On the guaranteed error correction capability of LDPC codes

On the guaranteed error correction capability of LDPC codes Shashi Kiran Chilappag ari, Dung V iet Nguyen , Bane V asi ´ c, Michael Marcellin Dept. of Electrical an d Comp uter Eng. University of Arizona T ucson, AZ 85721, USA Email: { shashic,nguyend v ,vasic,marcellin } @ece.arizona.edu Abstract — W e in vestigate the relation between the girth and the guaranteed error co rrection capability of γ -left regular LDPC codes when decoded using the bit flippin g (serial and p arallel) algorithms. A lower bound on the number of variable nodes which expand by a factor of at least 3 γ / 4 is found based on the Moore bound. An upper bound on the guaranteed correction capability is established by studying the sizes of smallest p ossible trapping sets. I . I N T RO D U C T I O N Iterative algorithms for decoding low-density parity -check (LDPC) codes have been the focus of research over th e past decade and most of their prop erties are well understood [1], [2]. These algorithm s operate by passing messages along the edges o f a gra phical re presentation of the code known as the T ann er grap h, and are op timal when the u nderlyin g graph is a tree. M essage passing de coders pe rform remark ably well which can be attr ibuted t o their ability to correct erro rs beyond the traditional bou nded distanc e decod ing capab ility . Howe ver , in contrast to boun ded d istance decoders (BDDs), the gu aranteed err or correctio n c apability of iterativ e decoders is largely unknown. The problem of recovering fr om a fixed num ber of erasures is solved fo r iterativ e decoding o n the b inary erasu re channel (BEC). If the T anner grap h of a co de does n ot contain any stopping sets [3] u p to size t ( or equiv alently the size of minimum stoppin g set is t + 1 ), th en the d ecoder is guar anteed to recover from any t erasures. Orlitsky et al. in [4] studied the relation between stopping sets and girth and deriv ed boun ds on the smallest stopping s et in any d - left regular bip artite graph and g irth g . An analogous result is unknown fo r de coding on other channels such as the bin ary symmetric chan nel (BSC) and the additive white Gaussian no ise (A WGN) channel. In this paper, we present a step to ward such result fo r har d dec ision decodin g algo rithms. Gallag er [5] p roposed two binar y mes- sage passing algorith ms, namely Gallager A and Gallager B, for decod ing over the BSC. He sho wed that for the co lumn- weight γ ≥ 3 and ρ > γ , there exist ( n, γ , ρ ) 1 regular low- density par ity-check (LDPC) codes for which the bit er ror probab ility asympto tically tend s to zero wh enever we operate below the threshold . T he minimu m distance was shown to 1 Precise definitions will be giv en in Section II increase linearly with th e cod e len gth, but correctio n of a linear fraction of err ors was not shown. Zyab lov an d Pinsker [6] an alyzed LDPC c odes un der a simpler deco ding algo rithm known as the bit flipping algorithm and showed that almost all the codes in the regular ensem ble with γ ≥ 5 c an correct a constant fraction of worst case er rors. Sipser and Spielman in [7] used expander grap h arguments to analyze two bit flippin g algorithm s, seria l and para llel. Specifically , they showed that these algorithm s can correct a fractio n of errors if the under lying T anner graph is a goo d expande r . Burshtein and M iller in [8] applied expander b ased arguments to show that me ssage passing algorith ms can also correct a fixed fraction of worst case erro rs when the degree of each variable node is mo re tha n five. Feldman et al. [9] showed that the linear programmin g decode r [10] is also capable of correcting a fraction o f errors. Recently , Bu rshtein in [1 1] showed that regular cod es with variable no des of degree fou r are capa ble of co rrecting a linear num ber of e rrors u nder bit flipping algorithm . He a lso showed tremen dous im provement in the fraction of corre ctable errors when the variable node degree is at least five. It is well known that a rando m grap h is a go od expa nder with high probability [ 7]. Howe ver , the f raction of n odes having the requir ed expansion is very small and hence the code length to gu arantee correction of a fixed numb er of errors must be large. Moreover, determining the expa nsion of a given graph is known to be NP ha rd [12], an d spectral gap me thods cannot g uarantee an expa nsion factor of more than 1 /2 [7]. On the other hand , co de par ameters such as co lumn weight and girth can be easily deter mined o r ar e assumed to be k nown for the code und er con sideration. The approa ch in th is paper is to determine the size of variable no de sets in a left r egular LDPC code which are guaran teed to h av e the expan sion requ ired by bit flipping algo rithms b ased on Moor e b ound [13, p.1 80]. The conseque nce of ou r results is that th e error correctio n capability grows expone ntially in g irth. Howe ver, we note that since th e g irth g rows logarithm ically in the cod e length , this result does not sho w that the bit flipping algorithm s can correct a linear f raction o f error s (the proo f and discussion are beyo nd the scop e of this pap er). T o find an upper boun d on the num ber of co rrectable e rrors, we study the size of sets o f variable no des which lead to decodin g failur es. A decodin g failur e is said to hav e occur red if the output of the deco der is not eq ual to the transmitted codeword [1 4]. Th e condition s tha t lead to decod ing failures are well unde rstood f or a variety o f decod ing algorithm s such as m aximum likeliho od deco ding, b ound ed distance d ecoding and iterative decodin g on the binary erasure chan nel. Ho wever , for iterative decodin g on the BSC an d A WGN chan nel, the understan ding is far from complete. T wo approaches have been taken in this direction, namely trapping sets [ 14] and pseudo- codewords [ 15]. W e adopt th e trapping set approach in th is paper to ch aracterize d ecoding failures. Richardson intro duced the notion of tr apping sets in [14] to estimate the erro r floor on th e A WGN channel. In [ 16], tra pping sets were used to estimate the frame erro r rate o f co lumn-w eigh-thre e LDPC codes. In this p aper, we d efine trapping sets with the help of fixed points for the bit flippin g algorithm s ( both ser ial and parallel). W e then find bou nds on th e size of trapping sets based on extremal graph s kn own as cag e gr aphs [17], there by finding an u pper b ound on the g uaranteed err or correctio n capability . The r est o f the paper is organized as fo llows. In Section II, we provide a brief intro duction to LDPC code s, decod ing algorithm s an d trapp ing sets [14]. I n Section III, we p rove our main theorem r elating th e co lumn weigh t and girth to the number of variable node s which expan d b y a factor o f a t least 3 γ / 4 . W e derive bounds on the size of trapp ing sets in Section IV, and con clude with a few remarks in Section V. I I . P R E L I M I N A R I E S In th is section, we first establish the notation an d then proceed to give a br ief introduction to LDPC codes and bit flipping algorith ms. W e then give t h e relation between the error correction cap ability of the code and the expansion o f the underly ing T anner graph. W e finally describ e trapp ing sets for the bit flippin g algo rithms. A. Graph Theory Notation W e adop t the standard no tation in graph theo ry (see [18] for examp le). G = ( U, E ) denotes a grap h with set of nodes U an d set of edges E . When there is no amb iguity , we simply denote the graph by G . An edge e is an unor dered pair ( u 1 , u 2 ) of no des an d is said to be in cident o n u 1 and u 2 . T wo no des u 1 and u 2 are said to be adjacent ( neighbo rs) if there is an edge e = ( u 1 , u 2 ) inciden t o n them. T he orde r o f the gr aph is | U | and the size of the gr aph is | E | . T he degree o f u , d ( u ) , is the nu mber of its neighb ors. A node with degree on e is called a leaf o r a pendan t node. A grap h is d -regular if all the no des have degree d . The average degree d of a grap h is defined as d = 2 | E | / | U | . T he girth g ( G ) o f a gr aph G , is the length of smallest cycle in G . H = ( V ∪ C , E ′ ) d enotes a bipartite graph with two sets of nod es; variable (left) nodes V and check (rig ht) nod es C and edge set E ′ . Nodes in V ha ve neighbo rs on ly in C and vice versa. A bipar tite grap h is said to b e γ -lef t regular if all variable no des have d egree γ , ρ -right regular if all chec k no des have d egree ρ and ( γ , ρ ) regular if all variable nod es h av e degree γ an d all check n odes have degree ρ . The girth o f a bipa rtite grap h is even. B. LDPC Codes an d Decoding Algorithms LDPC cod es [5] are a c lass of linear blo ck codes w hich can be d efined by spar se bipar tite gr aphs [19]. Let G be a bipartite graph with two sets o f nodes: n variable no des and m ch eck nodes. This graph defines a linear block code C o f length n and dimension at least n − m in the f ollowing way: Th e n variable nodes are associated to the n coordin ates of cod ew ord s. A vector v = ( v 1 , v 2 , . . . , v n ) is a cod ew ord if and only if for each check node , the mo dulo two sum of its n eighbo rs is zero. Such a graph ical rep resentation of a n LDPC code is called the T anner g raph [20] of the code. The adjacen cy m atrix of G giv es a p arity check matrix of C . An ( n, γ , ρ ) regular L DPC code h as a T ann er g raph with n variable n odes each of degree γ (colum n weigh t) a nd nγ / ρ c heck nodes each of degree ρ (row we ight). This code has length n and rate r ≥ 1 − γ / ρ [19]. W e n ow describe the para llel b it flipping algorithm [6], [7] to d ecode LDPC codes. As n oted ear lier , ea ch check node imposes a constraint o n the neigh boring variable n odes. A constraint (ch eck node) is said to be satisfied b y a setting o f variable nodes if the sum of the variable nodes in the constraint is even; otherwise the con straint is unsatisfied. Parallel Bit Flipping A lgorithm • In parallel, flip each variable that is in mor e u nsatisfied than satisfied con straints. • Repeat u ntil n o such variable remains. A serial version of the alg orithm is also d efined in [7] and all the results in this paper hold for the serial bit flipping algorithm also. The bit flippin g algorithm s are iterative in nature but do not b elong to the class of message p assing algo rithms. C. Expa nsion an d Err or Corr ection Capability Sipser and Spielman [ 7] analyzed the per forman ce of the bit flip ping algorithms u sing the expansion p roperties o f the underly ing T ann er graph of the code. W e s um marize the results from [7] below fo r the sake of completeness. W e start with the fo llowing defin itions fro m [7]. Definition 1 : Let G = ( U, E ) with | U | = n 1 . W e say that every set of at mo st m 1 nodes expands by a factor o f δ if, for all sets S ⊂ U | S | ≤ m 1 ⇒ |{ y : ∃ x ∈ S such that ( x, y ) ∈ E }| > δ | S | . W e co nsider bipartite graphs and expan sion of variable nod es only . Definition 2 : A graph is a ( γ , ρ, α, δ ) e xp ander if it is a ( γ , ρ ) regular b ipartite grap h in wh ich ev ery subset o f at most α fr action of the variable nod es expan ds by a factor of at least δ . The following theo rem from [7] relates the e x pansion and error correction c apability of an ( n, γ , ρ ) LDPC co de with T an ner graph G wh en decod ed using the p arallel bit flipping deco ding algorithm . Theor em 1: [7, The orem 11] Let G be a ( γ , ρ , α, (3 / 4 + ǫ ) γ ) expander over n variable nodes, f or any ǫ > 0 . Th en, the simple parallel decodin g algorithm will correc t any α 0 < α (1 + 4 ǫ ) / 2 f raction of erro rs after log 1 − 4 ǫ ( α 0 n ) d ecoding round s. Notes: 1) The ser ial bit flipping alg orithm can also cor rect α 0 < α/ 2 fraction o f error s if G is a ( γ , ρ, α, (3 / 4) γ ) e x- pander . 2) The results ho ld for any left regular cod e as we need expansion of variable nodes o nly . From the abov e discussion, we observe that finding the number of variable no des which are gu aranteed to expand by a factor of at least 3 γ / 4 , gives a lower b ound o n the guar anteed err or correction capability of LDPC cod es. D. Deco ding F ailu r es an d T rappin g Sets W e n ow defin e decoding failures of the bit flipping algo- rithms an d char acterize these failures usin g trap ping sets. Consider an LDPC code of leng th n and let x be th e b inary vector whic h is the in put to the hard decision decod er . Let S ( x ) be the sup port of x . The sup port of x is defined as the set o f all positions i where x ( i ) 6 = 0 . Th e set of variable nod es (bits) whic h differ f rom their or iginal value are refe rred to as corrup t variables. Definition 3 : x is a fixed poin t of the bit flip ping algor ithm if the set of corrup t variables remains unch anged after one round o f de coding. Definition 4 : [1 6] Let x be a fixed po int. Then S ( x ) is known as a trapp ing set. An ( a, b ) trap ping set T is a set of a variable nodes whose induced subgrap h h as b odd degree checks. The size of T denoted by |T | is the number of variable n odes in T . From th e d efinitions, it is clear th at if the input to the decoder is a fixed point th en th e ou tput is also th e same fixed point. Or in other words, if th e initial errors are in variable nodes co rrespon ding to a trapping set, then the d ecoder will not con verge to the original cod ew ord . H ence, b oundin g the size of trapp ing sets gives an upp er bo und on th e g uarantee d error correctio n capability . Apart from fixed p oint d ecoding failures, there exist other types of failures, the discussion of which is beyon d the scop e of this paper . I I I . E X PA N S I O N , C O L U M N W E I G H T A N D G I RT H In this section, we prove o ur main theorem which relates the colu mn weig ht and g irth of a co de to its error c orrection capability . W e show that the size of variable node sets wh ich have the required expansion is related to the well known Moore bou nd [13, p .180] . W e start with a f ew de finitions required to establish the main theor em. A. Defi nitions Definition 5 : Th e r edu ced graph H r = ( V ∪ C r , E ′ r ) of H = ( V ∪ C, E ′ ) is a g raph with vertex set V ∪ C r and edge set E ′ r giv en by C r = C \ C p , C p = { c ∈ C : c is a pe ndant no de } E ′ r = E ′ \ E ′ p , E ′ p = { ( v i , c j ) ∈ E : c j ∈ C p } Definition 6 : Let H = ( V ∪ C , E ′ ) be su ch th at ∀ v ∈ V , d ( v ) ≤ γ . T he γ a ugmented graph H γ = ( V ∪ C γ , E ′ γ ) is a gra ph with vertex set V ∪ C γ and ed ge set E ′ γ giv en by C γ = C ∪ C a , where C a = | V | [ i =1 C i a and C i a = { c i 1 , . . . , c i γ − d ( v i ) } . E ′ γ = E ′ ∪ E ′ a , where E ′ a = | V | [ i =1 E ′ i a and E ′ i a = { ( v i , c j ) ∈ V × C a : c j ∈ C i a } . Definition 7 : [7, Definition 4] The edge-vertex inciden ce graph G ev = ( U ∪ E , E ev ) o f G = ( U, E ) is the bip artite graph with vertex set U ∪ E and ed ge set E ev = { ( e, u ) ∈ E × U : u is an endpo int of e } . Notes: 1) The ed ge-vertex inciden ce gr aph is r ight regula r with degree two. 2) | E ev | = 2 | E | . 3) g ( G ev ) = 2 g ( G ) . Definition 8 : An in verse edge-verte x incidence gr aph H iev = ( V , E ′ iev ) of H = ( V ∪ C, E ′ ) is a graph with vertex set V and edge set E ′ iev which is obtained as follows. For c ∈ C r , let N ( c ) deno te the set of neigh bors of c . Label on e node v i ∈ N ( c ) as a roo t nod e. Then E ′ iev = { ( v i , v j ) ∈ V × V : v i ∈ N ( c ) , v j ∈ N ( c ) , i 6 = j, v i is a roo t nod e, for some c ∈ C r } . Notes: 1) Giv en a g raph, the inverse ed ge-vertex inciden ce grap h is no t uniq ue. 2) g ( H iev ) ≥ g ( H ) / 2 , | E ′ iev | = | E ′ r | − | C r | and | C r | ≤ | E ′ r | / 2 3) | E ′ iev | ≥ | E ′ r | / 2 with eq uality only if all ch ecks in C r have degree two. 4) The term in verse edg e-vertex in cidence is used for the following reason . Suppo se all checks in H have d egree two. Then the edge-vertex in cidence graph of H iev is H . The Moor e bou nd [13, p.180 ] denoted by n 0 ( d, g ) is a lower bound on the least number of vertices in a d -regular graph with girth g . It is given by n 0 ( d, g ) = n 0 ( d, 2 r + 1) = 1 + d r − 1 X i =0 ( d − 1) i , g odd n 0 ( d, g ) = n 0 ( d, 2 r ) = 2 r − 1 X i =0 ( d − 1) i , g even In [21], it was shown t h at a similar boun d holds for irregular graphs. Theor em 2: [21] Th e numb er of nodes n ( d, g ) in a graph of gir th g and average d egree at least d ≥ 2 satisfies: n ( d, g ) ≥ n 0 ( d, g ) Note that d need not be an in teger in the above theor em. B. The Ma in Theo r em W e n ow state and prove th e m ain theo rem. Theor em 3: Let C b e an LDPC code with γ -left r egular T an ner graph G . Let g ( G ) = 2 g ′ . Then for all k < n 0 ( γ / 2 , g ′ ) , any set of k v ariab le n odes e xp ands by a f acto r of at least 3 γ / 4 . Pr o of: Let G k = ( V k ∪ C k , E k ) denote the subg raph induced by a set of k variable nod es V k . Since G is γ -left regular , | E k | = γ k . Le t G k r = ( V k ∪ C k r , E k r ) be the r educed graph. W e have | C k | = | C k r | + | C k p | | E k | = | E k p | + | E k r | | E k p | = | C k p | | C k p | = γ k − | E k r | W e n eed to prove that | C k | > 3 γ k / 4 . Let f ( k , g ′ ) deno te the maximum num ber of edges in an arbitrary graph o f o rder k an d g irth g ′ . By Theorem 2, fo r all k < n 0 ( γ / 2 , g ′ ) , the average degree o f a grap h with k n odes and girth g ′ is less tha n γ / 2 . Hence, f ( k , g ′ ) < γ k / 4 . W e now hav e the fo llowing lem ma. Lemma 1: The number of ed ges in G k r cannot e xce ed 2 f ( k , g ′ ) i.e., | E k r | ≤ 2 f ( k , g ′ ) . Pr o of: Th e proof is by contradiction. Assume that | E k r | > 2 f ( k , g ′ ) . Consider G k iev = ( V k , E k iev ) , an inv erse e dge vertex incidence g raph of G k . W e have | E k iev | > f ( k , g ′ ) . This is a con tradiction as G k eiv is a graph o f order k and girth at least g ′ . W e now fin d a lower b ound on | C k | in terms of f ( k , g ′ ) . W e have the following lemma. Lemma 2: | C k | ≥ γ k − f ( k , g ′ ) . Pr o of: Let | E k r | = 2 f ( k , g ′ ) − j fo r so me integer j ≥ 0 . Then | E k p | = γ k − 2 f ( k , g ′ ) + j . W e claim tha t | C k r | ≥ f ( k , g ′ ) + j . T o see this, w e n ote that | E k iev | = | E k r | − | C k r | , or | C k r | = | E k r | − | E k iev | But | E k iev | ≤ f ( k , g ′ ) ⇒ | C k r | ≥ 2 f ( k , g ′ ) − j − f ( k , g ′ ) ⇒ | C k r | ≥ f ( k , g ′ ) − j Hence we have, | C k | = | C k r | + | C k p | ⇒ | C k | ≥ f ( k , g ′ ) − j + γ k − 2 f ( k , g ′ ) + j ⇒ | C k | ≥ γ k − f ( k , g ′ ) . The the orem now follows as f ( k , g ′ ) < γ k / 4 and th erefore | C k | > 3 γ k / 4 Cor o llary 1 : Let C be a n LDPC co de with column -weight γ and girth 2 g ′ . Then the bit flipp ing algorithm can correct any error pattern of weig ht less than n 0 ( γ / 2 , g ′ ) / 2 . I V . C AG E G R A P H S A N D T R A P P I N G S E T S In this section, we first gi ve nec essary a nd sufficient con - ditions for a g iv en set of variables to be a tr apping set. W e then pro ceed to d efine a class of interesting graph s k nown as cage g raphs and estab lish a relation b etween cag e graphs[1 7 ] and trapp ing sets. W e then g iv e an up per b ound o n the error correction c apability b ased on the sizes of cage graph s. The proof s in this section are along the same lines as in Section III. Hence, due to space c onsideration s, we on ly g iv e a sketch of the p roofs. Theor em 4: Let C b e an LDPC code with γ -left r egular T an ner gra ph G . Let T be a set co nsisting of V variable n odes with in duced sub graph I . L et the ch ecks in I b e partition ed into two disjoint subsets; O co nsisting of ch ecks with odd degree and E consisting of checks with ev en degree. Then T is a trappin g set for bit flip ping algorithm iff : (a) Every variable n ode in I has at least ⌈ γ / 2 ⌉ neigh bors in E , and (b ) No ⌊ γ / 2 ⌋ + 1 chec ks of O sha re a n eighbo r o utside I . Pr o of: W e first show that the condition s stated are sufficient. L et x T be the inpu t to the bit flipping algor ithm, with sup port T . The only unsatisfied constrain ts are in O . By the condition s o f th e theor em, we observe that no variable node is inv olved in more u nsatisfied con straints than satisfied constraints. Hence, no variable nod e is flipp ed and by defini- tion x T is a fixed p oint imp lying that T is a trapp ing set. T o see tha t the condition s are necessary , observe that f or x T to be a trapping set, n o variable n ode should be inv olved in mor e un satisfied constrain ts than satisfied co nstraints. Remark: T heorem 4 is a c onsequen ce of Fact 3 f rom [ 14]. T o d etermine whether a given set o f variables is a trap ping set, it is necessary to not o nly know th e indu ced subgra ph but also th e neig hbors of the o dd degree checks. However , in order to establish ge neral bound s o n the sizes of trapping sets giv en on ly the column weight a nd the girth, we conside r o nly condition (a) of Theor em 4 which is a necessary co ndition. A set o f variable n odes satisfying con dition (a) is kn own as a potential trapping set . A trapp ing set is a potential trapp ing set that satisfies con dition (b). Hence, findin g boun ds on the size of poten tial tra pping sets gives boun ds o n the size of trapp ing sets. Definition 9 : [1 7] A ( d, g ) - cage graph , G ( d, g ) , is a d - regular g raph with girth g ha v ing the minimum p ossible number of n odes. A lower boun d, n l ( d, g ) , on the nu mber of n odes n c ( d, g ) in a ( d, g ) -cag e g raph is given b y the Moo re b ound . An upper bound n u ( d, g ) on n c ( d, g ) (see [ 17] and ref erences therein ) is given by n u (3 , g ) =  4 3 + 29 12 2 g − 2 for g odd 2 3 + 29 12 2 g − 2 for g even n u ( d, g ) =  2( d − 1) g − 2 for g odd 4( d − 1) g − 3 for g even Theor em 5: Let C b e an LDPC code with γ -left r egular T an ner g raph G and girth 2 g ′ . L et T ( γ , 2 g ′ ) den ote the the smallest p ossible trappin g set o f C for the bit flipping algorithm . Then, |T ( γ , 2 g ′ ) | = n c ( ⌈ γ / 2 ⌉ , g ′ ) Pr o of: W e first find a lower bound o n |T ( γ , 2 g ′ ) | and then exhibit a potential tr apping set of size n c ( ⌈ γ / 2 ⌉ , g ′ ) . W e begin with the fo llowing lemm a. Lemma 3: |T ( γ , 2 g ′ ) | ≥ n c ( ⌈ γ / 2 ⌉ , g ′ ) . Pr o of: Let T 1 be a trapping set with |T 1 | < n c ( ⌈ γ / 2 ⌉ , g ′ ) and let G 1 denote the induced subg raph of T 1 . W e can construct a ( ⌈ γ / 2 ⌉ , g ′′ ) - cage graph ( g ′′ ≥ g ) with |T 1 | < n c ( ⌈ γ / 2 ⌉ , g ′ ) nod es by removing edges (if necessary ) from the in verse edge -vertex of G 1 which is a contr adiction. W e now exh ibit a p otential trappin g set o f size n c ( ⌈ γ / 2 ⌉ , g ′ ) . Let G ev ( ⌈ γ / 2 ⌉ , g ′ ) be the edge-vertex in cidence grap h of a G ( ⌈ γ / 2 ⌉ , g ′ ) . Note that G ev ( ⌈ γ / 2 ⌉ , g ′ ) is a left regula r bipartite graph with n c ( ⌈ γ / 2 ⌉ , g ′ ) variable nodes of de- gree ⌈ γ / 2 ⌉ an d all che cks h av e degree two. Now consider G ev,γ ( ⌈ γ / 2 ⌉ , g ′ ) , th e γ augm ented gra ph o f G ev ( ⌈ γ / 2 ⌉ , g ′ ) . It can be seen th at G ev,γ ( ⌈ γ / 2 ⌉ , g ′ ) is a potential tra pping set. Cor o llary 2 : L et C b e an LDPC co de with column -weight γ and girth 2 g ′ . Then the bit flipping algo rithm can not b e guaran teed to cor rect all err or patterns o f we ight more th an or equal to n c ( ⌈ γ / 2 ⌉ , g ′ ) . V . D I S C U S S I O N W e d erived lower bounds and uppe r bounds on the g uar- anteed err or c orrection capa bility of left regular LDPC codes. The lower bounds we deriv ed in this pap er are weak. Howe ver , extremal g raphs av oid ing three, f our and fiv e cycles have been studied in gr eat detail (see [22], [23]) and these results can be used to deri ve tig hter boun ds when th e girth is eight, ten or twelve. A lso, since an expan sion factor of 3 γ / 4 is no t necessary (see [7, Theorem 24 ]), it is possible that tighter lower bou nds can be der i ved for some cases. The results can be extended to Gallager A an d Gallager B alg orithms as well. It shou ld be no ted that the necessary and sufficient condition s for a set to be trapping set for Gallage r A/B algorith ms ar e similar ( depend ing on the message p assing ru les) to those in Theorem 4. Our ap proach can be used to derive bound s on the guaran teed erasure recovery capab ility fo r iterative decod ing on the BEC by finding number of variable nodes which expand by a factor of γ / 2 . In [4], the bounds on the guaranteed erasure recovery ca pability were derived based on the size o f smallest stopping set. Both ap proache s g iv e th e same bo und, which also coincide with the boun ds given by T an ner in [2 0] for the minimum d istance. A C K N OW L E D G M E N T This work is fund ed by NSF u nder Gr ant CCF-063 4969 , ITR-0325 979, ECCS-072540 5 and INSIC-EHDR pro gram. R E F E R E N C E S [1] T . J. Richardson and R. 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