On the Typical Minimum Distance ofProtograph-Based Non-Binary LDPC Codes

Dariush DivsalarJet Propulsion Laboratory

California Institute of TechnologyPasadena, CA 91109, USA

Email: Dariush.Divsalar@jpl.nasa.gov

Lara DolecekDepartment of Electrical EngineeringUniversity of California, Los AngelesLos Angeles, California 90095, USA

Email: dolecek@ee.ucla.edu

Abstract—This paper proves the existence of the typical min-imum distance for certain ensembles of the protograph-basednonbinary (PB NB) LDPC codes with degree-2 variable nodes.Using recently obtained ensemble weight enumerators for thePB NB LDPC codes, we show that asymptotically in the codelength N , the entries in the weight enumerators up to someweight d∗ = δ∗N (δ∗ > 0) vanish under certain conditions ondegree-2 nodes. As a consequence, the probability that the codeminimum distance is less than d∗ goes to zero as the code lengthgoes to infinity. Results of this type advance the understandingof structured non-binary LDPC codes.

I. INTRODUCTION

Low-density parity-check (LDPC) constitute an importantclass of linear codes. LDPC code defined over both binary andnon-binary fields were proposed by Gallager in 1963, [11].The quality of a linear code is commonly captured by itsweight enumerator polynomial (WE) that specifies the num-ber of codewords of each possible weight. A substantialamount of recent work has focused on characterizing thecodeword weight enumerators of various classes of binaryLDPC codes. Since it is generally difficult to characterizethe WE of a given code, the ensemble-wide approach isusually employed. Ensemble weight enumerators of unstruc-tured (ir)regular binary LDPC codes have been reported in[12], [27], [5], [6], [17], [18], and [3], and numerous ref-erences therein. Results on bounding the minimum distance,and on the linear minimum distance property of binary LDPCcodes were established in [16], [21], [20], and [28].

In addition to the characterization of unstructured binaryLDPC codes, recent research efforts have also been gearedtowards the design and analysis of binary LDPC with pre-scribed structures. Early work [22] and [23] introduced multi-edge type codes. Protograph-based binary LDPC codes [29]are a subclass of multi-edge LDPC codes. In [10] a methodfor the computation of the asymptotic (infinite block size)weight enumerators of binary LDPC codes with a protographstructure was proposed. In [7], [1], and [25] ensemble weightenumerators binary LDPC codes with a protograph structure

This research in part was carried out at the Jet Propulsion Labora-tory, California Institute of Technology, under a contract with NASA.This research in part was also carried out at the University of Cali-fornia Los Angeles (UCLA) under the grant CCF-1029030 from NSF.

were derived. results were too extended to the asymptoticsetting.

In contrast to the binary domain, much less is known aboutthe ensemble properties of non-binary structured LDPC codes.Recent results in [13], [15] characterized codeword weightdistributions of certain regular non-binary codes. Novel codedesign based on non-binary LDPC protographs was recentlydiscussed in [4] and [19]. In our recent work [8], we studied aclass of non-binary LDPC codes based on protographs, and wederived certain WEs of these codes. In the present paper, weupper bound the ensemble weight enumerators of protograph-based non-binary (PB NB) LDPC codes to prove the existenceof the typical minimum distance for a class of PB NB LDPCcodes with degree-2 variable nodes. The existence of a typicalminimum distance implies a linear growth of the minimumdistance with the code block length [11].

This paper proceeds as follows: In Section II, we summarizethe relevant background on PB NB codes [8]. In Section III wecompute an upper bound on weight enumerators. In SectionIV, we prove the existence of a typical relative minimumdistance for certain class of protograph-based nonbinary LDPCcodes. In Section V, we present a class of protograph-basednon-binary LDPC codes with a typical minimum distance.Finally, in Section VII we conclude the paper.

II. BACKGROUND AND PREVIOUS RESULTS

We first summarize protograph-based non-binary (PB NB)LDPC codes (simply referred by PB NB codes) and theirensembles introduced in [8].

A. PB NB Codes, Ensembles, and Their EnumeratorsA protograph-based non-binary code is constructed from

the constituent protograph by a copy-scale-and-permute proce-dure. When the protograph G = (V,C,E) is copied N timeseach variable node vi ∈ V (each constraint node ci ∈ C)produces the set Vi of variable nodes {vi,1, . . . , vi,N} (theset Ci of constraint nodes {ci,1, . . . , ci,N}) in the resultantgraph GN . Likewise, each edge ei ∈ E in the protographproduces the set Ei of edges in the resultant graph whereEi = {ei,1, . . . , ei,N}, and the edge ei,j for 1 ≤ j ≤ Nconnects the variable node vk,j and the constraint node cl,j ifthe edge ei connects the variable node vk and the constraint

Edges

Variable nodes

Check nodes

protograph Permute the edgesCopy-Scale 3 times

1C

2C

V1 V2 V3

11C

22C

12C

23C

13C

21C

V11V12V13 V21V22V23 V31V32V33

11C

22C

12C

23C

13C

21C

V11V12V13 V21V22V23 V31V32V33

s1

s2

s3

s4s5s6

s7

s8

s9s10 s15

s14

s13

s12

s11s1

s2

s3

s5s6

s7

s8

s9

s10s15

s14

s13

s12

s11s4

d1 d2 d3

d1 d121,[ ] d2 d221

,[ ]d23,

Fig. 1. An illustration of the steps in a PB NB code construction [8].

node cl in the mother protograph. We denote the resultantmatrix GN = (V N , CN , EN ).

Definition 1 (PB NB code): Given the mother protographG = (V,C,E), a (G,N, {sk}k, {πi}i) PB NB code isconstructed from the daughter graph GN = (V N , CN , EN )by scaling each edge k in GN by a nonzero element sk ofGF (q) for 1 ≤ k ≤ N · |E|, followed by permuting the edgesin the set Ei according to πi for each 1 ≤ i ≤ |E|. �

See Figure 1 for illustration.Definition 2 (PB NB code ensemble): The (G,N, q) PB

NB ensemble is the collection of all (G,N, {sk}k, {πi}i)PB NB codes with all possible choices of sk’s as non-zeroelements of GF (q) (for 1 ≤ k ≤ N × |E|) and {πi}’s as allpossible N -permutations (for 1 ≤ i ≤ N ). �

For convenience we quickly summarize the necessary no-tation from [8], [1] along with the main results regardingthe weight enumerators: Let Cj be the single parity checkcode induced by node cj , of degree mj . Let Kj = q(mj−1)

denote the number of codewords in Cj . Further, let MCj bethe Kj × mj matrix with the codewords of Cj as its rows(whose entries are by construction in GF (q)), and let MCj

b bethe Kj×mj binary matrix obtained by converting all nonzeroentries of MCj to 1. Note that by construction, some rows ofM

Cj

b may be the same.Let the set MCj

b represent all rows x of MCj

b , where x =[x1, x2, . . . , xmj ], xi ∈ {0, 1}, and let Kj,r ×mj denote thebinary matrix M

Cj

b,r as the submatrix of MCj

b that consists ofall distinct rows of M

Cj

b . Then, the number of rows in MCj

b,r

is Kj,r = 1 +�mj

i=2

�mj

i

�. Lastly, let the set MCj

b,r representthe rows xk = [xk,1, xk,2, . . . , xk,mj ], xk,i ∈ {0, 1}, for i =

1, 2, . . .mj , k = 1, 2, . . . ,Kj,r of MCj

b,r.To ease the notation, let c be the proxy for the constraint

node cj .Theorem 1: The weight-vector enumerator ACN

(w) of CN ,the code induced by N copies of the constraint node c, is givenby,

ACN

(w) =�

{n}

N !

n1!n2! . . . nKr !en·f

Tq , (1)

where {n} is the set of integer-vector solutions to w =n · MC

b,r, with n1, n2, . . . , nKr ≥ 0, and�Kr

k=1 nk = N .

The vector fq = [fq,1, fq,2 . . . , fq,Kr ] has entries fq,k =ln g(q, |xk|), where xk is the k-th element of MC

b,r, |xk| isthe weight of xk, and g(q, i) = q−1

q [(q − 1)i−1 + (−1)i].Let nk be the number of occurrences of the kth codeword

among the N copies of cj . Collect these counts into thevector n, where n = [n1, n2, . . . , nKj ]. Consider the weight-vector d = [d1, d2, . . . , dnv ], 0 ≤ di ≤ N , as a patternof nv weights corresponding to the nv length-N variablenode inputs satisfying the protograph constraints. Now, letdj =

�dj1 , dj2 , ..., djmj

�be the weight vector which describes

the weights of the N -bit words on the edges connectedto the constraint node cj , produced by the variable nodes{vj1 , vj2 , ..., vjmj

} neighboring cj .Theorem 2: The weight-vector enumerator of the PB NB

code averaged over the entire ensemble is

A(d) =

�nc

j=1 ACNj (dj)

�nv

i=1 (q − 1)di(ti−1)�Ndi

�ti−1 , (2)

where ACNj (dj) is the weight-vector enumerator of the code

CNj induced by the N copies of the constraint node cj . Here,

ti stands for the degree of the variable node vi. The elementsof dj are a subset of the elements of d = [d1, d2, ..., dnv ]obtained from the edge connections in the mother protographG.

Proof: The proofs of both theorems are provided in [8].

The average number of codewords of weight d in theensemble, denoted by Ad, equals the sum of A(d) over alld for which

�{di} di = d. This can be written as

Ad =�

d

A(d), (3)

under the constraint�

{di} di = d.The binary weight enumerators for the binary image of PB

NB codes can also be obtained using (3) and the results byEl-Khamy [9].

III. AN UPPER BOUND ON THE WEIGHT ENUMERATOR Ad

In order to prove the existence of typical minimum distancewe need to upper bound the Ad in (3). To do so we need the

following lemmas.Lemma 1: An upper bound on a multinomial

N(n2)!(n3)!...(nKj,r )!

can be expressed as

N

(n2)!(n3)! . . . (nKj,r )!

≤ e(1+ln(Kj,r−1))y+y ln Ny (4)

where y =�Kj,r

k=2 nk.Proof: First, note that

N !

(N − y)!≤ Ny. (5)

Using a lower bound on the Stirling formula for factorialswe have

1

(n2)!(n3)! . . . (nKj,r )!≤ 1

�Kj,r

i=2 ninie−ni

≤ eye−�Kj,r

i=2 ni lnni .

(6)

The second exponent in (6) can be written as

−Kj,r�

i=2

ni lnni = −y

Kj,r�

i=2

ni

y{ln ni

y+ ln y}

= −y

Kj,r�

i=2

ni

yln

ni

y− y ln y . (7)

We upper bound the first term on the right-hand side of (7) as

−Kj,r�

i=2

ni

yln

ni

y≤ ln(Kj,r − 1) . (8)

Thus, the multinomial N !(N−y)!(n2)!(n3)!...(nKj,r )!

can be up-per bounded as

N !

(N − y)!(n2)!(n3)! . . . (nKj,r )!

≤ e(1+ln(Kj,r−1))y+y ln Ny . (9)

Lemma 2: If n satisfies w = n ·MC j

b,r , then y =�Kj,r

i=2 ni

can be bounded as

max{w1, w2, . . . wmj} ≤ y ≤ 1

2

mj�

k=1

wk (10)

Proof: Note that for each constraint node cj of degreemj we have w = n · MCj

b,r. Post multiplying both side ofthis equation by 1T where 1 = [1, 1, . . . , 1] is the all-onesvector results in

�mj

k=1 wk =�Kj,r

i=2 ni|xi|. This equation thenimplies that 2y ≤

�mj

k=1 wk ≤ mjy, where y =�Kj,r

k=2 nk

(Kj,r = 1 +�mj

k=2

�mj

k

�≤ 2mj , where Kj,r is the number

codewords in MCj

b,r). Considering the k th component of w inw = n ·MCj

b,r we can write wk =�Kj,r

i=2 nixi,k where xi,k isthe kth component of the ith row in M

Cj

b,r. Since xi,k is either

0 or 1, we can upper bound xi,k ≤ 1. This implies that forany k = 1, 2, . . .mj , wk ≤ y. Thus, for cj we have

max{w1, w2, . . . wmj} ≤ 1

2

mj�

k=1

wk . (11)

Lemma 3: If n satisfies w = n ·MCj

b,r, then n · fTq can beupper bounded as

n · fTq ≤ (

mj�

i=1

wi − y) ln(q − 1) (12)

Proof: At this point we have to upperbound n · fTq . Notethat fq,k = ln g(q, |xk|), where xk is the k-th element ofMC

b,r, |xk| is the weight of xk. Then,

n · fTq =

Kj,r�

k=2

nk ln g(q, |xk|), (13)

since |x1| = 0, the summation can start at k = 2.

g(q, |xk|) =q − 1

q[(q − 1)|xk|−1 + (−1)|xk|] ≤ (q − 1)|xk|−1

(14)for all k ≥ 2. To show this relationship, note that (q−1)i−2 ≥(−1)i for all i ≥ 2. By algebraic manipulations, it follows that(13) can be upperbounded as

Kj,r�

k=2

nk ln g(q, |xk|) ≤Kj,r�

k=2

nk(|xk|− 1) ln(q − 1). (15)

Since we have already shown that�Kj,r

k=2 nk|xk| =�mj

i=1 wi and�Kj,r

k=2 nk = y, the following statement

n · fTq ≤ (

mj�

i=1

wi − y) ln(q − 1) (16)

follows.Lemma 4: If n satisfies w = n ·MCj

b,r, then the cardinality(support) of {n}, denoted by |{n}|, can be upper bounded as

|{n}| ≤ eKj,r+2e12

�mjk=1 wk . (17)

Proof: We can rewrite 2y ≤�mj

k=1 wk ≤ mjy as1mj

�mj

k=1 wk ≤ y ≤ 12

�mj

k=1 wk. Note that for each valueof y = y0, n1 takes the value N − y0. We get

�y0+Kj,r−1y0

�

possibilities for n2, n3, . . . nKj,r . Thus

|{n}| ≤� 12

�mjk=1 wk��

y0=� 1mj

�mjk=1 wk�

�y0 +Kj,r − 1

y0

�. (18)

Note that�y0+Kj,r−1

y0

�≤ [ e(y0+Kj,r−1)

y0]y0 =

ey0ey0 ln(1+Kj,r−1

y0). Using lnx ≤ x − 1, �x� ≤ x + 1,

and�i2

i=i1ei ≤ ei2+2, we obtain |{n}| ≤ eKj,r+2e

12

�mjk=1 wk .

Lemma 5: If n satisfies w = n · MCj

b,r, then the weight-vector enumerator ACN

(w) given by

ACN

(w) =�

{n}

C (N ;n1, n2, . . . , nKr ) en·fTq , (19)

in Theorem 1 can be upper bounded as

ACN

(w) ≤ νj

mj�

i=1

e2+m∗ ln 2+2(1−η) ln(q−1)

2 wi+ 12wi ln N

wi , (20)

where νj = e(Kj,r+2), η = 12 if q ≤ q∗, and η = 1

m∗ if q > q∗,q∗ = e1+m∗ ln 2 + 1.

Proof: Apply Lemma 2 result to Lemma 1 and note that

ey ln Ny ≤ e

12

�mjk=1 wk ln N

y

≤ e12

�mjk=1 wk ln N

wk . (21)

After using (21), we further upper bound (4) in Lemma 1 as

e(1+ln(Kj,r−1))y+y ln Ny

≤ e(1+m∗ ln 2)y+ 12

�mjk=1 wk ln( N

wk) .

(22)

In the upper bound we used ln(Kj,r−1) < mj ln 2 < m∗ ln 2where m∗ = maxj mj . Next we apply Lemma 3,

e(1+m∗ ln 2−ln(q−1))y+�mj

k=1 wk ln(q−1)+ 12

�mjk=1 wk ln N

wk . (23)

If q ≤ q∗ then

(1+m∗ ln 2−ln(q−1))y ≤ (1+m∗ ln 2−ln(q−1))1

2

mj�

k=1

wk,

(24)and if q > q∗ then we have to use the following upper bounds

(1 +m∗ ln 2)y ≤ (1 +m∗ ln 2)1

2

mj�

k=1

wk, (25)

and a looser upperbound

− ln(q − 1))y ≤ − 1

m∗

mj�

k=1

wk ln(q − 1). (26)

Note that the upper bound on each term in the summationin (19) is independent of n. Now, by Lemma 4, the upperbound in (20) follows.

Lemma 6: The weight-vector enumerator A(d) given by

A(d) =

�nc

j=1 ACNj (dj)

�nv

i=1 (q − 1)di(ti−1)�Ndi

�ti−1 , (27)

in Theorem 2 can be upper bounded as

A(d) ≤ νnv�

i=1

e12 (ti−2)di ln

diN +

ti(2+m∗ ln 2)−2(tiη−1) ln(q−1)2 di .

(28)

where ν =�nc

j=1 νj .Proof: Using the upper bound in Lemma 5, the numerator

in (27) can be written asnc�

j=1

ACNj (dj) = ν

nv�

i=1

ti�

j=1

e2+m∗ ln 2+2(1−η) ln(q−1)

2 di+ 12di ln N

di

(29)the denominator in (27) can be lower bounded as

nv�

i=1

(q − 1)di(ti−1)

�N

di

�ti−1

≥nv�

i=1

e(ti−1)di ln(q−1)e(ti−1)di ln( Ndi

)

(30)

Using the above results the desired upper bound in (28)follows.

IV. A GENERIC UPPER BOUND ON A(d)

We are going to show that for certain classes of PB NBLDPC codes to be discussed shortly, the upper bound on A(d)can be represented as

A(d) ≤ ν eαd ln dN +(αβ−1)d (31)

where d =�

i di = d1T , and α > 0. Equipped with thisupper bound, the weight enumerator Ad can be upper boundedas

Ad =�

{d}

A(d) ≤ ν|{d}|eαd ln dN +(αβ−1)d . (32)

We can show that |{d}| =�d+nv−1

d

�. Note that

�d+nv−1d

�≤

( e(d+nv−1)d )d = ed+d ln(1+nv−1

d ) ≤ ed+nv−1, where we usedthe fundamental inequality ln(x) ≤ x − 1. Now, the upperbound on Ad can be written as

Ad ≤ c eα[d ln dN +βd] . (33)

Lemma 7: If an upper bound on Ad can be represented asAd ≤ c eα[d ln d

N +βd] for α > 0, then there exists a δ∗ > 0such that

�N δ̃∗��

d=1

Ad ≤ kN−α, (34)

for some constant k independent of N .Proof: Define δ = d/N , and F (δ) = δ ln δ

δowhere δo =

e−β . Then the upper bound in (33) can represented as

Ad ≤ c eαNF (δ) . (35)

We note that for 0 ≤ δ ≤ δo the function F (δ) is convex, itis negative, and has a minimum at δ = δo/e. The minimumvalue at this point is F (δo/e) = −δo/e. For d = 1, δ = 1/Nwith F (1/N) = − ln(Nδo)/N .

Now we upper bound F (δ) with two straight lines, one thatconnects the point ( 1

N , F ( 1N )) to the point ( δoe , F ( δoe )). De-

note this line by L1(δ) = α1δ+β1 where α1 =− δo

e + 1N lnNδo

δoe − 1

N

,

β1 = −α1N − 1

N lnNδo. Then F (δ) ≤ L1(δ) for 1/N ≤δ ≤ δo/e. The other line connects the point ( δoe , F ( δoe )) tothe point (δo, F (δo)). Denote this line by L2(δ) = α2δ + β2

where α2 = 1e−1 , and β2 = − δo

e−1 . Thus F (δ) ≤ L2(δ) forδo/e ≤ δ ≤ δo. Then

��Nδo/e�−1d=1 Ad ≤ c N−α

δoα(1−eαα1 ) . (36)

Now for any δo/e < δ∗ < δo we have

��Nδ∗�d=�Nδo/e� Ad ≤ c e

αe−1 e

−αN(δo−δ∗)e−1

eα

e−1 −1(37)

We used �x� ≤ x and �x� ≥ x − 1 in the above equation.Choose δ∗ = δo − (e− 1) 1

N ln(Nδo). Then

��Nδ∗�d=�Nδo/e� Ad ≤ ≤ c

eα

e−1

eα

e−1 − 1

N−α

δoα . (38)

Therefore the total sum is��Nδ∗�

d=1 Ad ≤ cδoα ( 1

1−eαα1+ e

αe−1

eα

e−1 −1)N−α .

This upper bound goes to zero as N → ∞ .

V. PB NB LDPC CODES WITH THE TYPICAL RELATIVEMINIMUM DISTANCE

In this section, we investigate the existence of a typicalrelative minimum distance for PB NB LDPC codes. Thefollowing theorem states the existence of the typical relativeminimum distance as defined by Gallager [11].

Theorem 3: If an upper on A(d) can be represented as

A(d) ≤ ν eαd ln dN +(αβ−1)d (39)

for some α > 0, then there exists a δo > 0 and a small � > 0such that Pr{dmin ≤ �N (̃δo − �)�} → 0 as N → ∞. Theδo = e−β serves as a lower bound on the true typical relativeminimum distance.

Proof: We already proved that there exists a δo > 0 suchthat

��N(δo−�)�d=1 Ad ≤ kN−α. Using Markov’s inequality

we have Pr{dmin ≤ �N(δo − �)�} ≤��N(δo−�)�

d=1 Ad ≤kN−α → 0 as N → ∞.

These results allow us to determine whether or not thetypical minimum distance in the ensemble grows linearly withcodeword length.

A. Ensemble of PB NB LDPC codes with degree-2 nodesWe provide conditions on the connections of degree-2

variable nodes to constraint nodes for having a typical relativeminimum distance. Consider a class of PB NB LDPC codeswhere degree-2 subgraph of the protograph does not haveany cycles. This restrictions requires the number of degree-2variable nodes in the subgraph be strictly less than the numberof checks in the subgraph. Denote the total number of degree-2variable nodes by T2 ≤ nc−1. We call this class the degree-2-constrained class of PB NB LDPC codes, CT2 . This conditionrequires at least two check nodes in the subgraph induced bydegree-2 variable nodes to each have only one connection toa degree-2 variable node. The PB NB LDPC codes that do

not belong to this class and have cycles will have at mostlogarithmic minimum distance growth rate. Suppose the PBNB LDPC ensemble has T2 variable nodes with degree 2 andnv − T2 nodes with degree 3 or higher. Let us rename theweights of the degree-2 nodes by lk and the weights of theother nodes by ui. Then,

A(d) ≤ νnv−T2�

i=1

e(ti2 −1)ui ln

uiN +

ti(2+m∗ ln 2)−2(tiη−1) ln(q−1)2 ui

×T2�

k=1

e(2+m∗ ln 2−(2η−1) ln(q−1))lk .

(40)

The slope of ( ti2 −1)ui lnuiN + ti(2+m∗ ln 2)−2(tiη−1) ln(q−1)

2 ui

with respect to ti ≥ tm ≥ 3 is strictly negative when ui <Ne−[(2+m∗ ln 2)−2η ln(q−1)]. This inequality hold if d < Nδo =Ne−β (to be defined shortly). The tm is defined as tm =minti>2{ti}. Then we can upper bound A(d) as

A(d) ≤ νnv−T2�

i=1

etm−2

2 ui lnuiN + tm(2+m∗ ln 2)−2(tmη−1) ln(q−1)

2 ui

×T2�

k=1

e(2+m∗ ln 2−(2η−1) ln(q−1))lk . (41)

Let�T2

k=1 lk = L and�nv−T2

i=1 ui = u. Also note thatln ui

N ≤ ln uN and u = d− L. Then we obtain

A(d) ≤ νeE(d,L) , (42)

where

E(d, L) =tm − 2

2(d− L) ln

(d− L)

N

+tm(2 +m∗ ln 2)− 2(tmη − 1) ln(q − 1)

2(d− L)

+(2 +m∗ ln 2− (2η − 1) ln(q − 1))L (43)

At this point we want to obtain an upper bound on L beforefurther upper bounding (42). Using (11) we can show L ≤ γu,where γ = T2

2 t∗, and t∗ = maxi ti.Note that L ≤ γu = γ(d−L) implies that L ≤ γ

1+γ d. Nowwe go back to (42). Note that the first derivative of the functionE(d, L) with respect to L is tm−2

2 ln Ne(d−L) − tm−2

2 [(2 +m∗ ln 2)−2η ln(q−1)]. For 0 ≤ L ≤ γ

1+γ d, the second deriva-tive is tm−2

21

d−L > 0. Thus the function is convex. At thispoint we need to compare both boundary values at L = 0 andL = γ

1+γ d. As was mentioned before we are interested in theregion d ≤ Nδo. We will show that E(d, γ

1+γ d)−E(d, 0) ≥ 0.Therefore we proved max0≤L≤ γ

1+γ d E(d, L) = E(d, γ1+γ d)

and E(d, L) ≤ E(d, γ1+γ d). The weight vector enumerator

can be upper bounded as

A(d) ≤ νeE(d, γ1+γ d) . (44)

where E(d, γ1+γ d) = αd ln d

N + (αβ − 1)d. We obtain

α =tm − 2

2(1 + γ), (45)

and

β = α−1 + (1 + α−1)(2 +m∗ ln 2)− ln(1 + γ)

− [2η + (2η − 1)α−1] ln(q − 1). (46)

For a protograph-based NB LDPC code with no degree-2variable nodes we can set γ = 0.

For d ≤ Nδo = Ne−β we have ui ≤ u ≤ 11+γ d ≤

N δo1+γ = Ne−(β+ln(1+γ)). However, β + ln(1 + γ) >

(2 + m∗ ln 2) − 2η ln(q − 1). Thus the condition ui <Ne−[(2+m∗ ln 2)−2η ln(q−1)] is satisfied as it was required toshow.

Now define ∆E = E(d, γ1+γ d) − E(d, 0).

∆E = γαd(− ln dN −(2+m∗ ln 2)+2η ln(q−1)− 1

γ ln(1+γ)).Since d ≤ Ne−β , then − ln d

N ≥ β. Thus ∆E ≥d{γ − 1

2 ln(1 + γ) + γ[(2 +m∗ ln 2)− γ(2η − 1) ln(q − 1)]}.Using the fundamental inequality ln(1 + γ) ≤ γ, we get∆E ≥ γd

2 {1 + 2(2 +m∗ ln 2)− 2(2η − 1) ln(q − 1)} ≥ 0 asit was required to show.

VI. DISCUSSION

Our upper bounding technique shows that there existsδo > 0 that provides a proof for existence of the typical relativeminimum distance for certain class of PB NB LDPC codes.We note that as the degree of check nodes increases, (whichalso requires the degree of variable nodes other than degree-2to increase), q∗ also increases. The expression for δo revealsthat for q ≤ q∗, δo also increases with q up to q∗. However,when q > q∗, δo decreases as q increases beyond q∗. Since weused upper bounds on enumerators, δo only serves as a lowerbound on the true typical relative minimum distance δmin fora particular PB NB LDPC code within the class CT2 . The trueδmin within this class can be computed numerically from theasymptotic results provided in [8]. Although δo versus q showsan increase up to q∗ and then decrease such behavior may notrepresent the true functional behavior of δmin with respect to qdue to the upper bounding of Ad. However, similar behavior ofa typical relative minimum distance for regular and irregularnon-binary LDPC codes was observed by Kasai et al. [14].Such observations for weighted non-binary repeat multipleaccumulate codes were also made by Rosnes and Graell iAmat [24]. Fig. 2 shows the typical relative minimum distancefor rate 1/2 non-binary regular Gallager LDPC codes versus qbased on a numerical calculation. As it is seen from this figure,there is a maximum for δmin for some q. It is not exactly atq∗ but it is close. For very large q, δmin decreases towardszero. Andriyanova et al. also proved that the asymptotic binaryweight of non-binary regular LDPC codes, when the alphabetsize goes to infinity, has a large number of codewords ofpoor weight [2]. If we increase both degree of variable andcheck nodes such that the rate remains the same, then δmin

approaches the Gilbert Varshamov bound as it is seen in Fig.2.

The next item to discuss is the weight enumeration resultfor irregular non-binary LDPC codes. We know that if

Fig. 2. Typical relative minimum distance for rate 1/2 non-binary regularGallager LDPC codes.

λ�(0)ρ

�(1) < 1, then the typical relative minimum distance

exists where λ(x) and ρ(x) are degree distributions forvariables and check nodes. In this paper we have alreadyshown that if the number T2 of degree-2 variable nodes isless than or equal to the number of check nodes minus 1(provided that there is no cycles in the subgraph of degree-2nodes), then the typical relative minimum distance exits. Nowconsider a rate 1/2 protograph-based non-binary LDPC codewith T2 = nc − 1 degree-2 nodes and T2 + 2 degree-3 nodes.So we have nv = 2T2 + 2, and nc = T2 + 1. The degreedistribution of this code is λ(x) = 2T2

5T2+6x + 3T2+65T2+6x

2 andρ(x) = 5T2

5T2+6x4 + 6

5T2+6x5. Then λ

�(0)ρ

�(1) = 40T 2

2 +60T2

(5T2+6)2 ,For T2 = 2 this number is 1.09 > 1 and approaches 1.6 forvery large number of T2.

The other item to discuss is regarding some recent resultson use of binary image of non-binary LDPC with degree-2nodes. For example Savin and Declercq, have shown lineargrowing minimum distance for ultra-sparse non-binary cluster-LDPC codes with all degree-2 nodes [26]. We should notethat this ensemble is different than regular and irregular non-binary LDPC codes where the scalings on edges are selectedrandomly. Binary image of these non-binary codes resembleGLDPC codes. For binary GLDPC codes, were each constraintnode has the minimum distance strictly greater than 2, withuse of the same technique in this paper it might be possibleto show that at least for the protograph version, the typicalminimum distance exists under certain conditions. For non-binary LDPC codes, if we consider a check node with attachedscaling as a block code, then with a careful selection of scalesfor large enough q, the binary image of this block code shouldhave a minimum distance strictly greater than 2. Thus, suchblock code can be viewed as a constraint node in the GLDPCset up.

VII. CONCLUSION

In this paper we proved that under certain conditions thetypical minimum distance for PB NB LDPC codes exists.

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