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Reed Solomon Codes for Molecular

Communication with a Full Absorption Receiver

Maheshi B. Dissanayake, Yansha Deng, Arumugam Nallanathan, E. M. N. Ekanayake, and Maged Elkashlan

Abstract—Molecular communication (MC) has recently e-

merged as a novel paradigm for nano-scale communication

utilizing molecules as information carriers. In diffusion-based

molecular communication, the system performance is constrained

by the inter-symbol-interference (ISI) caused by crossover of

information carrying molecules in consecutive bits. To cope with

this, we propose the Reed-Solomon (RS) codes as an error

recovery tool, to improve the transmission reliability in diffusion-

based MC systems. To quantify the performance improvement

due to RS codes, we derive the analytical expression for the

approximate bit error probability (BEP) of the diffusion-based

MC system with the full absorption receiver. We further develop

the particle-based simulation framework to simulate the proposed

system with RS code to verify the accuracy of our derived

analytical results. Our results show that, as the number of

molecules per bit increases, the BEP of the system with RS

codes exhibits a substantial improvement than that of non-coded

systems. Furthermore, the BEP of the proposed system with

RS codes can be greatly improved by increasing the minimum

distance of the codeword.

Index Terms—Molecular communication, error correction

codes, Reed Solomon codes, particle-based simulation,

I. INTRODUCTION

WITH the advancements in nano-technology, new multi-

disciplinary research on communication between bio-

nanomachines via chemical signals is emerging [1]. This new

paradigm, namely Molecular Communication (MC), adopts

biologically-inspired techniques for information transmission

among nano-machines. The nature inspired molecular com-

munication can be observed in calcium signalling among

cells, and Deoxyribonucleic Acid (DNA) signalling among

DNA segments [2] In this paper, we concentrate on the

molecular communication via diffusion (MCvD) [3] to provide

the baseline for the communication design, which will serve as

an important foundation for designing more complicated MC

systems with drift. In MCvD, the transmitter modulates the

information onto the physical properties of the information

carrying molecules. Once emitted by the transmitter, the

molecules diffuse in the medium and certain portion of the

Manuscript received Sep. 6, 2016; revised Jan. 9, 2017 and Feb. 9, 2017;

accepted Feb. 16, 2017. The editor coordinating the review of this manuscript

and approving it for publication was Prof. Nghi Tran.

Maheshi B. Dissanayake is with the Department of Electrical and Electronic

Engineering, Faculty of Engineering, University of Peradeniya, Sri Lanka and

part of this work was carried out while at King0sCollege London, London,

WC2R 2LS, UK. (e-mail: maheshid@ee.pdn.ac.lk).

E.M.N. Ekanayake is with the Faculty of Engineering, University of

Peradeniya, Sri Lanka. (e-mail: neka@ee.pdn.ac.lk).

Y. Deng and A. Nallanathan are with King0sCollege London, London,

WC2R 2LS, UK (email:yansha.deng,arumugam.nallanathan@kcl.ac.uk).

M. Elkashlan is with Queen Mary University of London, London E1 4NS,

UK (email: maged.elkashlan@qmul.ac.uk).

transmitted molecules arrive at the receiver during the current

bit interval. The receiver then demodulates the transmitted

information using the number of molecules captured during

that bit interval. Due to the random walk of each molecule in

the ﬂuid medium, there exists high probability of crossover

between the molecules emitted in neighboring bits. These

molecules which arrive during different bit intervals, result

in the so-called inter-symbol-interference (ISI) [4]. Note that

ISI is regarded as one of the fundamental bottlenecks of MC

systems that degrades the system reliability.

In the literature, there exist many different approaches

for ISI mitigation in MC. As summarized in [1], borrowing

from traditional communication system designs, equalization

techniques like decision feedback ﬁlter technique, transmitter-

based novel modulation techniques such as molecular transi-

tion shift keying and Molecular ARray-based COmmunication

(MARCO), and energy efﬁcient transmitter and receiver-side

signal power adjustment methods have been proposed as

ISI mitigation techniques for MC environment. Although,

some of these techniques increase the complexity of the MC

system, they provide a signiﬁcant improvement in terms of ISI

mitigation. Furthermore, inspired by the biological behaviour

of MC system, different types of molecules with resistance or

kill effect on information carrying molecules have also been

proposed as another method to overcome the ISI effect [1].

Yet, this technique requires extra resources and intelligence

to handle the resistance molecular type and tends to increase

the amount of molecules in the medium most likely creating

collision.

Error correction codes (ECC) have been proposed as an

alternative way to mitigate the adverse effects of ISI in MC

systems by providing error correction capability at the receiver.

Hence, ECC can results in easy manipulation of the system

compared to using resistance molecules to overcome ISI. More

speciﬁcally, the bit error performance of established codes,

such as Hamming codes (HC) [5], Euclidean Geometry Low

Density Parity Check, and Cyclic Reed-Muller codes [5], have

been applied and investigated in MC systems. A new type

of ECC, called ISI free code, was proposed and designed

to prevent occurrence of ISI via the predeﬁned redundant bit

patterns in MC systems [6]. Furthermore, in [7], the authors

concentrate on improving the bit error performance via self-

orthogonal convolutional codes, which takes into account the

energy requirements at the receiver for decoding the error

correction information.

Different from existing literature, we introduce the Reed-

Solomon (RS) codes for error correction in the point-to-point

MC systems with a point transmitter and a full absorption

2

Fig. 1. Block Diagram of the Proposed MC system

receiver. The RS codes are non-binary block codes and are

highly effective against burst and random errors in many real

channels [8] compared to HC that only capable of correcting

single bit error. Also RS code is one of the few forward

error correction code, which can attain the theoretical limit

known as the Singleton bound. Since, RS codes utilize the

symbol based arithmetic, it can decode codewords with longer

block lengths with less decoding time, compared to HC [8].

Another advantage of RS codes is the existence of efﬁcient

decoding algorithm due to vast applications of RS codes in

data communication. More importantly, simpliﬁed RS codes

are relatively easy to implement in terms of hardware [9].

In this paper, we present an analytical model for the

point-to-point MC systems with RS codes and derive an

expression for the approximated bit error probability (BEP)

of the proposed system. Furthermore, the derived analytical

result is veriﬁed by the proposed simulation framework, which

captures the random Brownian motion of each molecule in

the diffusion-based MC system. The results obtained from

simulation are in close agreement with the analytical values.

Furthermore, it is observed that the BEP performance is greatly

improved via RS codes, and the performance gain increases

with the increase of the number of molecules per bit and

the minimum distance of the codeword. The remainder of the

paper is organized as follows. Section II presents the system

model, and Section III brieﬂy introduce RS codes. Section

IV presents an expression for the BEP of the proposed MC

system. Section V evaluates the performance improvement

achieved by the proposed RS codes using numerical and

simulation results. Conclusions are given in SectionVI.

II. SYSTEM MODEL

We consider a point-to-point diffusion-based MC system

with a full absorption receiver with radius rand a point

transmitter located distance daway from the surface of the

receiver. The block diagram of the proposed MC system is

outlined in Fig. 1. In the diffusion-based MC system, the

information is modulated on the number of the molecules

emitted by the transmitter at start of each bit interval, Tb.

Given that the molecules are released into a medium of large

extent compared to their size, collisions between messenger

molecules are neglected. Each molecule randomly diffuses

in the medium following Brownian motion, with constant

diffusion coefﬁcient D. Once a molecule reaches the surface of

the receiver, it will be absorbed by the receptors at the surface

of the receiver. The process of messenger molecules hitting

the body of the receiver is named as the hitting process.

For the modulation scheme, we adopt the Binary Concen-

tration Shift Keying (BCSK) as in [2]. At the transmitter,

the information is conveyed through the number of molecules

being emitted at the start of each bit interval. At the receiver, if

the total number of molecules absorbed during a bit interval

is above a threshold, the received bit is demodulated as 1,

and otherwise as 0. To facilitate error correction, RS encoded

redundant data is embedded into the message bit pattern at

the transmitter. These extra data is utilized at the receiver to

recover from bit errors, in turn improving the overall BEP.

III. REED SOLOMON CODES

In communication theory, ECC is used for error detection

and correction to achieve an acceptable level of accuracy for

the received information, when data is transmitted through

error prone transmission channels. As one type of ECC, the

Reed Solomon codes was ﬁrst proposed by I. S. Reed and G.

Solomon, which is a non-binary Bose, Chaudhuri, and Hoc-

quenghem (BCH) code, with a simple algorithm for the error

detection and correction [9]. Considering that the RS codes

are well suited for correcting burst errors, they are widely

used in digital data transmission and storage applications. We

denote RS code with nlength codeword and klength input

message block, as RS(n,k) and it is deﬁned over a Galois Field

(GF (pm)), where pis a prime, and mis a positive integer. The

length of the parity bits is n−kbits, which has the capacity

to correct up to n−k

2number of errors per codeword [9].

The RS codes can be implemented very easily at the encoder

using Linear Feedback Shift Registers and at the decoder using

Berlekamp algorithm. The systematic RS encoder, appends

parity information to the original message in such a way that

the constructed codeword is completely divisible by the gen-

erator polynomial using Galois Field algebra. This generator

polynomial is shared by both the RS encoder and decoder.

The RS decoding operates in two main stages. In the

ﬁrst stage, the decoder inspects the received codeword for

errors. The received codeword is identiﬁed as in error, if

it is not completely divisible by the generator polynomial.

This technique is called syndrome computation. If syndrome

calculation results in zero, the decoder terminates, otherwise

in the second stage, the system attempts to correct the error

by detecting the error position and error value. To correct the

identiﬁed errors, the decoder ﬁrst determines the error locator

polynomial using either Berlekamp-Massey algorithm [10] or

Euclidean algorithm [10]. The former leads to a more efﬁcient

implementation, while the latter is easier to implement. In the

next stage, the Chien search algorithm [10] is used to solve

the roots of the error locator polynomial gernerated, which

indicate the error locations in the receieved codeword. Then,

Forney algorithm [10] is used to estimate the error magnitudes.

Once the error location and the error values are estimated, then

a correction is applied to the received codeword to recover

from the error. In depth description of each algorithm used in

the implementation of RS codes can be found in [10] while RS

3

decoder implementation presented in this paper can be found

in [9].

One of the key bottlenecks in the implementation of ECC

in MC environment is the complexity and the processing

power requirements. Inspired by the extensive application of

RS codes in battery powered devices, many research are

conducted on optimizing the energy performance of the RS

coders. For instance, it is reported in [11] that the energy

consumption of RS coding can be reduced by 40% in low

activity environments and can be synthesized by using 45 nm

technology. The reduced complexity algorithms with accept-

able level of bit error rate are another method adopted by

many researchers to achieve low complexity and low power

requirements. For instance, [12] proposes to apply the soft

information based decoding to achieve better results in error

correction rates compared to hard decision decoding, while

keeping the complexity at lower level. In summary, it is

safer to assume that RS codes can be implemented with

reduced complexity in MC with the advanced research on low

complexity implementation of RS coders.

IV. BIT ER ROR PROBABI LI TY

It is known that in MC, the number of absorbed molecules

during one bit interval, Tb, can be modeled as binomial distri-

bution. For computational tractability, the binomial distribution

can be approximated using the normal distribution, when Nm

the number of molecules sent at the start of each bit interval

is large, and Phit (d,t) the fraction of molecules absorbed by

the receiver is not near 0 or 1. Thus, we express the total

number of absorbed molecules, in the ith bit Nhi t [i], during

[(i−1)Tb,iTb]in the absence of ISI as

Nhit [i]∼NNmPhit (d,Tb),NmPhit (d,Tb)[1 −Phit (d,Tb)].

(1)

Yet, due to ISI, the actual number of molecules absorbed

during Tb, account for the number of molecules recieved from

the current bit (ai), Nai, and from the previous bits, Na1:i−1.

However, prior work [13] shows that the most prominent effect

on ISI is contributed by the nearest past bit, ai−1, with an

appropriate bit interval, Tb. As such, in the presence of ISI,

the Nhit [i]can be approximated as

Nhit [i]=aiNai+ai−1Nai−1(2)

where

Nai∼NNmPh1,NmPh1[1 −Ph1](3)

and

Nai−1∼NNmPh2,NmPh2[1 −Ph2]

−NNmPh1,NmPh1[1 −Ph1].

(4)

In (3) and (4), Ph j denotes Phit (d,jTb) and it can be derived

as in [1], using Eq. (6), Eq. (7) and Eq. (8).

For an uncoded binary channel, the bit error probability, Pe,

over all possible combinations of aican be expressed as

Pe=Pr[e|ai=0]Pr[ai=0] +Pr[e|ai=1]Pr[ai=1],(5)

This can be further expanded by considering the possible

outcomes for both aiand ai−1as

Pe=Pr[e|ai=0,ai−1=0].Pr[ai=0].Pr[ai−1=0]

+Pr[e|ai=0,ai−1=1].Pr[ai=0].Pr[ai−1=1]

+Pr[e|ai=1,ai−1=0].Pr[ai=1].Pr[ai−1=0]

+Pr[e|ai=1,ai−1=1].Pr[ai=1].Pr[ai−1=1]

(6)

where Pr[e|ai,ai−1]is the conditional probability of error for

ith bit with known transmitted bits aiand ai−1. The conditional

probabilities in equation (6) can be approximated as

Pr[e|ai=0,ai−1=1] =Pr[Nhit ≥τ]=Pr[Nai−1≥τ]

≈Qτ−[(NmPh2)−(NmPh1)]

√(NmPh1[1 −Ph1])+(NmPh2[1 −Ph2])(7)

Pr[e|ai=1,ai−1=0] =Pr[Nhit < τ]=Pr[Nai< τ]

≈1−Qτ−(NmPh1)

√NmPh1[1 −Ph1](8)

Pr[e|ai=1,ai−1=1] =Pr[Nhit < τ]=Pr[Nai+Nai−1< τ]

≈1−Qτ−(NmPh2)

√(2NmPh1[1 −Ph1])+(NmPh2[1 −Ph2])(9)

and

Pr[e|ai=0,ai−1=0] =Pr[Nhit ≥τ]=0(10)

where τis the detection threshold and Q(.) denotes the tail

probability of the standard normal distribution.

The evaluation of the post-decoding BEP of block codes,

Pb, is complex, even classical coding books like [9] and [10]

fail to present a closed-form expression. As such, bounds for

post-decoding BEP of block codes can be obtained as

1

kPw≤Pb≤Pw(11)

following [10], where Pwrepresents the block error proba-

bility. When the complete weight distribution of the speciﬁc

RS (n,k) code is not available, the upper bound is used as

the post-decoding BEP. As such, we approximate the post-

decoding BEP of RS codes as [10]

Pb=

n

X

i=te+1 n

i!Pi

e(1−Pe)n−i(12)

where te=n−k

2and Peis the bit error probability for the un-

coded channel. Substituting (7), (8), (9), and (10) into (6), we

obtain the Pein (12).

V. PERFORMANCE EVAL UATION

A. Simulation Framework

To validate the expression of BEP derived in Section IV, we

present particle-based simulation framework extended from [2]

and [14]. This simulation framework takes into account the bit

pattern generation, the RS encoding, the signal modulation, the

propagation, the molecule reception, the signal demodulation,

and the RS decoding as outlined in Fig. 1. We simulate

the 2D environment due to the computational complexity in

4

10 20 30 40 50 60 70 80 90 100

10−7

10−6

10−5

10−4

10−3

10−2

10−

1

Bit Error Probability

Analytical with RS coding

Analytical without coding

Simulation with RS coding

Simulation without coding

Nm

Fig. 2. Bit error probability of RS(4,2) coded and uncoded bit stream.

particle-based simulation for 3D MC system with RS codes.

The information bits are randomly generated with simulation

parameters set as: d=1µm,D=79.4µm2/s,Tb=0.032 s,

τ=Nm

2, and the length of the bit sequence is 2400000 with

100 repetitions per each iteration. The analytical results are

plotted using (12), with the assumption that the occurrence of

bit 1 and 0 of the transmitted bit is equally likely.

B. Numerical Results

In this subsection, we present the simulation and analytical

results of the bit error probability of our proposed MC

system. The accuracy of the theoretical BEP approximation

is evaluated. Fig. 2 plots the impact of RS coding on the

BEP of proposed MC system. It is shown that the theoretical

results match with that of particle-based simulations, which

validate the accuracy of our derived expression. Compared

with uncoded case, the BEP is greatly improved with the help

of RS(4,2) applied in our proposed MC system. For instance,

with Nm=100, the BEP achieved by the RS(4,2) is 5×10−6,

which is a negligible level.

Fig. 3 plots the BEP of the proposed system with various

minimum distance of the RS code and HC. According to Fig.

3, RS codes is shown to provide a signiﬁcant level of BEP

improvement compared to HC. We also observe that the BEP

of the RS codes can be greatly improved by increasing the

minimum distance and the coding gain, k

n. This is because

increasing n−kincreases the error correction capacity of

the RS code given by n−k

2, in comparison to HC which is

only capable of correcting single bit error. In addition, since

particle-based simulations for very large input bit streams are

time consuming, only analytical results are presented.

VI. CONCLUSION

In this paper, we introduced RS code for error correction

of the diffusion-based MC system. We derived the bit error

probability of the proposed MC system with full absorption

receiver using theoretical approximations. The derived analyt-

ical results for the bit error probability are veriﬁed by particle-

based simulations. It is shown that the RS code substantially

without coding

with RS(4,2)

with RS(8,4)

with RS(10,2)

with RS(10,4)

with RS(16,8)

with HC(4,2)

with HC(8,4)

with HC(10,2)

with HC(10,4)

with HC(16,8)

10−8

10−6

10−4

10−2

100

Bit Error Probability

20 40 60 80 100 120 140

Nm

Fig. 3. Bit error probability of RS codes and Hamming Codes (HC).

improves the bit error probability of MC systems, compared to

uncoded MC systems. The bit error probability can be further

enhanced by increasing, the number of molecules emitted per

bit and the minimum distance of the RS codeword. These

results justify the effectiveness of RS codes in diffusion-

based MC systems. The extension of this work under adaptive

weighted threshold detection scheme and ﬁnding the optimum

combinations of (n,k) can be considered in future work.

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