LDPC Codes Based on Rational Functions

Abstract

In this paper, some affine and rational functions are applied to define a class of LDPC codes, called RLDPC codes, which can be classified in two types, type-I and type-II, depending on being equivalent or not with APM-LDPC codes, respectively. Then, for each type, some explicit methods are provided to generate RLDPC codes with girth at least 6. While, cyclotomic cosets are used to generate type-I RLDPC codes, normal and diameter RLDPC codes are proposed as a class of type-II RLDPC codes which are analyzed for the existence of 4-cycles. Finally, simulation results show that the constructed type-II RLDPC codes outperform the randomly constructed QC LDPC codes, APM-LDPC codes and the LDPC codes based on PEG.

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LDPC Codes Based on Rational Functions
Mohammad Gholami & Akram Nassaj
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IETE JOURNAL OF RESEARCH
https://doi.org/10.1080/03772063.2021.1951365
LDPC Codes Based on Rational Functions
Mohammad Gholami 1,2and Akram Nassaj1
1Department of Mathematical Sciences, Shahrekord University, P. Code 88186-34141, Shahrekord, Iran ; 2School of Computer Science, Institute
for Research in Fundamental Sciences (IPM), P. Code 19538-33511, Tehran, Iran
ABSTRACT
In this paper, some affine and rational functions are applied to define a class of LDPC codes, called
RLDPC codes, which can be classified in two types, type-I and type-II, depending on being equivalent
or not with APM-LDPC codes, respectively. Then, for each type, some explicit methods are provided
to generate RLDPC codes with girth at least 6. While, cyclotomic cosets are used to generate type-I
RLDPC codes, normal and diameter RLDPC codes are proposed as a class of type-II RLDPC codes which
are analyzed for the existence of 4-cycles. Finally, simulation results show that the constructed type-II
RLDPC codes outperform the randomly constructed QC LDPC codes, APM-LDPC codes and the LDPC
codes based on PEG.
KEYWORDS
Linear codes; parity check
codes; error correction codes;
matrices; Bipartite graph;
performance analysis
1. INTRODUCTION
Low-density parity-check (LDPC) codes are the most
promising class of linear block codes which for many data
transmission and storage channels [1] perform very close
to the Shannon capacity.Bya(J,L)-regular LDPC code,
we mean the LDPC code whose parity-check matrix
(PCM) has row and column weight Land J,respectively.
To each PCM Hof an LDPC code, a Ta n n e r grap h [2],
denoted by TG(H), is associated which is helpful to iter-
atively share the results of the local node decoding by
passing them along the edges. The girth of an LDPC code
isthelengthofthesmallestcycleinitsTannergraph.The
construction of LDPC codes with large girth is interest-
ing, because of the accuracy of belief propagation, known
as sum-product algorithm [3].
LDPC codes are constructed into two main methods:
random [4,5]andalgebraicstructured[6–16]meth-
ods. While random methods are generally based on
a computer search, structured methods combine some
algebraic techniques along with some computer search
methods. Among random techniques, progressive edge-
growth (PEG) algorithm [5]isoneofthepromising
methods to construct a PCM with a large girth. On
the other hand, quasi-cyclic (QC) LDPC codes [7]are
among the most prominent structured methods, because
of the low encoding complexity and performing well
rather than random LDPC codes with moderate block
lengths.
Recently, a class of LDPC codes from ane permuta-
tion matrices,namedAPM-LDPC codes [13], have been
consideredbecausetheyhavesomeadvantagesinthe
cycle distribution, minimum-distance and error-rate per-
formance than QC-LDPC codes. Among the class of
APM-LDPC codes, anti-circulant and circulant permu-
tation matrices have been used to identify AQC-LDPC
codes [17], which outperform some QC and APM LDPC
codes with an explicit construction [11,14].
In this paper, rational LDPC (RLDPC) codes are dened
as a class of LDPC codes based on some ane and
rational functions which are bijections on a nite group.
Then, Type-I and Type-II RLDPC codes are presented
which are equivalent and non-equivalent with APM-
LDPC codes, respectively. For each type, some RLDPC
codeswithanexplicitmethodarepresentedwithgirthat
least 6. Although, cyclotomic cosets are used to construct
type-I LDPC codes, two explicit methods, normal and
diameter RLDPC codes, are proposed to dene a class of
Type-II RLDPC codes which outperform QC, APM and
PEG LDPC codes.
2. PRELIMINARIES
Let G={g1,...,gn}be a nite ordered group of order
nwith the identity element id(G)and f:G→Gbe
a bijective function, i.e a one-to-one correspondence
between the elements of G.ByIf,wemeanthen×n
© 2021 IETE

2 M. GHOLAMI AND A. NASSAJ: LDPC CODES BASED ON RATIONAL FUNCTIONS
the permutation matrix (pi,j)1≤i,j≤n,inwhichpi,j=1if
and only if f(gi)=gj. Some of the properties of such
permutation matrices are as follows [13].
Proposition 2.1: (1) (If)T=If−1,where(If)Tis the
transpose of Ifand f −1is the inverse function of f.
(2) For two bijections f1and f2,wehaveIf1×If2=
If2◦f1in which f2◦f1is the composition of f2and f1.
(3) (If)n=If(n),wheref
(n)is the n times composition
of f.
Let Jand L,J<L,besomepositiveintegers.Fortheempty
function ∅on G,wesetI∅to be the n×nzero matrix.
By a (J,L)bijective matrix,wemeanaJ×Larray of
some bijective functions or the empty function on G.For
a(J,L)bijective matrix F=(fi,j)J×L,thecorresponding
(J,L)LDPC code with exponent function matrix (EFM) F
of length nL,lifting-degree n and rate at least 1 −J
Lcan be
dened as the protograph LDPC code with the following
parity-check matrix H=H(F).
H=⎛
⎜
⎝
If1,1 ··· If1,L
.
.
.....
.
.
IfJ,1 ··· IfJ,L
⎞
⎟
⎠(1)
Theorem 2.2 ([7]): H(F)canbeconsideredasthePCM
of a QC-LDPC code if, under the function composi-
tion operator, each two elements of F interchanges with
each other, i.e. fi1,j1◦fi2,j2=fi2,j2◦fi1,j1,foreach(i1,j1)=
(i2,j2),1≤i1,i2≤Jand1≤j1,j2≤L.
The following Theorem investigates a two-way condition
for H=H(F)to have 2lcycles in TG(H).
Theorem 2.3 ([13]): Each 2l−cycleinTG(H)corre-
sponds to a chain (i0,j0);(i1,j1);··· ;(il−1,jl−1);(il,jl)=
(i0,j0),ik= ik+1and jk= jk+1for each 0≤k≤l−1,for
which the function
f=fil,jl◦f−1
il,jl−1◦fil−1,jl−1◦f−1
il−1,jl−2◦··· ◦fi1,j1◦f−1
i1,j0(2)
has a xed point in G.
TheLDPCcodewithPCM(1)iscalledconventional,if
the corresponding (J,L)bijective matrix Fdoes not con-
tain the empty function, i.e. Hdoes not contain the zero
block, otherwise it is called unconventional.
The following main theorem gives a necessary condi-
tion to construct LDPC codes equivanent with a primary
code.
Theorem 2.4: For F =(fi,j)J×Las the EFM on the group
GofanLDPCcodeCwith PCM H(F),ifCis the LDPC
code with EFM F=(f
i,j)J×L,inwhichf

i,j=ri◦cj◦fi,j,
where ri,1≤i≤J, and cj,1≤j≤Laresomebijective
functions on G, then Cis equivalent with C.
Proof: Let Ri=Iriand Cj=Icjand H1and H2be the
PCMs derived from multiplication of Riand Cjon the
ith row-block and jth column-block of H,respectively.
Clearly, EFMs corresponding to the LDPC code with
PCMs H1and H2are those from combining the bijec-
tive functions riand cjwith the functions in the ith row
and jth column of F,respectively,whichcompletesthe
proof. 
3. RLDPC CODES
Clearly, for each e∈G,theane function f1(x)=ex and
rational function f2(x)=ex−1are bijective functions on
G, then, for simplicity of the notations, If1and If2are
denoted by A(e)and R(e),respectively,whenGis known.
Moreover, we accept this convention that for e=∞,A(e)
and R(e)are the zero matrix of order n.ByProposi-
tion 2.1, some of the properties of the matrices A(e)and
R(e)can be simplied as follows.
Lemma 3.1: For each e,e1,e2∈G, we have:
(1) (R(e))−1=(R(e))T=R(e),
(2) R(e1)×R(e2)=A(e2e−1
1),
(3) A(e1)×A(e2)=A(e2e1),
(4) R(e1)×A(e2)=R(e2e1),
(5) A(e1)×R(e2)=R(e2e−1
1),
Then, by the above lemma, the set G={A(e),R(e):e∈
G}isagroupwhichisnotabelian(commutative)ingen-
eral. By following lemma, the order of Gmay be less than
2|G|
Lemma 3.2: A(e1)=R(e2),forsomee
1,e2∈G, if and
only if e1=e2and for each x ∈G, we have x =x−1.
It is noticed that there exist some groups satised in
Lemma 3.2, for example the set of all symmetric permu-
tation matrices of the same order.
For positive integers J,L,J<L,letE=(ei,j)be a J×L
array of some elements of G∪{∞}.Bya(J,L)rational

M. GHOLAMI AND A. NASSAJ: LDPC CODES BASED ON RATIONAL FUNCTIONS 3
LDPC,orbriey(J,L)RLDPC,codewithexponent
matrix E=(ei,j), we mean the LDPC code having the
PCM H=H(E), with lifting degree |G|,inwhich(i,j)th
element of the corresponding EFM is the ane function
ei,jxor the rational function ei,jx−1. To simplify of the
notations, the element ei,jis denoted by eA
i,jand eR
i,jfor
ane and rational functions, respectively.
Example 3.3: For positive integer m,letG=Z∗
mbe the
group of nonnegative integers less than mwhich are
coprime relative to m, i.e. Z∗
m={i∈Zm|gcd(i,m)=
1}.Clearly,Z∗
mis of order φ(m)in which φ(m)is the
Euler’s phi function. Now, for m=8and(2, 3)exponent
matrix E=1A1A1A
5A3R7R, the corresponding RLDPC code
has the following PCM.
H=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
100010001000
010001000100
001000100010
000100010001
001001000001
000110000010
100000010100
010000101000
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3)
Dene G={aA,aR,∞:a∈G}.Clearly,Gisanon-
abelian group with the following product operation:
1)aA∗bA=(ab)A2)aA∗bR=(ab)R
3)aR∗bA=(ab−1)R4)aR∗bR=(ab−1)A
5)∞∗aR=aR∗∞=∞
6)aA∗∞=∞∗aA=∞
Now, for a given exponent matrix Eof a (J,L)RLDPC
code with the elements belong to G,byanextension
of E, we mean the exponent matrix derived by mul-
tiplyingtherowsorcolumnsofEby some elements
of G\{∞}. By Lemma 3.1, it can be seen easily that
the RLDPC code with the exponent matrix Eis equiv-
alent with an APM-LDPC code if and only if there
is an extension of E,sayEe, such that all of the ele-
ments of Eebelong to the set {aA,∞:a∈G}.Incon-
tinue, type-I and type-II RLDPC codes are precedented
which are and are not equivalent with APM-LDPC codes,
respectively.
4. TYPE-I RLDPC CODES
Let E=(eR
i,j),inwhichei,j∈Gand H=H(E)be the
PCM of the corresponding RLDPC code. Clearly, Ee=
(aA
i,j),aA
i,j=eR
i,1 ∗eR
i,j, is an extension of E,sointhis
case, the RLDPC code with PCM His equivalent with
the APM-LDPC code with PCM H(Ee).Then,using
Theorem 2.3 and Theorem 3 in [13], pursuing the cycles
in the class of constructed RLDPC LDPC codes can be
summarized as follows.
Theorem 4.1: Each 2l−cycle in TG(H)corresponds to a
chain (i0,j0);(i1,j1);··· ;(il−1,jl−1);(il,jl)=(i0,j0),ik=
ik+1and jk= jk+1,0≤k≤l−1,suchthat
l−1

k=0
eik,jk=
l−1

k=0
eik+1,jk(4)
Especially, for 4-cycles, we have the following conse-
quence.
Corollary 4.2: For l =2and G =Z∗
min Theorem 4.1,
it can see easily that the Tanner graph of H=H(E)is
free of 4-cycle, if and only if each 2×2submatrix of E has
nonzero determinant.
Moreover, the following theorem investigates the condi-
tion in which the constructed RLDPC code is equivalent
with a QC-LDPC code.
Theorem 4.3: H(E)canbeconsideredasthePCMof
aQC-LDPCcodeife
2
i1,j1=e2
i2,j2in group G for each
(i1,j1)= (i2,j2)with 1≤i1,i2≤Jand1≤j1,j2≤L.
Here, we propose a class of RLDPC codes based on
cyclotomic cosets which are free of 4-cycles.
4.1 Cyclotomic Coset
For prime integer mand prime power q,(m,q)=1, let
Fqbe the nite eld of order qand xm−1∈Fq[x], where
Fq[x]isthesetofnitedegreepolynomialshavingcoef-
cients in Fq.Thesplitting eld of xm−1overFqis Fqs,
in which sisthesmallestpositiveintegersatisfyingin
m|qs−1. The roots of xm−1inthesplittingeldFqs
are called the mth roots of unity over Fq.Forxm−1∈
Fq[x], let βbe a primitive element of the splitting eld
Fqs.Clearly,ω=β
qs−1
misaprimitivemth root of unity,
that is, the mth roots of unity are {1, ω,ω2,...,ωm−1}.
For 0 ≤i<m,theconjugates of ωiare
ωi,ωiq,ωiq2,...,ωiqd−1(5)
where dis the smallest positive integer for which iqd=
iin modulo m.Foreachi,0≤i<m,thesetCi=

4 M. GHOLAMI AND A. NASSAJ: LDPC CODES BASED ON RATIONAL FUNCTIONS
{i,iq,...,iqd−1}, is referred to as the ith cyclotomic coset
of qmodulo m.Sincemis prime and sis the order of q
in modulo m,wehaves|m−1, i.e. m−1=ls, for a pos-
itive integer l,thenZ∗
mcan be partitioned to lcyclotomic
cosets Ci1,Ci2,··· ,Cil,forsomei1=1, i2,...,il∈Z∗
m.
Clearly, for each k,1≤k≤l,wehaveCik=ikC1and
|Cik|=s.Now,deneM=(M1M2...Ml),inwhich
Mk,1≤k≤l, is the following circulant matrix.
Mk=⎛
⎜
⎜
⎜
⎝
ikq0ikq1··· ikqs−1
ikqs−1ikq0··· ikqs−2
.
.
..
.
.....
.
.
ikq1ikq2··· ikq0
⎞
⎟
⎟
⎟
⎠s×s
(6)
It is veried in [15]thatforeach2×2submatrixab
cd

of M,wehavea+d= b+cmod m.Now,forM=
(μi,j),letCMbe the RLDPC code with lifting degree
qs−1andtheexponentmatrixE=(eR
i,j),ei,j=ωμi,j.
The following lemma conrms that CMisa4-cyclefree
RLDPC code.
Lemma 4.4: CMhas girth at least 6.
Proof: By Corollary 4.2, it is sucient to verify that
each 2 ×2submatrixE1=ei1,j1ei1,j2
ei2,j1ei2,j2of the exponent
matrix Ehas a nonzero determinant. However, det(E1)=
0ifandonlyifei1,j1ei2,j2=ei1,j2ei2,j1in Fqs,whichleadsto
μi1,j1+μi2,j2=μi1,j2+μi2,j1mod mandthisisacon-
tradiction with denition of the matrix M=(μi,j).
Example 4.5: for n=11 and q=3, we have l=2,
s=5andM=(M1M2),where
M1=
⎛
⎜
⎜
⎜
⎜
⎝
13954
41395
54139
95413
39541
⎞
⎟
⎟
⎟
⎟
⎠
,
M2=
⎛
⎜
⎜
⎜
⎜
⎝
2 6 7108
826710
108267
710826
671082
⎞
⎟
⎟
⎟
⎟
⎠
Then, CMis a (5, 10)RLDPC code with lifting degree 242
and girth 6.
Adding an all-zero row-vector to the matrix M,and
then adding an all-zero column vector to the left side
of the obtained matrix gives the following (s+1)(sl +1)
matrix extension Mex.
Mex =00 0··· 0
0M1M2··· Ml
Moreover, if V=(Ci1Ci2···Cil)be the vector from the
concatenation of the cyclotomic cosets Cik,1≤k≤l,
then Mkr can be dened as the Kronecker product of VT,
the transpose of V,withVmodulo m, i.e. Mkr :=VT⊗
V. Similar to Lemma 4.4, CMex and CMkr are RLDPC
codes having girth at least 6.
5. TYPE-II RLDPC CODES
AnotherclassofRLDPCcodesarethosewhicharenot
equivalent with APM-LDPC codes, i.e. each extension of
the PCM contains some elements corresponding to ratio-
nal functions. Unfortunately, in this case, there are not
a regular relation, other than Theorem 2.3, to represent
the cycles. On the other hand, there are some dierent
explicit methods to construct such RLDPC codes. Here,
we just consider two classes of (conventional) RLDPC
codes which are not equivalent with APM-LDPC codes
and they are 4-cycle free.
5.1 Normal RLDPC Codes
By a normal RLDPC code, we mean the RLDPC code
with the following exponent matrix.
E=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
eA
1,1 eA
1,2 ··· eA
1,L
eA
2,1 eR
2,2 ··· eR
2,L
.
.
..
.
.....
.
.
eA
J,1 eR
J,2 ··· eR
J,L
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(7)
in which the rst row and column correspond to ane
functions and the other elements correspond to ratio-
nal function. For example, the exponent matrix of the
RLDPC code in Example 3.3 is in the normal form.
Now, using Theorem 2.3 for l=2, each 4-cycle in a nor-
mal RLDPC code with the exponent matrix in Equation 7
corresponds to one of the following cases.
(1) a−1
i1,j0ai0,j0x=ai1,j1a−1
i0,j1x−1,Figure1,Part(a).
(2) ai1,j0a−1
i0,j0x−1=ai1,j1a−1
i0,j1x−1,Figure1,Part(b).
(3) a−1
i1,j0ai0,j0x=ai1,j1a−1
i0,j1x,Figure1,Part(c).
(4) ai1,j0a−1
i0,j0x=ai1,j1a−1
i0,j1x,Figure1, Part (d).
5.2 Diameter RLDPC Codes
By a diameter RLDPC code,wemeananRLDPCcode
with the exponent matrix E=(ei,j)J×L,inwhichei,j

M. GHOLAMI AND A. NASSAJ: LDPC CODES BASED ON RATIONAL FUNCTIONS 5
Figure 1: All of the possible 4-cycles appeared in the EFM of normal and diameter RLDPC codes.
corresponds to an ane function if and only if i=jmod
J. For example, the exponent matrix of a (3, 7)diameter
RLDPCcodeisinthefollowingform.
E=⎛
⎜
⎜
⎝
aA
1,1 aR
1,2 aR
1,3 aA
1,4 aR
1,5 aR
1,6 aA
1,7
aR
2,1 aA
2,2 aR
2,3 aR
2,4 aA
2,5 aR
2,6 aR
2,7
aR
3,1 aR
3,2 aA
3,3 aR
3,4 aR
3,5 aA
3,6 aR
3,7
⎞
⎟
⎟
⎠
For diameter RLDPC codes, each 4-cycle corresponds to
one of the following cases.
Figure 2: RLDPC codes against QC, APM and PEG codes.
(1) ai1,j0a−1
i0,j0x−1=a−1
i1,j1ai0,j1x,Figure1,Part(e).
(2) ai1,j0a−1
i0,j0x−1=ai1,j1a−1
i0,j1x,Figure1,Part(f).
(3) ThiscaseissameasCase2fornormalRLDPCcodes.
(4) ThiscaseissameasCase4fornormalRLDPCcodes.
To constructed normal and diameter RLDPC codes with
girth at least 6, the exponent matrices are constructed
such that the corresponding relations 1–4 in Section (A)
or Section (B) are avoided.
5.3 Simulation Results
Forsimulationresults,wehaveusedanadditivewhite
Gaussian noise (AWGN) channel, using software avail-
able online [18]. The decoding algorithm is sum-product
with iteration number 20 and block number 10000.
Figure 2is provided a bit error performance compari-
son between the normal and diameter RLDPC codes with
girth 6, denoted by RLDPC-T1 and RLDPC-T2, respec-
tively, on one hand and randomly constructed QC-LDPC
codes, APM-LDPC codes [13]andLDPCcodesbasedon
PEG [5], on the other hand. As the gure conrms, the
constructed type-II RLDPC codes outperform QC, APM
andPEGcodeswiththesamelength,rateandgirth.
ACKNOWLEDGMENTS
Theauthorswouldliketothanktheanonymousrefereefor
their helpful comments.
DISCLOSURE STATEMENT
No potential conict of interest was reported by the
author(s).
FUNDING
This work was supported by the research council of Shahrekord
University.
ORCID
Mohammad Gholami http://orcid.org/0000-0002-3174-0138

6 M. GHOLAMI AND A. NASSAJ: LDPC CODES BASED ON RATIONAL FUNCTIONS
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IEEE Trans. Inform. Theory, Vol. 53, pp. 2885–2891, Aug
2007.
10. M. Gholami, M. Samadieh and Gh. Raeisi, “Column-
weight three QC LDPC codes with girth 20,” IEEE Commun.
Lett., Vol. 17, pp. 1439–1442, July 2013.
11. M. Karimi and A. H. Banihashemi, “On the girth of quasi-
cyclic protograph LDPC codes,” IEEE Trans. Inform. Theory,
Vol. 59, pp. 4542–4552, July 2013.
12. S. Sonavane, D. P. Rathod, S. Sukhdeve and A. Patil, “Con-
struction of irregular LDPC code using ACE algorithim,”
Int. J. Adv. Res. Computer Sci. Softw. Eng.,Vol.3,no.3,
pp. 650-–653, March 2013.
13. M. Gholami and M. Alinia, “High-performance binary
and non-binary low-density parity-check codes based on
ane permutation matrice,” IET Commun.,Vol.9,no.17,
pp. 2114–2123, Nov. 2015.
14. M. Gholami and M. Alinia, “Explicit APM-LDPC codes
with girths 6, 8, and 10,” IEEE Signal Process. Lett.,Vol.24,
no. 6, pp. 741–745, 2017.
15. M.Esmaeili,M.NajaanandA.T.Gulliver,“Structured
quasi-cyclic low-density parity-check codes based on cyclo-
tomic cosets,” IET Commun., Vol. 9, no. 4, pp. 541–547, Jan
2015.
16. M. Gholami and A. Nassaj, “Row and column extensions of
4-cyclefreeLDPCcodes,”IEEE Commun. Lett., Vol. 20, no. 1,
pp. 25–28, 2016.
17. Z. Gholami and M. Gholami, “Anti quasi-cyclic LDPC
codes,” IEEE Commun. Lett., Vol. 22, no. 6, pp. 1116–1119,
Jun. 2018.
18. Radford M. Neal, “Software for Low Density Parity
Check (LDPC) codes,” copyright © 1995–2012. Available:
http://www.cs.utoronto.ca/ ∼radford/ldpc.software.html
AUTHORS
Mohammad Gholami wasbornin27
July 1979, Isfahan, Iran, received the M.S.
degree in Mathematics in 2003 from Sharif
University of Technology, Tehran, Iran,
and the Ph.D degree in Mathematics
(Coding Theory) in 2009 form Isfahan
University of Technology, Isfahan, Iran.
His research interest includes algebraic
coding theory, LDPC codes and iterative decoding algorithms.
Since September 2009 he has been with the Department of
Mathematical Sciences at Shahrekord University, Shahrekord,
Iran, where he is now an Associate Professor.
Corresponding author Email: gholami-m@sci.sku.ac.ir; gho-
lamimoh@gmail.com
Akram Nassaj was born in Iran, she
received the B.E. degree in Mathemati-
calfromKashanUniversityofIranand
M.E. degree in mathematical from Sharif
University of Technology of Tehran. She
is now a doctor candidate of coding. Her
research interests include LDPC codes.
Email: akramnassaj@gmail.com.