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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 ane 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 dened

as a class of LDPC codes based on some ane 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 dene 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

dened 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),ifCis 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,theane 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 simplied 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 satised 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,orbriey(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 ane 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

ane 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)

Dene G={aA,aR,∞:a∈G}.Clearly,Gisanon-

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]isthesetofnitedegreepolynomialshavingcoef-

cients in Fq.Thesplitting eld of xm−1overFqis Fqs,

in which sisthesmallestpositiveintegersatisfyingin

m|qs−1. The roots of xm−1inthesplittingeldFqs

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,deneM=(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 veried in [15]thatforeach2×2submatrixab

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 conrms that CMisa4-cyclefree

RLDPC code.

Lemma 4.4: CMhas girth at least 6.

Proof: By Corollary 4.2, it is sucient to verify that

each 2 ×2submatrixE1=ei1,j1ei1,j2

ei2,j1ei2,j2of 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 denition 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 dened 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 dierent

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 ane

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 ane 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 conrms, the

constructed type-II RLDPC codes outperform QC, APM

andPEGcodeswiththesamelength,rateandgirth.

ACKNOWLEDGMENTS

Theauthorswouldliketothanktheanonymousrefereefor

their helpful comments.

DISCLOSURE STATEMENT

No potential conict 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

REFERENCES

1. R. G. Gallager, “Low-density parity-check codes,” IEEE

Trans. Inform. Theory, Vol. IT-8, no. 1, pp. 21–28, Jan. 1962.

2. R. M. Tanner, “Arecursive approach to low complexity

codes,” IEEE Trans. Inform. Theory, Vol. IT-27, pp. 533–547,

Sept 1981.

3. F. R. Kschischang, B. J. Frey and H. A. Loeliger, “Factor

graphs and the sum-product algorithm,” IEEE Trans. Inform.

Theory, Vol. 47, no. 2, pp. 498–519, Feb. 2001.

4. D. J. C. MacKay and R. M. Neal, “Near Shannon limit per-

formance of low density parity-check code,” IEEE Electron.

Lett., Vol. 32, no. 18, pp. 1645, Aug. 1996.

5.X.Y.Hu,E.EleftheriouandD.M.Arnold,“Regular

and irregular progressive edge-growth tanner graphs,” IEEE

Trans. Inform. Theory, Vol. 51, no. 1, pp. 386–398, Jan.

2005.

6. J. Thorpe“Low-dencity parity-check (LDPC) codes con-

structed from protographs,” IPN Progress Report, Aug. 2003.

pp. 42–154.

7. Marc P. C. Fossorier, “Quasi-cyclic low-density parity-check

codes from circulant permutation matrices,” IEEE Trans.

Inform. Theory, Vol. 50, no. 8, pp. 1788–1793, 2004.

8. H.Song,J.LiuandB.V.K.V.Kumar,“Largegirthcyclecodes

for partial response channels,” IEEE Trans. Magn., Vol. 40,

pp. 3084–3086, 2004.

9. S. Kim, J-S. No, H. Chung and D-J. Shin, “Quasi-cyclic

low-density parity-check codes with girth larger than 12,”

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

ane 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.NajaanandA.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.