On the existence of typical minimum distance for protograph-based LDPC Codes

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1 On the existence of typical minimum distance for protograph-based LDPC Codes Shadi Abu-Surra Samsung Telecommunications America Dariush Divsalar Jet Propulsion Laboratory California Institute of Technology William E. Ryan University of Arizona Abstract In this paper we prove that, for a certain class of protograph-based LDPC codes with degree-two variable nodes, a typical minimum distance exists. We obtain a tight bound on the sum of weight enumerators, up to some weight d, for the ensemble of finite-length protograph LDPC codes. Then we prove that this sum goes to zero as the block length goes to infinity. Finally, we prove that Pr(d < d ) goes to zero as the block length goes to infinity. This typical minimum distance exists if degreetwo nodes have certain connections to the check nodes. This is also important in practice since it identifies a certain class of protograph LDPC codes that have typical minimum distances. I. ITRODUCTIO Low-density parity-check (LDPC) codes were proposed by Gallager [1] in Ensemble weight enumerators for unstructured irregular LDPC codes and turbo-like codes have been reported in [10], [11], [1], [13], [14], [15], [16], [17], [18], [], [3]. Recently, researchers became interested in the design of LDPC codes with imposed sub-structures, starting with the introduction of multi-edge type codes in [7] and [8]. Protograph-based LDPC codes are a subclass of multi-edge LDPC codes. In [9] a method for the computation of asymptotic (infinite block size) weight enumerators for LDPC codes with protograph structure has been proposed. In [4] ensemble weight enumerators for finite block size LDPC codes with a protograph structure was obtained. The results then were extended to the asymptotic case as the block size goes to infinity. Weight enumerators for specific codes are useful for bounding or estimating the decoding error probability of channel codes. As noted by Gallager [1], it is generally impractical to calculate the weight enumerator for a given code. Given this, Gallager and others have calculated the average performance for ensembles of codes. Gallager derived asymptotic weight enumerators for the ensembles of regular LDPC codes. This result was extended to the irregular LDPC ensembles (see the above references). In this present paper, we upper bound the ensemble weight enumerators of protograph-based LDPC codes to prove the existence of a typical minimum distance for a class of protograph LDPC codes with degree- variable nodes. The existence of a typical minimum distance implies linear growth of the minimum distance with the code block length [1]. For any protograph LDPC code, we use a random permutation per each edge. Then we obtain the weight enumerators by averaging over all possible permutations. This is equivalent to using a uniform interleaver [6] per each edge of the protograph. There was few research work on bounding the minimum distance, and linear minimum distance property for example see [], [3], [4], [5] and references there. Recently in [0] and later in [1], the minimum distance was upper bounded using circulant permutations. The results show that if circulant permutations are used, the minimum distance will not grow linearly with the code block length. This paper proceeds as follows: In Section II, we define a protograph LDPC code. In Section III, we define our notation and provide the ensemble weight enumerators for protographbased LDPC codes as background. In Section IV, we define a class of protograph LDPC codes with typical minimum distance. Finally, in Section V, we prove the existence of typical minimum distance for this class of protograph LDPC codes. II. PROTOGRAPH-BASED LDPC CODES A protograph is a Tanner graph with a relatively small number of variable nodes (Vs) and check nodes (Cs) [5], [19], [6]. A protograph G = (V, C, E) consists of a set of variable nodes V = {v 1, v,..., v nv }, a set of check nodes C = {c 1, c,..., c nc }, and a set of edges E. Each edge e E connects a variable node v e V to a check node c e C. Parallel edges are permitted, so the mapping e (v e, c e ) V C is not necessarily 1:1. Each edge in the base protograph represents an edge type. For multi-edge LDPC codes, a group of edges (number of edges in each group can be different) represents an edge type. For unstructured irregular LDPC codes, there is only one edge type. Having the base protograph, we can obtain a larger graph by a copy-andpermute operation. This operation consists of first making copies of the protograph, and then permuting the endpoints of each edge type among the variable and check nodes connected to the set of edges copied from the same edge type in the protograph. The derived or lifted graph is the graph of a code times as large as the code corresponding to the protograph, with the same rate and the same distribution of variable and check node degrees. Denote by q vi the degree of variable node v i. Denote by q cj the degree of check node c j. The code rate for the protograph is R c = nv nc n t provided that the parity check matrix of the derived or lifted graph is full

2 v 1 v v n v Fig. 1.! 1!! j! 3! E Vectorized protograph. c 1 c c nc rank; n t is the number of transmitted variable nodes, n t n v. Here for simplicity we assume n t = n v. The copy-andpermute process can be simply represented by replacing each node with a vector of nodes of the same type and replacing each edge with a bundle of (permuted) edges of the same type. This vectorized protograph is depicted in Fig. 1. III. ESEMBLE WEIGHT EUMERATORS FOR FIITE-LEGTH PROTOGRAPH-BASED LDPC CODES ow consider the LDPC code constructed from a protograph G by making replicas of G and using uniform interleavers, each of size, to permute the edges among the replicas of the protograph. We treat the Vs and Cs as constituent codes in a concatenated coding scheme. More specifically, the group of Vs of type v i is considered to be a constituent (repetition) code with a weight-d i input of length and q vi length- outputs. Also, the group of Cs of type c j is considered to be a constituent code with q cj inputs, each of length, and a fictitious output of weight zero. Let A(d) be the average (over the ensemble) number of codewords having weight vector d = [d 1, d,..., d nv ] corresponding to the n v V -groups and satisfying the protograph constraints. A(d) is the weight vector enumerator for the ensemble of codes of length n v described by the protograph. Let us further define A vi (w i ) = ( ) d δdi,w i i,1 δ di,w i,qvi = the weight vector enumerator for the type-v i (V) constituent code for a weightd i input, where w i = [ ] w i,1, w i,,..., w i,qvi is a weight vector describing the constituent code s output, and A cj (z j ) = the weight vector enumerator for the] type-c j (C) constituent code and z j = [z j,1, z j,,..., z j,qcj, where z j,l = w i,k if the l th edge of C c j is the k th edge of V v i. As shown in [4, Eq. ], by exploiting the uniform interleaver [6] property, we may write A(d) = w m,u = nc (w Avi i ) n c j=1 (z Acj j ) qvs ( ) s=1 r=1 w s,r j=1 (d Acj j ) ( ) qvi 1 (1) d i where the summation in the first line is over all weights w m,u, m = 1, ]..., n v and u = 1,..., q vm, and d j = [d j1, d j,..., d jqcj is a weight vector which describes the weights of the -bit words on the edges connected to C c j, produced by the Vs neighboring c j. The elements of d j comprise a subset of the elements of d. Then the average number of codewords of weight d in the ensemble, denoted by A d, equals the sum of A(d) over all d for which {d i:v i V } d i = d. otationally, A d = A(d) () {d i:v i V } under the constraint {d i:v i V } d i = d. To evaluate A d in (), one first needs to compute the weight vector enumerators, A cj (d j ), for the check nodes c j, as seen in (1). Consider a check node c with degree 3. We need to find its weight vector enumerator A c (w), where w = [w 1, w, w 3 ] is the weight vector at the input to a degree-3 check node. Following [4], the A c (w) may be easily found as the coefficients of the multi-dimensional z-transform of {A c (w)} as A c (w 1, w, w 3 ) = ( s ) s! (s w 1 )!(s w )!(s w 3 )! where s = w1+w+w3. This is true if w 1 +w +w 3 is even and max{w 1, w, w 3 } s, otherwise A c (w 1, w, w 3 ) = 0. The partial weight enumerators for checks with degree higher than 3 can be obtained from the result for a check with degree 3 by concatenation. For example A c (w 1, w, w 3, w 4 ) can be obtained as A c (w 1, w, w 3, w 4 ) = l=1 (3) A(w 1, w, l)a(w 3, w 4, l) ( ) (4) l The weight enumerators for higher degree checks can be obtained in a similar way. IV. A CLASS OF PROTOGRAPH LDPC CODES WITH TYPICAL MIIMUM DISTACE Consider a class of protograph-based LDPC codes where the connections between degree- Vs and Cs in the protograph have certain restrictions to be defined. Any protograph with degree- nodes that satisfies the following criterion belongs to this class. First consider the set of degree- V nodes for which each node in this set is only connected to a C node through a single edge connection. Remove from the set each degree- node and the two edges connected to this degree- node. Repeat this process for the

3 Fig.. A member of the class in which no degree- loops exist (Only degree- Vs and check nodes connected to them are shown) remaining degree- V nodes. If at the end of this process no degree- node is left, then the protograph-based LDPC code belongs to the class of protograph-based LDPC code ensembles with typical minimum distance. In this class, no degree- node is allowed to be connected to a C node through double edges. This constraint also implies that no loop should exist in the graph between degree- nodes and the checks connected to these nodes. A member of this class is shown in Fig. This class of protograph LDPC codes with degree- variable nodes is also similar to the class of protograph-based LDPC codes that are derived from a protograph with variable node degrees at least 3 using the check split operation as described in [4] and [19]. V. EXISTECE OF TYPICAL MIIMUM DISTACE ow we will prove that class of protograph-based LDPC code ensemble with degree- nodes that was described in the previous section has a typical minimum distance. We first prove that there exists a δ > 0 such that δ d=1 A d 0 as, where the block size n = n v. Then, using Markov s inequality we can show P r{d min < δ } 0 as. Proof: We have already computed A c (w) for a check c (SPC) with degree 3. If w 1 + w + w 3 is even and max{w 1, w, w 3 } u, then A c! (w 1, w, w 3 ) = ( s)!(s w 1 )!(s w )!(s w 3 )! (5) otherwise A c (w 1, w, w 3 ) = 0, where s = w1+w+w3. First we obtain an upper bound to A c (w 1, w, w 3 ). ote that! ( s)! s (6) 1 (s w 1 )!(s w )!(s w 3 )! s! s! s! 1 = (s w 1 )! (s w )! (s w 3 )! (s!) 3 (7) s! (s w i )! swi ; i = 1,, 3. (8) s! s s+ 1 e s Further, s w i, i = 1,, 3, implies ln s ln w i. 3 A c (w 1, w, w 3 ) w i 3 e wi 1 wi ln(wi) 1 U(wi) ln(wi) (10) where U(w i ) is a unit step function, i.e., for w i 0, U(w i )=1, otherwise it is zero. The unit step function was introduced to cover the cases when w i =0. An upper bound on A c (w) for a check c with degree 4 can be obtained using the check split method discussed in [4] and [19]. If we split the check node into two checks and connect them with a degree- variable node, we get A c (w 1, w, w 3, w 4, l) = A(w 1, w, l)a(w 3, w 4, l) ( ) (11) l Using the upper bound (10) for a degree-3 check node, we get A c (w 1, w, w 3, w 4, l) 4 w i e 3 wi 1 wi ln(wi) 1 U(wi) ln(wi) (9) e 3l U(l) ln(l) (1) We can obtain A c (w 1, w, w 3, w 4 ) by summing over l as A c (w 1, w, w 3, w 4 ) = ote that A c (w 1, w, w 3, w 4, l) (13) l=0 l min{(w 1 +w ), (w 3 +w 4 )} w 1 + w + w 3 + w 4 l max Then A c (w 1, w, w 3, w 4 ) 4 w i e 3 wi 1 wi ln(wi) 1 U(wi) ln(wi) l w 1 +w +w 3 +w 4 e 3l U(l) ln(l) (14)

4 But the summation can be upper bounded as l w 1 +w +w 3 +w 4 e 3l U(l) ln(l) l max e 3lmax ln(lmax) = e 3lmax The upper bound can be written as A c (w 1, w, w 3, w 4 ) 4 (15) w i 3 e wi 1 wi ln(wi) 1 U(wi) ln(wi) (16) A c (w 1, w, w 3,..., w qc, l 1, l,..., l qc 3) qc w i e 3 wi 1 wi ln(wi) 1 u(wi) ln(wi) 1 U(wi) ln(wi) q c 3 j=1 e3lj U(lj) ln(lj) (18) Summing over l j, j = 1,..., (q c 3), we get A c (w 1, w, w 3,..., w qc ) = A c (w 1, w, w 3,..., w qc, l 1, l,..., l qc 3) l l qc 3 l 1 (19) w 1 w w 3 w 4 l 1 l For each check in the Figure 3 we have l 1 w 1 + w, l j l j 1 + w j+1 forj =,..., q c 3, l j l j+1 + w j+ forj = 1,..., q c 4, and l qc 3 w qc 1 + w qc. These inequalities imply l j j+1 k=1 w k and l j q c k=j+ w k. Thus, l j min{ j+1 k=1 w qc k, k=j+ w k} 1 qc k=1 w k l max. From these inequalities we can also conclude that for any check of degree q c we should have w i < q c j=1,j i w j or max{w 1, w,... w qc } 1 qc j=1 w j. This bounding techniques for the weight of degree- nodes can be used to prove that the total weight L of degree- nodes in the class of protograph codes with typical minimum distance can be upper bounded as w qc - w qc -1 w qc l qc -4 l qc -3 L u, (0) where u is the total weight of the other variable nodes with degree at least 3. ow using the above results l j l max e 3lj u(lj) ln(lj) l max e 3lmax ln(lmax) = e 3lmax (1) The upper bound can be written as Fig. 3. Check split method to compute enumerators for a check node with degree q c An upper bound on A c (w) for a check c with degree q c can be obtained either by using mathematical induction or directly applying the check split method to a check with degree q c. Here we use the latter. ow suppose we use the check split method for a check node of degree q c and we generate q c 3 degree- nodes. Then we have A c (w 1, w, w 3,..., w qc, l 1, l,..., l qc 3) = A(w 1,w,l 1)A(l 1,w 3,l )...A(l qc 3,w qc 1,w qc ) ( l 1 )( l )...( l qc 3 ) Using the upper bound for a degree-3 check, we have (17) A c (w 1, w,..., w qc ) q c w i 3 e (qc )wi 1 wi ln(wi) 1 u(wi) ln(wi) () Using the above results, the weight vector enumerator A(d) = nc j=1 (d Acj j ) ( ) qvi 1 d i (3) can be upper bounded. The numerator n c j=1 Acj (d j ), which is a product over the check nodes, with some manipulation can be upper bounded using the above result for check nodes as n v 1 3 qv i di e (Pqv i j=1 (q c (i) j ))d i 1 qv i di ln(di) 1 qv u(di) ln(di) i

5 In the above result the product now is over the variable nodes and the checks c (i) j (j = 1,... q v i ) are adjacent to variable node v i (neighbors of v i ). ow we lower bound the denominator n v ( ) qvi 1 d i in the weight vector equation by n v (qv i 1)di e (qv i 1)di ln(di). Using 1 q v i u(d i ) ln(d i ) 0 the weight vector enumerator now can be upper bounded as n v A(d) 1 e (qv i )di ln( d i )+ 3 (Pqv i j=1 (q c (i) ))d i j (4) Let qc max be the maximum check degree in the protograph. Let t be the number of degree- variable nodes. We separate the degree- nodes. The upper bound can be written as A(d) n v t e 1 (qv i )di ln( d i )+ 3 qv i (qmax c ))d i t e 3(qmax c ))d k k=1 (5) In the above bound q vi 3. Consider the following function of q vi 1 (q v i )d i ln( d i ) + 3 q v i (qc max ))d i This function is decreasing in q vi if di c ) is true. We can prove that this inequality is true for a distance region of interest, namely, d d o (d o will be defined shortly). Thus, the above function is maximum when q vi = 3. Define the sum of weights of all variable nodes with at least degree 3 as u = n v t d i, and the sum of weights of all degree- variable nodes as L = t k=1 d k. The total weight of all nodes is d = u + L. In (1) we had L u where depends on the particular connections of degree- nodes to checks in our class of protograph codes. However, we can obtain the worst-case upper bound per weight of a degree- node as l 1 nv t q vi d i. With further upper bounding we get a worstcase value for as 1 qmax v t, where qv max is the maximum variable node degree. Then L u = (d L). This implies that L d. With the above results, we have 1+ n v t A(d) e 3(qmax c ))L Using ln d i ln u, we get A(d) e 3(qmax c e 3(qmax e 1 di ln( d i )+ 9 (qmax c ))d i ))L e 1 u ln( u )+ 9 (qmax c ))u (6) (7) or where E(d, L) 1 A(d) e E(d,L) (8) L (d L) ln(d )+ 9 (qmax c )(d 1 L) (9) 3 We further upper bound the above as A(d) e E(d,L) e max L E(d,L) (30) The slope of the function E(d, L) with respect to L is 1 d L ln 3 qmax c + 5. This slope is positive for d d o (d o to be defined shortly) and 0 L 1+ d. The weight vector enumerator can thus be upper bounded as A(d) e E(d, 1+ d) (31) The weight enumerator now can be computed as A d = {d} A(d) {d} e E(d, 1+ d) (3) We can show that {d} = ( d+n v 1) ( n v 1 = d+nv 1) d. We have ( d+nv 1) d ( e(d+n v 1) d ) d = e d+d ln(1+ nv 1 d ) e d+nv 1. We used the fundamental inequality ln(x) x 1. ow the upper bound on A d can be written as A d e nv 1 e d+e(d, 1+ d) (33) The exponent in the upper bound d + E(d, 1+ d) = α(d ln d 1 + βd), where α = (1+), and β = 3(qmax c )(3+ ) + (1 + ) ln(1 + ). For the protograph LDPC codes with degree-3 variable nodes and higher, we should set = 0. Define δ = d/, and F ( δ) = δ ln δ δo where δ o = e β. Also define d o = δ o. Then A d e nv 1 αf ( δ) e (34) We note that for 0 δ δ o the function F ( δ) is convex, it is negative, and has a minimum at δ = δ o /e. The minimum value at this point is F ( δ o /e) = δ o /e. For d =, δ = / with F (/) = ln( δ o /)/. We assume δ o >> 1. For δ δ o, or equivalently d d o, we have d i u d δ o = e β. However, β > 3(qc max ). Thus the condition d i < e 3(qmax c ) is satisfied as it was required to show. It is also easy to show that when d d o the slope of E(d, L) with respect to L is positive. ow we upper bound F ( δ) with two lines, namely, one that connects the point (, F ( )) to the point ( δ oe, F ( δ oe )). Denote this line by where α 1 = L 1 ( δ) = α 1 δ + β1 ln δ o + δ o /e δ o /e = 1 + ln δ o e δ o e 1

6 and 0 δ o /e δ ' α L 1 α L Fig. 4. β 1 = α 1 δ o α L 3 Corresponding to Upper bound α F( d ) ln A d δ min upper bounding by lines ln δ o Then F ( δ) L 1 ( δ) for / δ δ o /e. The other line connects the point ( δ oe, F ( δ oe )) to the point ( δ o, F ( δ o )). Denote this line by L ( δ) = α δ + β where and α = 1 e 1 β = δ o e 1 Thus, F ( δ) L ( δ) for δ o /e δ δ o. Then δo/e 1 d= A d e nv 1 eα(α 1 +β 1 ) e α(α 1 δo/e+β1 ) 1 e αα 1 e nv 1 eα(α 1 +β 1 ) α nv 1 1 e e αα 1 ( δo ) α (1 e αα 1 ) ow for any δ o /e < δ < δ o, we have δ d= δ A d δ o/e d= δ env 1 e α d δo e 1 o/e = e nv 1 e e 1 α α( δo δ ) e e 1 e e 1 α 1 α δo e e Choose δ = δ o (e 1) ln( δ o ). Then δ d= δ o/e A d e nv 1 e e nv 1 Therefore the total sum δ d= A d e e 1 α e e 1 α 1 env 1 ( δo ) α ( 1 α α( δo δ ) e 1 e e 1 α e α e 1 1 d (35) (36) ( δo ) α (37) 1 e αα 1 + e α e 1 e α e 1 1 ) α This upper bound goes to zero as. Using Markov s inequality we can show P r{d min < δ } 0 as. Here we have shown existence of a typical minimum distance. However, due to the upper bounding technique, δ o might be smaller than the typical minimum distance δ min that can be obtained through numerical calculation. ote that δ min > 0 is the zero crossing of lim sup ln A d as becomes very large. If ln A d is in fact a negative and convex function for 0 d δ min, then by upper bounding ln A d by the line αl 3 as shown in Fig. 4, we can show δ min ɛ δ A d also goes to zero as, where δ δ o corresponds to the tangent point. This says that not only a typical minimum distance exists, but also δ min is the accurate value for such typical minimum distance. ote that the non-normalized δ min = δ min /n v. Sometimes we take the liberty of saying that if ln A d as is negative for 0 δ δ min, then δ min is a typical minimum distance. Having a negative function, say G( δ), there is no guaranty that δ d= eg( δ) goes to zero as. For example, take the function G( δ) = δ( δ δ ). It is negative between 0 δ δ. It is convex and has a minimum at δ = δ /, but the sum will not go to zero as. VI. COCLUSIO In this paper we proved that, for a certain class of protograph-based LDPC codes with degree- variable nodes, a typical minimum distance exists. ACKOWLEDGMET This research was supported in part by grant X09AL75G from ASA Goddard Space Flight Center. This research in part was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with ASA. REFERECES [1] R. G. Gallager, Low-density parity-check codes. Cambridge, MA: MIT Press, [] S. Litsyn and V. Shevelev, Distance distributions in ensembles of irregular low-density parity-check codes, IEEE Trans. on Inform. Theory, vol. 49, pp , December 003. [3] D. Burshtein and G. Miller, Asymptotic enumeration methods for analyzing LDPC codes, IEEE Trans. on Inform. Theory, vol. 50, pp , June 004. [4] D. Divsalar, Ensemble weight enumerators for protograph LDPC codes, IEEE Int. Symp. on Inform. Theory, pp , July 006. [5] J. Thorpe, Low-density parity-check (LDPC) codes constructed from protographs, Tech. Rep , IP Progress Report, August 003. [6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding, IEEE Trans. on Inform. Theory, vol. 44, pp , May 1998.

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