Doubly-Generalized LDPC Codes: Stability Bound over the BEC

Doubly-Generalized LDPC Codes: Stability Bound o v er the BEC Enrico Paolin i, Member , IEEE , and Marc Fo ssorier , F ellow , IEEE , and Marco Chiani, Senior Member , IEEE . Corresponding Address: Marco C hiani DEIS, Univ ersity of Bologna V .le Risorg imento 2 40136 Bologna, IT AL Y T el: +39-051-2093084 Fax: +39-051-2093540 e-mail: ma rco.chia ni@unibo .it Enrico Paolini and Marco C hiani are with DEIS/ W iLAB, Univ ersity of Bologna, V .le Risorgimento 2, 40136 Bologna, IT AL Y . E-mail: e.paolini@uni bo.it , mfossorier2@yahoo.com , marco.chiani@unibo.it . DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 1 Abstract The iterativ e d ecoding thr eshold o f low-density p arity-ch eck (LDPC) co des over th e binary erasure channel ( BEC) fulfills an u pper bou nd dep ending only o n th e variable and check nodes with minimu m distance 2. This boun d is a conseque nce of the stability c ondition , and is here referred to as stability bound . I n this paper, a stability bound over the BEC is de velop ed fo r doub ly-gen eralized LDPC co des, where the variable and the ch eck n odes can b e gen eric linear block cod es, assumin g maximu m a posteriori erasur e corre ction a t each node. It is p roved that in th is generalized co ntext as well the bou nd depend s o nly on th e variable and check comp onent codes with minim um distance 2. A cond ition is also developed, namely the deriv ative matching cond ition, un der which the bound is achie ved with eq uality . I . I N T R O D U C T I O N LDPC codes [1] ha ve been i ntensively studied in the last decade due to th eir capabil ity t o approach the Shannon li mit under iterative belief-propagatio n decoding. An LDPC code of length N and dimension K can be graphically represented as a bipartite graph, known as T ann er graph, with N var iable nodes (VNs) and M ≥ N − K check nodes (CNs) [2]. In the T anner graph, the degree of either a VN or a CN is defined as the n umber of edges connected to it. A degree- n VN of an LDPC code can be i nterpreted as a length- n repetition code, i.e., as a ( n, 1) li near block code repeating n times its onl y informatio n bit tow ards the check node decoder (CND). Instead, a d egree- n CN of an L DPC code can b e interpreted as a length - n single parity-check (SPC) code, i.e., as a ( n, n − 1) li near block code. An extension of the concept of LDPC code i s represented by doubly-generalized LDPC (D- GLDPC) codes [3], where the VNs and the CNs are allowed t o be generic ( n, k ) linear block codes ins tead of repetition and SPC codes, respectively . If onl y the CND is generalized while all t he VNs are repetition codes, then t he code i s s aid a g eneralized LDPC (GLDPC) code, or a T anner cod e [2]. In a D-GLDPC code t he codes used as VNs and CNs are called component codes . In thi s work each comp onent code is supp osed to be a linear block code having a mi nimum distance d min ≥ 2 . The VNs and the CNs which are n ot repetiti on or SPC codes, are referred to as generalized nodes . The corresponding code structure is depi cted in Fig. 1. An ( n, k ) generalized VN is characterized by n connectio ns towards the CND; mo reove r , k of the N D-GLDPC encoded 2 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y P S f r a g r e p l a c e m e n t s n 2 edges k 1 bits ( n 2 , k 2 ) generalized CN n 2 − k 2 equations SPC CN rep. VN ( n 1 , k 1 ) generalized VN | {z } encoded bits n 1 edges Fig. 1. Structure of a D-GLDP C code. bits are associat ed wi th the VN, and in terpreted by the VN as its k information bit s. Then, the code word length of a D-GLDPC code with N V VNs i s N = P N V i =1 k i ( k i being the dimension of the i -th VN). An ( n, k ) generalized CN i s characterized by n connecti ons towards the variable node decoder (VND), and is ass ociated wit h n − k independent parity check equations. Then, th e number of parity-check equations for a D-GLDPC code with N C CNs is M = P N C i =1 ( n i − k i ) ( k i and n i being the di mension and length of t he i -th check no de, respectiv ely). For a descriptio n of a D-GLDPC code iterative decoder over the A WGN channel and BEC we refer to [3] and [4], respectiv ely . For LDPC code ensembles, an important rol e is played by a theorem known as the stabil ity condition [5]–[7]. T he most im portant consequence of the s tability con dition i s the p ossibilit y to u pper bound t he asymptot ic i terativ e decoding threshold. If the commun ication channel is a BEC with erasure probabilit y q , the stabilit y condi tion leads t o the following upper bound on the asympto tic thresho ld q ∗ for the LDPC ensembl e: q ∗ ≤ [ λ ′ (0) ρ ′ (1)] − 1 . (1) The inequali ty (1) is refer red to as sta bility bound i n this paper 1 . In (1), λ ′ (0) = λ 2 is th e fraction 1 W e use the nomenclature stability bound as, e ven though (1) is sometimes referred to as the stability condition, strictly speaking it is a consequence of it . For more details we refer to [7, T heorem 3.66] and the related discussion. DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 3 of edges connected to the length-2 repetiti on VNs, while ρ ′ (1) is t he deriv ativ e (computed at x = 1 ) of the L DPC CNs degree dis tribution ρ ( x ) = P j ≥ 2 ρ j x j − 1 , where ρ j is the fraction of edges connected to SPC CNs of lengt h j . The bound (1) was first de veloped from densi ty e volution. Next w e propose a sim ple graphical interpretation of (1) using extrinsic i nformation transfer (EXIT) charts [8]. L et us denote by I A the av erage a pri ori mutual information i n input to the VND or to the CND. Furthermore, let us denote by I E ,V ( I A , q ) and I E ,C ( I A ) the avera ge extrinsic information for the VND and CND respectively (these functions are us ually referred to as EXIT function s ). Th en, (1) is equivalent to the following condition: for q = q ∗ , the deriv ativ e of the VND EXIT functi on I E ,V ( I A , q ) , with respect to I A and e valuated at I A = 1 , must be smaller than the deriv ative of the inv erse CND EXIT function I − 1 E ,C ( I A ) e valuated at I A = 1 . That is (1) is equivalent to requiring ∂ I E ,V ( I A , q ∗ ) ∂ I A    I A =1 ≤ d I − 1 E ,C ( I A ) d I A    I A =1 . (2) There exist LDPC degree dis tributions achieving the bo und (1) with equality , so t hat their threshold over th e BEC assumes t he simp le closed form q ∗ = [ λ ′ (0) ρ ′ (1)] − 1 . For such LDPC distributions, the fir st occurrence of a tangenc y point between the VND EXIT function I E ,V ( I A , q ) and the in verse CND EXIT function I − 1 E ,C ( I A ) appears at I A = 1 , i.e.    I E ,V (1 , q ∗ ) = I − 1 E ,C (1) ∂ I E ,V ( I A , q ∗ ) ∂ I A    I A =1 = d I − 1 E ,C ( I A ) d I A    I A =1 . (3) For LDPC code ensem bles characterized by VNs and CNs with degree at least 2, the first equality is always sati sfied as both terms are equal to 1. This occurs also for D-GLDPC codes with all variable and check component codes ha ving a m inimum dis tance d min ≥ 2 [9], whi ch is an assumpti on of this paper . Then, only the second equality is consi dered in the sequel, and is referred to as the derivative matching condition . In this paper , th e stability upper bound (1) and the deriv ative matchin g condition are extended to D-GLDPC codes (and to GLDPC codes as a sub-case). O ur deriv atio ns lead to the conclusion that only the check and variable component codes with m inimum di stance d min = 2 , incl uding length-2 repetition codes and SPC codes, contribute to the stabilit y bound. W e also show that 4 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y for D-GLDPC codes sati sfying t he deri vati ve m atching con dition the asymptotic thresh old over the BEC can be expressed by a si mple formula. The paper i s organized as fol lows. Some definiti ons and t he notation used in the paper are introduced in Section II. In Section III the possi bility to reduce the rank of a linear block code generator matrix by colum n eliminatio n is discussed. Usi ng th ese results, in Section IV and Section V the stability bound is de veloped for GLDPC codes and for D-GLDPC codes, respectiv ely . Fin al remarks are given in Section VI. I I . D E FI N I T I O N S A N D B A S I C N O T A T I O N W e assume as transm ission channel a BEC w ith erasure prob ability q . For a bi partite graph with random conn ections, the extrinsic channel (that is th e channel over which the messages are exchanged between the VND and the CND duri ng the iterativ e decoding process) is modelled as a s econd BEC with erasure probabilit y p d epending on the decoding iteration [10], where it is readily proved that I A = 1 − p . Since we express bo th the VND and the CND EXIT functions as functions of p (and q for the VND), t heir deriv ativ es are ev aluated at p = 0 (corresponding to I A = 1 ). In th is case (2) becomes ∂ I E ,V ( p, q ∗ ) ∂ p    p =0 ≥ d I E ,C ( p ) − 1 d p    p =0 . (4) Under th e hypo thesis of a random bipartite graph, the VND and CND EXIT functions can be expressed as I E ,V ( p, q ) = I V X i =1 λ i I ( i ) E ,V ( p, q ) (5) and I E ,C ( p ) = I C X i =1 ρ i I ( i ) E ,C ( p ) , (6) respectiv ely , where I V and I C are the num ber of d iffe rent VN and CN ty pes, I ( i ) E ,V ( p, q ) and I ( i ) E ,C ( p ) are t he EXIT functions for t he i -th VN type and for the i -th CN type, respectively , and λ i and ρ i are the fractions of edg es towards the VNs of type i and the CNs of type i , respecti vely . DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 5 For the sake of clarity , i t is useful to isolate the contribution of the repetit ion com ponent cod es in (5) and the con tribution of the SPC com ponent codes in (6), so th at I E ,V ( p, q ) = (rep) X j ≥ 2 λ (rep) j · (1 − q p j − 1 ) + (gen) X i λ i I ( i ) E ,V ( p, q ) = (rep) X j ≥ 2 λ (rep) j − q λ rep ( p ) + (gen) X i λ i I ( i ) E ,V ( p, q ) (7) I E ,C ( p ) = (SPC) X j ≥ 2 ρ (SPC) j · (1 − p ) j − 1 + (gen) X i ρ i I ( i ) E ,C ( p ) = ρ SPC (1 − p ) + (gen) X i ρ i I ( i ) E ,C ( p ) . (8) In (7), j is the length of the g eneric repetitio n VN, λ (rep) j is the fraction of edges connected to the repetition VNs of l ength j , and λ rep ( x ) , P j ≥ 2 λ (rep) j x j − 1 . W e use i n (7) the well k nown EXIT function expression over the BEC for a ( j, 1 ) repetition VN, i .e. I E ( p, q ) = 1 − q p j − 1 . The second summ ation in (7) is ove r all the generalized VN typ es. Analogou sly , in (8) j is the length of th e generic SPC CN, ρ (SPC) j is the fraction of edges towa rds t he SPC CNs of length j , ρ SPC ( x ) , P j ≥ 2 ρ (SPC) j x j − 1 , and we us e the well-known EXIT function expression over the BEC for a ( j , j − 1) SPC CN, i.e. I E ( p ) = (1 − p ) j − 1 . The EXIT function of an ( n, k ) generalized VN over the BEC, when m aximum a posteriori (MAP) erasure correction is performed at the VN, can be expressed as I E ( p, q ) = 1 − 1 n n − 1 X t =0 k X z =0 a t,z p t (1 − p ) n − t − 1 q z (1 − q ) k − z , (9) which can be readily ob tained from [10, eq. 36] with a t,z = [( n − t ) ˜ e n − t,k − z − ( t + 1) ˜ e n − t − 1 ,k − z ] . The parameter ˜ e g ,h (with g = 0 , . . . , n and h = 0 , . . . , k ) is known as the ( g , h ) -th un-norm alized split information functi on, d efined as explained next. Considering a representation G of th e generator matrix for t he ( n, k ) VN, and appending to it the ( k × k ) identity m atrix I k , ˜ e g ,h is equal to the summatio n of the ranks over all the possible submatrices ob tained selectin g g columns out of G and h colum ns out of I k . W e remark that the spl it information functions for a generalized VN, and therefore its MAP EXIT fun ction (9), depend on the chosen generator 6 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y matrix representation [9]. Then, the performance of the overall D-GLDPC code depends on the code representati on used for th e variable com ponent codes. F or the same reason, two generalized VNs associat ed with t he same code, but with diff erent generator m atrices (i.e. dif ferent m appings between information words and codew ords) must be regarded in (7) as VNs of diffe rent types. The EXIT function of a generalized ( n, k ) CN over t he BEC, when MAP decoding is performed at the CN, can be obtained by letting q → 1 in (9). The obtained expression, equi valent to [10, eq. 40], is I E ( p ) = 1 − 1 n n − 1 X t =0 a t p t (1 − p ) n − t − 1 , (10) with a t = ( n − t ) ˜ e n − t − ( t + 1) ˜ e n − t − 1 For g = 0 , . . . , n , ˜ e g is known as t he g -th un-normalized i nformation functi on of the ( n, k ) code, a concept first introduced in [11]. It is defined as the summ ation of t he ranks over all t he possible sub matrices obt ained by selecting g column s out of t he g enerator m atrix G . As op posed to the split i nformation function s ˜ e g ,h , the information functi ons ˜ e g are ind ependent of the code representation. Thus, different check compon ent code representations are associated with the same EXIT functi on for the generalized CN. The performance o f a GLDPC or D-GLDPC code is then independent of the specific representation of its generalized check component codes. Let us supp ose that a generic VN is a ( n, k ) l inear block code C , with generator m atrix G . W e denote by C ′ the ( n + k , k ) linear block code generated by [ G | I k ] . The generic codeword of C is deno ted by c , while the generic cod e word of C ′ by c ′ . W e hav e c ′ = [ c | u ] , where c and u must sat isfy c = u G : the code C ′ then d epends o n th e chosen generator matrix representation for C . It is readily s hown that d ′ min ≥ d min + 1 , where d min and d ′ min are the m inimum dist ances of C and C ′ , respective ly . I I I . R E D U C I N G A G E N E R A T O R M A T R I X R A N K B Y C O L U M N E L I M I NA T I O N For a given ( n, k ) l inear b lock code C and for a given representation G of i ts generator matrix, we denote by S t a generic su bmatrix obtained by selecting t columns out of G , and by S t the submatrix composed of the n − t remaining columns . DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 7 Definition 1: W e say that S t cove rs a non-null codew ord c ∈ C when th ere are no ‘1’ posi tions of c correspondin g to columns belonging to S t . Example 1: L et us consider a (7 , 3) simplex code wi th generator matrix G =      1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1      , and let us denote by S 2 the submatri x compos ed of the last two col umns of G . Then, the onl y non-null codeword covere d by S 2 is [0 , 1 , 1 , 1 , 1 , 0 , 0] . The following theorem states that in order to reduce the rank of a given generator matrix by column elimi nation, i t is necessary and sufficient that t he removed pattern of columns covers at least one non-null codeword. Theor em 1: Let us consi der an ( n, k ) linear block code C . For any generator matrix repre- sentation, we have rank( S t ) < k i f and only if S t cove rs at least one codeword. Pr oof: [ Sufficiency ] Suppose that S t cove rs a codew ord ˆ c , and consi der a representation ˆ G of the generator m atrix where ˆ c is one of the rows. It follows that removing from ˆ G the n − t columns associated with S t reduces t he rank b ecause at l east one of the rows becomes an al l-zero row , so that rank ( S t ) < k . Since any representation of the generator matrix can be obt ained from any other representatio n by row summations only , and si nce row summati ons cannot mod ify the rank of submatrices composed of g enerator matrix columns, we hav e ra nk( S t ) < k also for any representation other than ˆ G . [ Necessity ] Con versely , let u s suppose that rank( S t ) < k fo r a given generator matrix repre- sentation. Using the same argument as for the sufficienc y , we observe that this inequ ality must be satisfied also for any other representation of the generator matrix. A s removing S t from any generator matri x leads t o a ( k × t ) matrix with reduced rank, it m ust be poss ible to obtain (from any generator matrix representation) a generator m atrix where one or more rows hav e only ‘0’ in those posi tions corresponding to S t . All t hese rows are n on-null code words of C cover ed by S t . Cor ollary 1: W e have rank( S t ) = k for all S t if and onl y if n − t < d min . 8 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y Pr oof: [ Sufficiency ] Let us suppo se that ra nk( S t ) = k fo r all S t . By applying Theorem 1 it foll ows that no submatrix S t (composed o f n − t col umns) can cover any cod e word. Then n − t < d min . [ Necessity ] Con versely , let us suppose that n − t < d min . Then, no submatrix S t (composed of n − t colum ns) can cov er any code word. By applying Theorem 1 we conclude that rank( S t ) = k for all S t . Example 2: A ll the cod e words of the (7 , 3) simplex code of Example 1 have Hamming weight 4. As one of these code words is [1 , 1 , 1 , 0 , 0 , 0 , 1] , Theorem 1 guarantees that if we remove the first three and the last column from G given in Example 1 (or from any m atrix obtained performing row summations on G ) we obtain a (3 × 3) matrix with rank s maller than 3. On t he ot her hand, by Coroll ary 1 we know that, ev en if we remove any set of three or less columns , the rank o f G remains unchanged. In [9] the concept of independent set was introduced. Giv en a ( k × n ) rank- r binary matrix, an independent set of size s is defined as any set of s columns such that removing these columns from the matrix leads to a ( k × ( n − s )) matrix wi th a rank smaller t han r . By Theorem 1 we now state that a necessary and sufficient condit ion for a set of s columns to be an in dependent set of a ( k × n ) generator matrix is that the s colu mns cover at least one code word. Mo reove r , by Corollary 1 we recognize that any set of s < d min columns cannot form an independent set for the generator m atrix. I V . S T A B I L I T Y B O U N D A N D D E R I V A T I V E M A T C H I N G F O R G L D P C C O D E S In GLDPC codes all the v ariable component codes are repetition codes, which in (7) leads to P (rep) j ≥ 2 λ (rep) j = 1 . The EXIT function over t he BEC for th e VND is then giv en by I E ,V ( p, q ) = 1 − q λ rep ( x ) . It follows ∂ I E ,V ( p, q ) ∂ p    p =0 = − q λ (rep) 2 . (11) From (8), the deriv ative of I E ,C ( p ) at p = 0 is d I E ,C ( p ) d p    p =0 = − ρ ′ SPC (1) + (gen) X i ρ i d I ( i ) E ,C ( p ) d p    p =0 . (12) DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 9 In order to develop (12) it is necessary to explicit the deriv ativ e of each generalized CN type EXIT function. This t ask can be performed by exploiting Corollary 1, as explained next. Consider an ( n, k ) generalized CN wi th EXIT function I E ( p ) in the form (10). It is readily shown that d I E ( p ) d p    p =0 = ( n − 1) a 0 − a 1 n . W e have a 0 = 0 if and only i f the generalized CN has minimum distance d min ≥ 2 . In fact, the generator matrix of the check compo nent code is full rank (rank = k ) by definition , so ˜ e n = k . Furthermore, from Corollary 1, removing any single column from the generator matrix does n ot reduce the rank if and only if d min ≥ 2 , i n wh ich case we o btain ˜ e n − k = n k , so that a 0 = n ˜ e n − ˜ e n − 1 = n k − n k = 0 . As recalled in Section I, the hypothesis d min ≥ 2 is alwa ys assumed in this paper . Then, we can ass ume a 0 = 0 . If d min ≥ 2 for the CN we obtain d I E ( p, q ) d p    p =0 = − a 1 n , where a 1 = ( n − 1) ˜ e n − 1 − 2 ˜ e n − 2 = k n ( n − 1) − 2 ˜ e n − 2 . By app lying again Corollary 1, we obtain a 1    = 0 if d min ≥ 3 > 0 if d min = 2 . (13) If the CN exhibits a minim um di stance d min ≥ 3 , then removing any p air of col umns from th e generator matrix does not af fect the rank. In this case 2 ˜ e n − 2 = 2 k  n 2  = k n ( n − 1) , hence a 1 = 0 . According to these results, the only generalized CNs th at contribute to the summat ion in the second term of (12 ) are thos e characterized by d min = 2 . By recalling that all the SPC codes hav e mini mum distance 2, we conclu de that (12) only depends on t he check com ponent codes with d min = 2 . The d eri vati ve at p = 0 o f the CND EXIT functio n can be then expressed as d I E ,C ( p ) d p    p =0 = − ρ ′ SPC (1) − [2] X i ρ i k i n i ( n i − 1) − 2 ˜ e n i − 2 n i = − ρ ′ SPC (1) − [2] X i ρ i 2∆ ( i ) n − 2 n i (14) 10 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y where the n otation P [2] is adopted to indicate the summation over those g eneralized CN typ es with minimum di stance 2. In (14), we ha ve denoted by ∆ ( i ) n − 2 the expression k i n i ( n i − 1) / 2 − ˜ e n i − 2 , that does not depend on the chosen representation for the i -th generalized CN type. The next t heorem s tates t hat ∆ ( i ) n − 2 is equal to the multip licity A ( i ) 2 of th e weight -2 codewords for the CNs of type i . Theor em 2: For any linear block check component code with minimum dis tance d min = 2 , the parameter ∆ n − 2 equals t he multi plicity A 2 of the CN codew ords wit h Ham ming weigh t 2 , i.e. ∆ n − 2 = A 2 . Pr oof: Let S n − 2 be th e generic ( k × ( n − 2)) m atrix obtained by removing 2 columns from (any representation of) t he CN generator matrix. By Corollary 1 we hav e that either rank( S n − 2 ) = k or rank( S n − 2 ) = k − 1 : consid ering a CN wi th d min = 2 , removing any sin gle column cannot reduce the rank so that removing two columns can reduce the rank at m ost b y one. W e have ∆ n − 2 = k n ( n − 1) 2 − ˜ e n − 2 = X S n − 2 k − X S n − 2 rank ( S n − 2 ) = X S n − 2 ( k − rank ( S n − 2 )) , where we know that each term in the summation is either equal to 0 or to 1. By Theorem 1 any such term is equal t o 1 if and only if S n − 2 cove rs a (necessarily weight-2) codeword. The deriv ativ e at p = 0 of the in verse CND EXIT function I − 1 E ,C ( p ) is given by 1 / d I E ( p ) / d p | p =0 . Combining (11), (14) and Theorem 2, for GLDPC codes, (4 ) becomes q ∗ ≤ h λ (rep) 2  ρ ′ SPC (1) + [2] X i ρ i 2 A ( i ) 2 n i i − 1 . (15) DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 11 W e can furth er simplify (15) by noting that ρ ′ SPC (1) = (SPC) X j ρ j ( j − 1) = (SPC) X j ρ j 2 A ( j ) 2 n j , as for a SPC CN n j = j and A ( j ) 2 =  j 2  = j ( j − 1) / 2 . Hence, (15) can be written in the more compact form q ∗ ≤   λ (rep) 2 [2] X i 2 ρ i n i A ( i ) 2   − 1 = h λ (rep) 2 C i − 1 (16) where C = [2] X i ρ i C i with C i = 2 A ( i ) 2 n i , and where now P [2] indicates the summ ation over all the d min = 2 check component codes, both SPC and generalized. For GLDPC codes sati sfying th e deriva tiv e m atching cond ition (3) (the first occurrence of a tangency po int between I E ,V ( p, q ) and I − 1 E ,C ( p ) appears at p = 0 ), th e threshol d assumes t he simple clo sed form q ∗ = [ λ (rep) 2 C ] − 1 . If onl y generalized CNs with d min ≥ 3 are us ed, th en (15) becomes q ∗ ≤ [ λ (rep) 2 ρ ′ SPC (1)] − 1 . If the deri vati ve matching conditi on is fulfilled in t his case, we obtain q ∗ = [ λ (rep) 2 ρ ′ SPC (1)] − 1 . V . S TA B I L I T Y B O U N D A N D D E R I V A T I V E M A T C H I N G F O R D - G L D P C C O D E S The deriv ativ e at p = 0 of the CND EXIT function of D-GLDPC codes is t he same as for GL DPC codes, that is d I − 1 E ( p ) / d p | p =0 = − 1 /C . The partial deriv ative of the VND EXIT function with respect to p and ev aluated at p = 0 , i s de veloped next. It follows from (7) that ∂ I E ,V ( p, q ) ∂ p    p =0 = − q λ (rep) 2 + (gen) X i λ i ∂ I ( i ) E ,V ( p, q ) ∂ p    p =0 . (17) In order to develop the summ ation ove r the generalized VN types in the second part of (17), we hav e to explicit the partial deriv ative respect to p of each generalized VN type EXIT function, 12 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y e valuated at p = 0 . T o this end, let us consider an ( n, k ) generalized VN whose EXIT function is giv en by (9). After defining f ( p ) = n − 1 X t =0 a t,z p t (1 − p ) n − 1 − t we hav e ∂ I E ( p, q ) ∂ p    p =0 = − 1 n k X z =0  d f ( p ) d p    p =0  q z (1 − q ) k − z = k X z =0 ( n − 1) a 0 ,z − a 1 ,z n q z (1 − q ) k − z , (18) as it is readily sh own that that d f ( p ) / d p | p =0 = − ( n − 1) a 0 ,z + a 1 ,z . The expression (18) can be further d e veloped by in voking Corollary 1. Since any variable component code has minimu m distance d min ≥ 2 by hypothesis, removing any si ngle column from the generator matrix G of the variable component code cannot reduce the rank of G . It follows a 0 ,z = n ˜ e n,k − z − ˜ e n − 1 ,k − z = k n  k k − z  − k n  k k − z  = 0 , thus leading to ∂ I E ( p, q ) ∂ p    p =0 = − k X z =0 a 1 ,z n q z (1 − q ) k − z . Corollary 1 can be in vok ed agai n in order to show t hat a 1 ,z    = 0 ∀ z if d min ≥ 3 > 0 ∀ z if d min = 2 , (19) where d min is the variable com ponent code m inimum distance. In fact, under the h ypothesis d min ≥ 3 , removing any singl e colum n or any pair of columns from (any representation of) G cannot reduce its rank. Under this hypothesis a 1 ,z = ( n − 1 ) ˜ e n − 1 ,k − z − 2 ˜ e n − 2 ,k − z = k n ( n − 1)  k k − z  − 2 k  n n − 2   k k − z  = 0 . DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 13 Hence, the o nly generalized variable compo nent codes cont ributing to (17) are thos e with minimum distance d min = 2 . This i s coherent with the fact that, among the repetition component codes, only those w ith d min = 2 (i.e. t he l ength-2 repetit ion codes) giv e a non-null contribution to (17). Then, (17) can be developed as ∂ I E ,V ( p, q ) ∂ p    p =0 = − q λ (rep) 2 − [2] X i λ i k i X z =0 k i n i ( n i − 1)  k i k i − z  − 2 ˜ e n i − 2 ,k i − z n i q z (1 − q ) k i − z = − q λ (rep) 2 − [2] X i λ i k i X z =0 2∆ ( i ) n − 2 ,k − z n i q z (1 − q ) k i − z . (20) In the previous expression t he symbol P [2] indicates the sum mation over those generalized VN types with mi nimum distance 2. Moreover , ∆ ( i ) n − 2 ,k − z is defined as k i n i ( n i − 1) 2  k i k i − z  − ˜ e n i − z ,k i − z . As opposed to ∆ ( i ) n − 2 in (14), ∆ ( i ) n − 2 ,k − z in (20) depends on th e component code representation. Using (20) we can express (4) for a D-GLDPC code as q ∗ λ (rep) 2 + [2] X i λ i k i X z =0 2 ∆ ( i ) n − 2 ,k − z n i ( q ∗ ) z (1 − q ∗ ) k i − z ≤ 1 C . (21) In t he reminder of this section, we prove that (21 ) can be written as an explicit upper bound to the decoding threshol d q ∗ . W e start by proving the fol lowing theorem. Theor em 3: Let us cons ider an ( n, k ) linear b lock variable component code with minimum distance d min = 2 . W e have k X z =0 2 ∆ n − 2 ,k − z n ( q ∗ ) z (1 − q ∗ ) k − z = k X u =1 2 A 2 ,u n ( q ∗ ) u , (22) where A 2 ,u is the num ber o f the VN weig ht-2 codew o rds generated by weight- u i nformation words. Pr oof: Let C be the ( n, k ) variable component code and let G be the chos en g enerator matrix for C . Moreover , let S n − 2 ,k − z be the generic ( k × ( n − 2 + k − z )) matrix obtain ed by selecting n − 2 colum ns in G and k − z columns in the ( k × k ) id entity matrix. Let us apply Theorem 1 t o the code C ′ introduced at the end of Section II . Each codeword c ′ ∈ C ′ is comp osed of the concatenation of a codew ord c ∈ C with one of the p ossible 2 k sequences o f k bit s (where by the linearity of C the all -zero length - k sequence is always concatenated with th e all-zero codeword of C ). Com bining t his observation with Theorem 1 14 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y and introd ucing t he notation S n − 2 ,k − z = [ S G n − 2 |S I k − z ] we o bserve that a necessary (thou gh not suffi cient) condition for having rank( S n − 2 ,k − z ) < k i s that S G n − 2 cove rs a weight-2 codew ord of C . Next, we d e velop ∆ n − 2 ,k − z as ∆ n − 2 ,k − z = k n ( n − 1 ) 2  k k − z  − ˜ e n − 2 ,k − z = X S n − 2 ,k − z k − X S n − 2 ,k − z rank( S n − 2 ,k − z ) = X S n − 2 ,k − z ( k − rank( S n − 2 ,k − z )) = X S G n − 2 X S I k − z  k − rank  [ S G n − 2 |S I k − z ]  = [2] X c X S I k − z  k − rank  [ S G n − 2 |S I k − z ]  , where P [2] c is used to i ndicate the s ummation over those S G n − 2 such that S G n − 2 cove rs a weight-2 code word of C . Th en, we can write k X z =0 2 ∆ n − 2 ,k − z n ( q ∗ ) z (1 − q ∗ ) k − z = 2 n k X z =0 [2] X c X S I k − z  k − rank  [ S G n − 2 |S I k − z ]  ( q ∗ ) z (1 − q ∗ ) k − z = 2 n [2] X c k X z =0 X S I k − z  k − rank  [ S G n − 2 |S I k − z ]  ( q ∗ ) z (1 − q ∗ ) k − z . (23) By hyp othesis there are no VNs wit h min imum distance 1. Then, for a giv en weight-2 code word c ∈ C , any s ubmatrix S n − 2 ,k − z is such t hat S n − 2 ,k − z can cover at most one code word of C ′ , i.e. the codeword [ c | u c ] s ubject to c = u c G . If we denote by w H ( u c ) t he Ham ming weight of u c , for each weight-2 cod e word c ∈ C the s ummation over z in (23) can always start from w H ( u c ) . In fact, for z = 0 , . . . , w H ( u c ) − 1 it i s not possib le for S n − 2 ,k − z to cover the code word [ c | u c ] , so t hat k − rank  [ S G n − 2 |S I k − z ]  = 0 . That all ows writi ng th e second mem ber DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 15 in (23) as 2 n [2] X c k X z = w H ( u c ) X S I k − z  k − rank  [ S G n − 2 |S I k − z ]  ( q ∗ ) z (1 − q ∗ ) k − z . (24) For given z ≥ w H ( u c ) , the codew ord [ c | u c ] i s covered by exactly  k − w H ( u c ) z − w H ( u c )  matrices S n − 2 ,k − z . Hence, there are exactly  k − w H ( u c ) z − w H ( u c )  non-null terms in X S I k − z  k − rank( S G n − 2 |S I k − z )  ( q ∗ ) z (1 − q ∗ ) k − z . Deleting from G two colum ns corresponding t o a weight-2 codew ord of C reduces the rank of this m atrix by one, leading to a rank k − 1 . In fact, cons idering the VN minimum distance d min = 2 , removing t he first colum n cannot reduce the rank (Corollary 1) and removing the second column reduces t he rank (Theorem 1) necessarily by one. W e can then conclude t hat each of the  k − w H ( u c ) z − w H ( u c )  non-null terms in the summatio n P S I k − z  k − rank  [ S G n − 2 |S I k − z ]  is equal to one, independentl y of z . Then we can further de velop (24) as 2 n [2] X c k X z = w H ( u c )  k − w H ( u c ) z − w H ( u c )  ( q ∗ ) z (1 − q ∗ ) k − z . (25) W e next observe that t hose weight -2 codewords c ∈ C associated wi th the same w H ( u c ) (i.e. generated by information words ha ving the same weight) produce the same contri bution in (25), since onl y t he Hamm ing weight of the informati on words u c matters. This observation allows us to write (25) as k X u =1 2 A 2 ,u n k X z = u  k − u z − u  ( q ∗ ) z (1 − q ∗ ) k − z , (26) where A 2 ,u is the n umber of weight -2 codew ords c ∈ C such th at w H ( u c ) = u . In general, A 2 ,u depends on the variable com ponent code representation. By no ting t hat k X z = u  k − u z − u  ( q ∗ ) z (1 − q ∗ ) k − z = ( q ∗ ) u , we finally obtai n (22). Theorem 3 allows us to writ e the first member o f (21) as q ∗ λ (rep) 2 + [2] X i λ i k i X u =1 2 A ( i ) 2 ,u n i ( q ∗ ) u . 16 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y The length -2 repetition VNs can be embedded int o the summation over the generalized VN t ypes with minimum distance 2. In fact, th e only weight-2 cod e word of a length-2 repetition VN is c = [1 , 1] , wh ich is generated by a weight-1 informat ion word. T hen, for a length-2 repeti tion VN we have λ (rep) 2 k X u =1 2 A 2 ,u n ( q ∗ ) u = λ (rep) 2 q ∗ . Hence, (21) can be put i nto the more compact form [2] X i λ i k i X u =1 2 A ( i ) 2 ,u n i ( q ∗ ) u ≤ 1 C , (27) where now the su mmation P [2] is over all the VN types wit h minimum distance 2, both repetition and generalized. The first p art o f (27) is a real polynomial P ( · ) in the variable q ∗ . This p olynomial can be written as P ( x ) = P [2] i λ i P i ( x ) , where P i ( · ) is a degree- k i real polynom ial associated with the d min = 2 type- i VNs. Each P i ( · ) is a monotonically increasing functi on (since all i ts coeffi cients are p ositive). Consequently , P ( · ) is a mo notonically increasing function and its in verse P − 1 ( · ) exists. W e hav e then proved t he following theorem, whi ch is the m ain contribution of t his paper . Theor em 4 (St ability bound over the BEC for D-GLDPC codes): The asymptotic threshol d q ∗ of a D-GLDPC code ensemb le over the BEC, assuming MAP erasure correction at each com- ponent code, fulfills q ∗ ≤ P − 1  1 C  , (28) where P ( x ) = [2] X i λ i P i ( x ) with P i ( x ) = k i X u =1 2 A ( i ) 2 ,u n i x u and C = [2] X i ρ i C i with C i = 2 A ( i ) 2 n i . For an L DPC code ensemble (28) returns q ∗ ≤ [ λ ′ (0) ρ ′ (1)] − 1 , i.e. the well-k nown stability bound for LDPC codes. DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 17 Pr operty 1: For a l ength-2 repetiti on VN (i.e. for a con ventional LDPC degree-2 VN) we hav e P i ( x ) = x . Hence, if the only d min = 2 VNs are length-2 repetition codes, we hav e P − 1 ( x ) =  1 /λ (rep) 2  x . This is t he case for GLDPC codes. Pr operty 2: For a lengt h- n i SPC CN (i.e. for a con ventional LDPC degree- n i CN) we have C i = n i − 1 . Pr operty 3: Any lengt h- n i and weight-2 binary sequence is a codew ord for a l ength- n i SPC CN. Then, C i for a length- n i CN with min imum dist ance 2 is maximum when t he CN is a SPC code. In other words, C i fulfills C i ≤ n i − 1 , where the equality hol ds when the CN is a SPC code. Pr operty 4: For any VN wi th mi nimum dist ance d min = 2 , the value of P i ( x ) depends on the chosen g enerator matrix through t he coefficients A ( i ) 2 ,u . This is t rue for all x , except at x = 0 and at x = 1 where we have P i (0) = 0 and P i (1) = 2 A ( i ) 2 n i = C i respectiv ely . Independentl y of th e VN representation, the value assumed by P i ( x ) at x = 1 is equal to the value o f C i for the same d min = 2 linear block code when used as a CN. Pr operty 5: For a D-GLDPC code ensemble satisfyin g the deriv ative m atching condition, the iterativ e decoding threshol d over the BEC assumes the s imple fo rm q ∗ = P − 1 (1 /C ) . It should be noted that i n general P − 1 (1 /C ) is not a clo sed form for the threshold. Howe ver , there are simple cases in which P − 1 ( · ) can be explicited. An e x ample is pro v ided in the appendix. V I . C O N C L U S I O N In this paper , a stability bou nd over the BEC has been dev eloped for D-GLDPC codes. It generalizes t he inequalit y q ∗ ≤ [ λ ′ (0) ρ ′ (1)] − 1 , v alid for LDPC code ensembles. W e have shown 18 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y that for D-GLDPC codes, as for LDPC codes, the only variable and check compon ent codes contributing to the bound are thos e having minimum d istance 2. A deriv ative matching cond ition suffi cient t o achieve the bound with equalit y has also been d efined. If the deriv ative matching condition is fulfilled, then the decoding thresho ld over the BEC for D-GLDPC codes i s expressed by a sim ple formul a, although in general not in closed-form. For GLDPC codes t his form ula alwa ys leads to a closed-form threshold expression. A P P E N D I X I D - G L D P C C O D E S W I T H S P C V A R I A B L E N O D E S GLDPC codes employing strong generalized CNs (such as H amming or BCH CNs) represent a possible solution for obtaining a good compromise between w aterfall performance and error floor . Examples of such GLDPC code constructi ons are described in [12]–[16]. In general, increasing the fraction of strong generalized CNs can be very fa vorable from the point of vi e w of the overa ll code minim um dist ance and then of the error floor , but presents drawbacks. A first drawback is represented by an overall code rate loss which makes GLDPC codes with l ar ge fractions of strong generalized CNs of in terest only for low or very low rate [17]. The reason is briefly revie wed n ext. Let us consider a more general code structure, namely a D-GLDPC code. If we denote by r V , i and by r C,j the code rate of the type- i VNs and of the type- j CNs, respectively , the overall design rate is R = 1 − P j ρ j (1 − r C,j ) P i λ i r V , i , (29) which is mon otonically increasing respect to any r V , i and to any r C,j . A generalized CN of lengt h n has a code rate smaller than th e code rate of a length- n SPC CN. Then, a l ar ge fraction of strong generalized CNs determines an overall rate loss. In GLDPC cod es t his rate loss is di f ficult to comp ensate eve n usi ng large fractions of lengt h-2 repetition VNs (which are th e highest rate VNs av ailable if all the node in the T anner graph h a ve mini mum distance at least 2) so th at usually the ove rall GL DPC code remains of low rate. A second drawback is that GLDPC codes with large fractions of strong generalized CNs and large fractions of lengt h-2 repetit ion VN are typically characterized by a p oor asymptotic threshold due to t he lar ge area g ap b etween t he EXIT curves in t he EXIT function (see t he Area Theorem in [18]). Allowing the generalization of the VND t ogether with t he generalization of the CND provides an increased flexibility in the code design, that can be exploited to ov ercome the above ment ioned DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 19 limitatio ns. In particul ar , the rate loss due to the generalized CNs can be com pensated using generalized VNs wi th a code rate larger than 1/2. In t his cont ext, a special class of generalized VNs is represented by ( n, n − 1) SPC VNs each one having n edges towards th e CND and associated with n − 1 encoded bit s. It is shown in [19] that t hese codes can be effecti vely exploited for the desi gn o f D-GLDPC codes with good waterf all and error floor performance. In this appendix, we develop th e polynom ial P i ( · ) defined in Theorem 4 for such VNs when represented i n both systematic and cyclic form . W e als o propose a num erical example ill ustrating the capabilities offere d by D-GLDPC codes with SPC VNs. A. SPC V ariable Nodes in Syst ematic F orm Let us suppose that the VNs o f t ype- i are length - n i SPC codes in systematic form, i.e., represented by the (( n i − 1) × n i ) generator matrix G i =           1 0 0 . . . 0 1 0 1 0 . . . 0 1 0 0 1 . . . 0 1 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 1 1           . Each of these VNs has  n i 2  weight-2 code words. Specifically , there are n i − 1 w eight-2 code words generated by weigh t-1 in formation words,  n i − 1 2  = ( n i − 1)( n i − 2) 2 weight-2 codewords generated by weight-2 information words and no weight-2 codewords generated by informati on words of weight larger than 2. Then A ( i ) 2 ,u =          n i − 1 if u = 1 ( n i − 1) ( n i − 2) / 2 if u = 2 0 if u = 3 , . . . , n i − 1 so that P i ( x ) = 2 n i · ( n i − 1) x + 2 n i · ( n i − 1)( n i − 2) 2 x 2 = 2( n i − 1) n i x  1 + n i − 2 2 x  . (30) 20 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y B. SPC V ariable Nodes in Cyclic F orm Let the VNs of type- i be ( n i , n i − 1) SPC codes in cyclic form , i.e. generated by G i =           1 1 0 . . . 0 0 0 1 1 . . . 0 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 1 1           . In this case we obtain an expression of P i ( x ) dif ferent from (30). In fa ct, it is readily shown that in a SPC code represented in cyclic form, an inform ation word o f weight u generates a weight-2 code word if and only i f all its ‘1’ positi ons are contig uous. Then, for all u = 1 , . . . , n i − 1 we hav e A ( i ) 2 ,u = n i − u , from which P i ( x ) = n i − 1 X u =1 2 ( n i − u ) n i x u = 2 n i − 1 X u =1 x u − 2 n i n i − 1 X u =1 u x u = 2 x x n i − 1 − 1 x − 1 − 2 x n i · 1 − n i x n i − 1 + ( n i − 1) x n i ( x − 1) 2 = 2 x [ x n i − n i ( x − 1) − 1] n i ( x − 1) 2 . (31) If n i = 2 or n i = 3 , then (30) coincid es with (31) as expected. Specifically , from both (30 ) and (31) we obtain P i ( x ) = x and P i ( x ) = 2 3 x 2 + 4 3 x for n i = 2 and n i = 3 , respectively . C. Compari son between Systematic and Cyclic F orm Let us denote by P s ( · ) and by P c ( · ) the poly nomial P i ( · ) of a length- n SPC VN in sy stematic and cyclic form , respecti vely . W e show next that if n > 3 P s ( x ) − P c ( x )          > 0 if 0 < x < 1 = 0 if x = 1 < 0 if x > 1 . DOUBL Y -GENERALIZED L DPC CODE S: ST ABILITY BOUND O VER THE BEC 21 In fact, we hav e P s ( x ) − P c ( x ) = 2 n  ( n − 1) x + ( n − 1)( n − 2) 2 x 2  − 2 n n − 1 X u =1 ( n − u ) x u = 2 x 2 n " ( n − 2)( n − 3) 2 − n − 1 X u =3 ( n − u ) x u − 2 # . (32) It is readily shown that P n − 1 u =3 ( n − u ) = ( n − 2)( n − 3) 2 . Then, P s (1) − P c (1) = 0 , a result which is consistent wit h Property 4. For 0 < x < 1 we must have P n − 1 u =3 ( n − u ) x u − 2 < ( n − 2)( n − 3) 2 which leads to P s ( x ) − P c ( x ) > 0 ; analogo usly , for x > 1 we m ust have P n − 1 u =3 ( n − u ) x u − 2 > ( n − 2)( n − 3) 2 which leads to P s ( x ) − P c ( x ) < 0 . D. D-GLDPC Codes with Length-2 Repetiti on VNs and SPC VNs in Syst ematic F orm Let us consider (28 ). Alth ough in g eneral it is not poss ible to express P − 1 ( · ) in an explicit closed form, this is p ossible i n special cases. For instance, obtaining a clos ed form expression of P − 1 ( · ) is poss ible when the on ly d min = 2 va riable component codes are length-2 repetit ion codes and lengt h- n SPC codes i n systematic form. Let λ be the fraction of edges connected to the length-2 repetition VNs and µ the fraction of edges connected to the lengt h- n SPC VNs (so λ + µ is the to tal fraction of edges connected t o d min = 2 VNs). W e have P ( x ) = λ x + µ 2( n − 1) n x  n − 2 2 x + 1  . By solving for posit iv e y the equation P ( y ) = x , we obt ain P − 1 ( x ) = − [ n λ + 2 ( n − 1) µ ] 2 ( n − 2) ( n − 1) µ + q [ n λ + 2 ( n − 1) µ ] 2 + 4 ( n − 2) ( n − 1 ) n µ x 2 ( n − 2) ( n − 1) µ . (33) In Fig. 2, (33 ) is plotted for different values of µ , assuming λ + µ = 0 . 3 and SPC VNs of length n = 7 . Each curve is associated wi th a di f ferent value of µ , i .e., with a dif ferent proportio n between length-2 repetition VNs and lengt h-7 SPC VNs in the VN D. Hence, the curve labelled as 0 . 0 corresponds to the presence o f only length-2 repetition VNs, while the curve labelled as 0 . 3 to the p resence of only SPC VNs. Hence modifying µ provides a wide variety of opt ions. 22 SUBMITTED TO IEEE TRANS. INFORM. THEOR Y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.25 0.3 0.0 0.05 0.1 0.15 P S f r a g r e p l a c e m e n t s x P − 1 ( x ) Fig. 2. Plot of P − 1 ( · ) for a D-GLDPC code where the only d min = 2 VNs are length-2 repetition and length-7 SPC VNs. The total fraction of edges connected to d min = 2 VNs i s λ + µ = 0 . 3 , and each curve is associated with a specific value of µ . E. Distribution Optimi zation W e consider t he opti mization prob lem of a GLDPC and of a D-GLDPC code ensemb le for design rate R = 1 / 2 . In both cases we constrain the optimi zation process by all owing the repetition VN degree t o range o nly between 2 and 15 and the SPC CN degree on ly between 5 and 15. M oreover , we use (31 , 21) BCH CNs, imp osing a mi nimum fraction of edges connected to t he BCH CNs equal to 0 .7. For the D-GLDPC code ensemble, we allow also length-15 SPC CNs in cyclic form. The output of an opti mization process over the BEC performed with diffe rential e volution [20], [21] is reported in T able I (from an edge perspective). For each of the two optimi zed distributions the threshold and the stability bo und (28) are shown. While for the GLDPC code ensemble it is necessary to use only lengt h-2 repetit ion VNs to com pensate the rate los s introduced by the large fraction of BCH CNs wi th an overall poor threshold, for the D-GLDPC code ensemble the use of SPC VNs allows obtaini ng a mu ch lar ger threshold . 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