields and Their Applications, 92:102278, 2023.
  * [9] Sven Puchinger, Johan Rosenkilde, and Irene Bouw.  Improved power decoding of interleaved one-point Hermitian codes.  Designs, Codes and Cryptography, 87:589–607, 2019.
  * [10] Sven Puchinger, Johan Rosenkilde, and Grigory Solomatov.  Improved power decoding of algebraic geometry codes.  In 2021 IEEE International Symposium on Information Theory (ISIT), pages 509–514, 2021.
  * [11] Welington Santos.  On fractional decoding of Reed-Solomon codes.  In 2019 IEEE International Symposium on Information Theory (ISIT), pages 1552–1556, 2019.
  * [12] Henning Stichtenoth.  Algebraic Function Fields and Codes.  Springer Publishing Company, Incorporated, 2nd edition, 2008.
  * [13] Itzhak Tamo, Min Ye, and Alexander Barg.  Fractional decoding: Error correction from partial information.  In 2017 IEEE International Symposium on Information Theory (ISIT), pages 998–1002, 2017.
  * [14] Min Ye and Alexander Barg.  Explicit constructions of MDS array codes and RS codes with optimal repair bandwidth.  In 2016 IEEE International Symposium on Information Theory (ISIT), pages 1202–1206, 2016.