FOUNTAIN codes [3], [4] provide an efficient solution

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1 Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design Francisco Lázaro, Stdent Member, IEEE, Gianligi Liva, Senior Member, IEEE, Gerhard Bach, Fellow, IEEE arxiv: v1 [cs.it 19 Jn 217 Abstract In this paper we analyze LT and Raptor codes nder inactivation decoding. A first order analysis is introdced, which provides the expected nmber of inactivations for an LT code, as a fnction of the otpt distribtion, the nmber of inpt symbols and the decoding overhead. The analysis is then extended to the calclation of the distribtion of the nmber of inactivations. In both cases, random inactivation is assmed. The developed analytical tools are then exploited to design LT and Raptor codes, enabling a tight control on the decoding complexity vs. failre probability trade-off. The accracy of the approach is confirmed by nmerical simlations. Index Terms Fontain codes, LT codes, Raptor codes, erasre correction, maximm likelihood decoding, inactivation decoding. I. INTRODUCTION FOUNTAIN codes [3, [4 provide an efficient soltion for data delivery to large ser poplations over broadcast channels. The reslt is attained by encoding the data object (e.g., a file) throgh an (n, k) linear code, where the nmber of otpt (i.e., encoded) symbols n can grow indefinitely, enabling a simple rate adaptation to the channel conditions. De to this, fontain codes are often regarded as rateless codes. Fontain codes were originally conceived for transmission over erasre channels. In practice, different sers experience a different channel qality reslting in a different erasre probability. When an efficient fontain code is employed, after a ser has received m = k +δ otpt symbols, with δ small, the original inpt symbols can be recovered with high probability. Lby transform (LT) codes were introdced in [5 as a first example of practical fontain codes. In [5 an iterative erasre decoding algorithm was introdced that performs remarkably Francisco Lázaro and Gianligi Liva are with the Institte of Commnications and Navigation of the German Aerospace Center (DLR), Menchner Strasse 2, Wessling, Germany. {Francisco.LazaroBlasco, Gianligi.Liva}@dlr.de. Gerhard Bach is with the Institte for Telecommnication, Hambrg University of Technology, Hambrg, Germany. Bach@thh.de. Corresponding Address: Francisco Lázaro, KN-SAN, DLR, Menchner Strasse 2, Wessling, Germany. Tel: , Fax: , Francisco.LazaroBlasco@dlr.de. This work has been presented in part at the 54th Annal Allerton Conference on Commnication, Control, and Compting, Monticello, Illinois, USA, September 215 [1, and at the 214 IEEE Information Theory Workshop, Hobart, Tasmania, November 214 [2. This work has been accepted for pblication in IEEE Transactions on Commnications, DOI: 1.119/TCOMM c 217 IEEE. Personal se of this material is permitted. Permission from IEEE mst be obtained for all other ses, in any crrent or ftre media, inclding reprinting /repblishing this material for advertising or promotional prposes, creating new collective works, for resale or redistribtion to servers or lists, or rese of any copyrighted component of this work in other works well when the nmber of inpt symbols k is large. Raptor codes [6 [8 address some of the shortcomings of LT codes. A Raptor code is a serial concatenation of an (h, k) oter linear block code with an inner LT code. Most of the literatre on LT and Raptor codes considers iterative decoding (see e.g. [9 [15). In [9, [14, [16 LT codes nder iterative decoding were analyzed sing a dynamic programming approach. This analysis models the iterative decoder as a finite state machine and it can be sed to derive the probability of decoding failre of LT codes nder iterative decoding. Iterative decoding is particlarly effective for large inpt block sizes, with k at least in the order of several tens of thosands symbols [8. However, in practice, moderate and small vales of k are often sed, de to memory limitations at the receiver side, or de to the fact that the piece of data to be transmitted is small. For example, the recommended vale of k for the Raptor codes standardized in [17 is between 124 and 8192 symbols. In this regime, iterative decoding of LT and Raptor codes trns to be largely sb-optimm. In actal implementations, different decoding algorithms are sed. In particlar, an efficient maximm likelihood (ML) decoding algorithm sally referred to as inactivation decoding [18, [19 is freqently employed. Inactivation decoding belongs to a large class of Gassian elimination algorithms tailored to the soltion of large sparse linear systems [2 [25. The algorithm can be seen as an extension of iterative decoding, where whenever the iterative decoding process stops, a variable (inpt symbol) is declared as inactive (i.e., removed from the eqation system) and iterative decoding is resmed. At the end of the process, the inactive variables have to be solved sing Gassian elimination. If a niqe soltion is fond, it is possible to recover all the remaining inpt symbols by back-sbstittion (i.e., sing iterative decoding). A few works addressed the performance of LT and Raptor codes nder inactivation decoding. The decoding failre probability of several types of LT codes was thoroghly analyzed via tight lower and pper bonds in [26 [29. In [3 the distance spectrm of fixed-rate Raptor codes is characterized, which enables the evalation of their performance nder ML erasre decoding (e.g., via pper nion bonds [31, [32). Upper bonds on the failre probability for binary and non-binary Raptor codes are introdced in [33. However, none of these works addressed inactivation decoding from a complexity viewpoint. In fact, the complexity of inactivation decoding grows with the third power of the nmber of inactivations [24. It is hence of large practical interest to develop a code design which leads (on average) to

2 2 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS a small nmber of inactivations. In [25, inactivation decoding of low-density parity-check (LDPC) codes was analyzed, providing a reliable estimate of the expected nmber of inactivations. The approach of [25 relies on bridging the inactivation decoder with the Maxwell decoder of [34. By establishing an eqivalence between inactive and gessed variables, it was shown how to exploit the Area Theorem [35 to derive the average nmber of inactivations. Thogh, the approach of [25 cannot be extended to LT and Raptor codes de to their rateless natre. In [36 the athors proposed a new degree distribtion design criterion to design the LT component of Raptor codes assming inactivation decoding. However, when this criterion is employed there is not a direct control on the average otpt degree. In [37, the athors present a finite length analysis of batched sparse codes nder inactivation decoding that provides the expected nmber of inactivations needed for decoding. In this paper we provide a thorogh analysis of LT and Raptor codes nder inactivation decoding. A first order analysis is introdced, which allows estimating the expected nmber of inactivations for an LT code, as a fnction of the otpt distribtion, the nmber of inpt symbols and the decoding overhead. The analysis is then extended to the calclation of the distribtion of the nmber of inactivations. In both cases, random inactivation is assmed. The developed analytical tools are then exploited to design LT and Raptor codes, enabling a tight control on the decoding complexity vs. failre probability trade-off. The work presented in this paper is an extension of the work in [1. In particlar, the different proofs in [1 have been modified and extended. Frthermore, in this work we not only consider LT codes bt also a class of Raptor Codes. An analysis based on a Poisson assmption approximation is presented in the Appendix 1. The rest of the paper is strctred as follows. In Section II we present the system model considered in this paper. Section III contains a detailed description of inactivation decoding applied to LT codes. In Section IV we focs on the analysis of LT codes nder inactivation decoding. Section V shows how to exploit the analysis of the LT component of a Raptor code to estimate the nmber of inactivations for the Raptor code. A Raptor code design methodology is presented too. The conclsions are presented in Section VI. II. PRELIMINARIES Let v = (v 1, v 2,..., v k ) be the k inpt symbols, ot of which the LT encoder generates the otpt symbols c = (c 1, c 2,..., c n ), where n can grown indefinitely. Each otpt symbol is generated by first sampling a degree d from an otpt degree distribtion Ω = (Ω 1, Ω 2, Ω 3,... Ω dmax ), where d max is the maximm otpt degree. In the following, we 1 In the Appendix we provide a simplified analysis, which, thogh related to the one in [2, is flly developed here on the basis of the analysis provided in Section IV of this paper. The simplified analysis is based on a Poisson approximation, which is analyzed in depth in the Appendix, discssing its limitations and comparing the analytical reslts with Monte Carlo simlations. m Fig. 1. Strctre of the matrix G T. will make se of a polynomial representation of the degree distribtion in the form Ω(x) := Ω d x d where x is a dmmy variable. Then, d distinct inpt symbols are selected niformly at random and their x-or is compted to generate the otpt symbol. The relation between otpt and inpt symbols can be expressed as d k c = vg where G is the generator matrix of the LT code. The otpt symbols are transmitted over a binary erasre channel (BEC) at the otpt of which the receiver collects m = k + δ otpt symbols, where δ is referred to as absolte receiver overhead. Similarly, the relative receiver overhead is defined as ɛ := δ/k. Denoting by y = (y 1, y 2,..., y m ) the m received otpt symbols and by I = {i 1, i 2,..., i m } the set of indices corresponding to the m non-erased symbols, we have y j = c ij. This allows s to express the dependence of the received otpt symbols on the inpt symbols as G T vt = y T (1) with G given by the m colmns of G with indices in I. ML erasre decoding reqires the soltion of (1). Note that decoding is sccessfl (i.e., the niqe soltion can be fond) if and only if rank [G = k. III. INACTIVATION DECODING OF LT CODES Inactivation decoding is an efficient ML erasre decoding algorithm, which can be sed to solve the system of eqations in (1). We will illstrate inactivation decoding by means of an example and with the aid of Figres 1 and 2. In the example, we fix k = 5 and m = 6. The strctre of G T is given in Figre 1 (in the figre, dark sqares represent the non-zero elements of G T ). Inactivation decoding consists of 4 steps.

3 FRANCISCO LÁZARO ET AL.: INACTIVATION DECODING OF LT AND RAPTOR CODES: ANALYSIS AND CODE DESIGN 3 k α α m k + α k α (a) Trianglation process. k α α (b) Zero matrix procedre. k α α (c) Gassian elimination. k α α (d) Back-sbstittion. Fig. 2. Strctre of G T as inactivation decoding proceeds. Step 1 (Trianglation). Initially, G T is pt in an approximate lower trianglar form by means of colmn and row permtations only. Given that no operation is performed on the rows or colmns of G T, the density of GT is preserved. At the end of trianglation process, G T can be partitioned in for sbmatrices as [ A D B C and as depicted in Figre 2a. In the pper left part we have the (k α) (k α) lower trianglar matrix A. The matrix nder sbmatrix A is denoted by B and it has dimension (m k + α) (k α). The pper right part is given by the (k α) α sbmatrix D, while the lower right sbmatrix is denoted by C and it has dimension (m k + α) α. The α rightmost colmns of G T (corresponding to matrices C and D) are sally referred to as inactive colmns. Note that the maniplation of G T is typically performed throgh inactivation or pivoting algorithms (see e.g. [24, [38), which aim at redcing the nmber of inactive colmns. In the rest of this work, we will assme that the se of the random inactivation algorithm [38, as it will be detailed later in the section. Step 2 (Zero matrix procedre). Matrix A is pt in a diagonal form and matrix B is zeroed ot by means of row sms. As a reslt, on average the density of the matrices C and D increases (Figre 2b). Step 3 (Gassian elimination). Gassian elimination (GE) is applied to solve the system of eqations associated with C. Being C in general dense, the complexity of this step is cbic in the nmber of inactive colmns α. As observed in [24, this step drives, for large eqation systems, the complexity of inactivation decoding. After performing GE, matrix G T has the strctre shown in Figre 2c. Step 4 (Back sbstittion). If Step 3 scceeds, after determining the vale of the inactive variables (i.e., of the inpt symbols associated with the inactive colmns), backsbstittion is applied to compte the vales of the remaining variables in v. This is eqivalent to setting to zero all elements of matrix D in Figre 2c. At the end G T shows a diagonal strctre as shown in Figre 2d, and all inpt symbols are recovered. Recalling that a niqe soltion to the system of eqations exists if and only if G is fll rank, we have that decoding scceeds if and only if the rank of C at Step 3 eqals the nmber of inactive variables. Given that the nmber of inactive variables is determined by the trianglation step, and that the GE on C dominates the decoding complexity for large k, a first analysis of inactivation decoding can be addressed by focsing on the trianglation procedre. A. Bipartite Graph Representation of Inactivation Decoding We introdce next a bipartite graph representation of the LT code from the receiver perspective. The graph comprises m + k nodes, divided in two sets. The first set consists of k inpt symbol nodes, one per inpt symbol, while the second set consists of m otpt symbol nodes, one per received otpt symbol. We denote the set of inpt symbol nodes as V, and the inpt symbol nodes in V as v 1, v 2,..., v k. Similarly, we denote the set of otpt symbol nodes as Y with otpt symbol nodes denoted by y 1, y 2,..., y m. Here, we prposely referred to inpt and otpt symbol nodes with the same notation sed for their respective inpt and otpt symbols to emphasize the correspondence between the two sets of nodes and the sets of inpt and received otpt symbols. For simplicity, in the following we will refer to inpt (otpt) symbol nodes and to inpt (otpt) symbols interchangeably. An inpt symbol node v i is connected by an edge to an otpt symbol node y j if and only if the corresponding inpt symbol contribtes to the generation of the corresponding otpt symbol, i.e., if and only if the (i, j) element of G is eqal to 1. We denote by deg(v) (or deg(y)) the degree of a node v (or y), i.e., the nmber of its adjacent edges. Trianglation can be modelled as an iterative prning of the bipartite graph of the LT code. At each iteration, a redced graph is obtained, which corresponds to a sb-graph of the original LT code graph. The sb-graph involves only a sbset of the original inpt symbols, and their neighboring otpt symbols. The inpt symbols in the redced graph will be referred to as active inpt symbols. We shall se the term redced degree of a node to refer to the degree of a node in the redced graph. Obviosly, the redced degree of a node is less than or eqal to its original degree. We denote by red(v)

4 4 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS (or red(y)) the redced degree of a node v (or y). Let s now introdce some additional definitions that will be sed to model the trianglation step. Definition 1 (Ripple). We define the ripple as the set of otpt symbols of redced degree 1 and we denote it by R. The cardinality of the ripple is denoted by r and the corresponding random variable as R. Definition 2 (Clod). We define the clod as the set of otpt symbols of redced degree d 2 and we denote it by C. The cardinality of the clod is denoted by c and the corresponding random variable as C. Initially, all inpt symbols are marked as active, i.e., the redced sb-graph coincides with the original graph. At every step of the process, one active inpt symbol is marked as either resolvable or inactive and leaves the graph. After k steps no active symbols are present and trianglation ends. In order to keep track of the steps of the trianglation procedre, the temporal dimension will be added throgh the sbscript. This sbscript corresponds to the nmber of active inpt symbols in the graph. Given the fact that the nmber of active inpt symbols decreases by 1 at each step, trianglation will start with = k active inpt symbols and it will end after k steps with =. Therefore, the sbscript decreases as the trianglation procedre progresses. Trianglation with random inactivations works as follows. Consider the transition from to 1 active inpt symbols. Then, If the ripple R is not empty (r > ), the decoder selects an otpt symbol y R niformly at random. The only neighbor of y is marked as resolvable and leaves the redced graph, while all edges attached to it are removed. If the ripple R is empty (r = ), one of the active inpt symbols v is chosen niformly at random 2 and it is marked as inactive, leaving the redced graph. All edges attached to v are removed. The inpt symbols marked as inactive are recovered sing Gassian elimination (step 3). After recovering the inactive inpt symbols, the remaining inpt symbols, those marked as resolvable, can be recovered by iterative decoding (back sbstittion in step 4). Example 1. We provide an example for an LT code with k = 4 inpt symbols and m = 4 otpt symbols (Figre 3). i. Transition from = 4 to = 3. Initially, there are two otpt symbols in the ripple (r 4 = 2) (Figre 3a). Ths, one of the inpt symbols in the ripple is randomly selected and marked as resolvable. In this case symbol v 1 is selected and all its adjacent edges are removed. The graph obtained after the transition is shown in Figre 3b. Observe that the nodes y 1 and y 4 have left the graph since their redced degree is now zero. ii. Transition from = 3 to = 2. As shown in Figre 3b the ripple is now empty (r 3 = ). Ths, an inactivation 2 This is certainly neither the only possible inactivation strategy nor the one leading to the least nmber or inactivations. However, this strategy makes the analysis tractable. For an overview of the different inactivation strategies we refer the reader to [38. has to take place. Node v 2 is chosen (according to a random selection) and is marked as inactive. All edges attached to v 2 are removed from the graph. As a conseqence, the nodes y 2 and y 3 that were in the clod C 3 enter the ripple R 2 (i.e., their redced degree is now 1), as shown in Figre 3c. iii. Transition from = 2 to = 1. Now the ripple is not empty (r 2 = 2). The inpt symbol v 3 is selected (again, according to a random choice) from the ripple and is marked as resolvable. All its adjacent edges are removed. The nodes y 2 and y 3 leave the graph becase their redced degree becomes zero (see Figre 3d). iv. Transition from = 1 to =. The ripple is now empty (Figre 3d). Hence, an inactivation takes place: node v 4 is marked as inactive and the trianglation procedre ends. Note that in this example inpt symbol v 4 has no neighbors, i.e., the matrix G is not fll rank and decoding fails. IV. ANALYSIS UNDER RANDOM INACTIVATION In this section we analyze the trianglation procedre with the objective of determining the average nmber of inactivations, i.e., the expected nmber of inactive inpt symbols at the end of the trianglation process. We will then extend the analysis to obtain the probability distribtion of the nmber of inactivations. A. Average Nmber of Inactivations Following [9, [14, [16, we model the decoder as a finite state machine with state at time given by the clod and the ripple sizes at time, i.e. S := (C, R ). We aim at deriving a recrsion for the decoder state, that is, to obtain Pr{S 1 = (c 1, r 1 )} as a fnction of Pr{S = (c, r )}. We do so by analyzing how the ripple and clod change in the transition from to 1. In the transition exactly one active inpt symbol is marked as either resolvable or inactive and all its adjacent edges are removed. Whenever edges are removed from the graph, the redced degree of one or more otpt symbols decreases. Conseqently, some of symbols in the clod may enter the ripple and some of the symbols in the ripple may become of redced degree zero and leave the graph. We focs first on the symbols that leave the clod and enter the ripple dring the transition at step, conditioned on S = (c, r ). Since for an LT code the neighbors of the otpt symbols are selected independently and niformly at random, in the transition each otpt symbol may leave the clod and enter the ripple independently from the other otpt symbols. Hence, the nmber of symbols leaving C and entering R 1 is binomially distribted with parameters c and P with P : = Pr{Y R 1 Y C } = Pr{Y R 1, Y C } Pr{Y C } (2)

5 FRANCISCO LÁZARO ET AL.: INACTIVATION DECODING OF LT AND RAPTOR CODES: ANALYSIS AND CODE DESIGN 5 active symbols active symbols active symbols v 1 v 2 v 3 v 4 r v 2 v 3 v 4 r i v 3 v 4 y 1 y 2 y 3 y 4 y 2 y 3 y 2 y 3 ripple, R 4 clod, C 4 ripple, R 3 = clod, C 3 ripple, R 2 clod, C 2 = (a) = 4. (b) = 3. (c) = 2. active symbols r i r v 4 r i r i ripple, R 1 = clod, C 1 = ripple, R = clod, C = (d) = 1. (e) =. Fig. 3. Trianglation procedre example where random variable Y represents a randomly chosen otpt symbol. We first consider the nmerator of (2). Conditioning on the original degree of Y, we have the following proposition. Proposition 1. The probability that an otpt symbol Y belongs to the clod at step and enters the ripple at step 1, condition to its original degree being d, is Pr{Y R 1, Y C deg(y) = d} = ( ) ( ) 1 k k ( 1) if 2 d k +2, d 2 d (3) otherwise. Proof: For an otpt symbol Y of degree d to belong to the clod at step and to the ripple at step 1 we need that the otpt symbol has redced degree 2 before the transition and redced degree 1 after the transition. For this to happen two events mst take place: Y R 1 one of the d edges of otpt symbol Y is connected to the symbol being marked as inactive or resolvable at the transition, Y C another edge is connected to one of the 1 active symbols after the transition and at the same time the remaining d 2 edges connected to the k not active inpt symbols (inactive or resolvable). The joint probability of Y R 1 and Y C can be derived as Pr{Y R 1,Y C } = Pr{Y R 1 } Pr{Y C Y R 1 } Let s focs first on Y R 1. We consider an otpt symbols of degree d, ths this is simply probability that one the d edges of Y is connected to the symbol being marked as inactive or resolvable at the transition, Pr{Y R 1 } = d/k. Let s now focs on Pr{Y C Y R 1 }. Since we condition on Y R 1 we have that one of the d edges of Y is already assigned to one inpt symbol. We now need to consider the remaining d 1 edges and k 1 inpt symbols. The probability that one of the d 1 edges is connected to the set of 1 active symbols is (d 1)( 1)/(k 1). At the same time we mst have exactly d 2 edges going to the k inpt symbols that were not active before the transition. Hence, we have Pr{Y C Y R 1 } = (d 1) 1 ( ) ( k k 2 k 1 d 2 d 2 By mltiplying the two probabilities, applying few maniplations, and making explicit the conditioning on deg(y) = d, we get ) 1 ( ) ( ) 1 k k Pr{Y R 1, Y C deg(y) = d} = ( 1). d 2 d We shall anyhow observe that if deg(y) < 2, the otpt symbol cannot belong to the clod. Moreover, one needs to impose the condition d 2 k, since otpt symbols choose their neighbors withot replacement (an otpt symbol cannot have more than one edge going to an inpt symbol). Hence, for d < 2 and d > k + 2, the probability Pr{Y R 1, Y C deg(y) = d} is zero. By removing the conditioning on the degree of Y in (3), we

6 6 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS have Pr{Y R 1, Y C } = ( 1) k +2 d=2 Ω d ( k d 2 ) ( ) 1 k. d Let s now focs on the denominator of (2) which gives the probability that a randomly chosen otpt symbol Y is in the clod when inpt symbols are still active. This probability is provided by the following proposition. Proposition 2. The probability that the randomly chosen otpt symbol Y is in the clod when inpt symbols are still active is Pr{Y C } = 1 k +1 d=1 k d=1 Ω d ( k d 1 Ω d ( k d ) ( ) 1 k + d ) ( ) 1 k. (4) d Proof: The probability of Y not being in the clod is given by the probability of Y having redced degree or being in the ripple. Given that the two events are mtally exclsive, we can compte sch probability as the sm of the probabilities of the two events, Pr{Y C } = 1 Pr{Y R red (Y) = } = 1 Pr{Y R } Pr{red (Y) = } (5) where red (Y) denotes the redced degree of otpt symbol Y when inpt symbols are still active. We first focs on the probability of Y being in the ripple. Let s assme Y has original degree d. The probability that Y has redced degree 1 eqals the probability of Y having exactly one neighbor among the active inpt symbols and the remaining d 1 neighbors among the k non-active (i.e., solved or inactive) ones. Ths, we have Pr{Y R deg(y) = d} = d k ( k ) d 1 ( k 1 d 1 ( ) ( ) 1 k k ) =. (6) d 1 d The probability of Y having redced degree is the probability that all d neighbors of Y are in the k non-active symbols. The total nmber of different edge assignments is ( k d), and the total nmber of edge assignments in which all d neighbors of Y are in the k non-active symbols is ( k ) d. Ths we have ( ) ( ) 1 k k Pr{red (Y) = deg(y) = d} =. (7) d d By removing the conditioning on d in (6) and (7) and by replacing the corresponding reslts in (5) we obtain (4). The expression of P proposition. is finally given in the following Proposition 3. The probability P that a randomly chosen otpt symbol leaves the clod C enters the ripple R 1 after the transition from to 1 active inpt symbols is d=1 k +2 ( ( 1) Ω k ( k d d 2) d d=2 d=1 ) 1 P =. k +1 ( 1 Ω k ( k ) 1 k ( d d 1) d Ω k ) ( k ) 1 d d d Proof: The proof follows directly from (2) and Propositions 1 and 2. Let s now focs on the nmber a of symbols leaving the ripple dring the transition from to 1 active symbols. We denote by A the random variable associated with a. Two cases shall be considered. In a first case, no inactivation takes place becase the ripple is not empty. Ths, an otpt symbol Y is chosen at random from the ripple and its only neighbor v is marked as resolvable and removed from the graph. By removing v from the graph, other otpt symbols that are connected to v and that are in the ripple leave the ripple. Ths, for r > we have Pr{A = a R = r } = ( ) ( ) a 1 ( r ) r a (8) a 1 with 1 a r. In the second case, the ripple is empty (r = ) and an inactivation takes place. Since no otpt symbols can leave the ripple, we have { 1 if a = Pr{A = a R = } = (9) if a >. We are now in the position to derive the transition probability Pr{S 1 = (c 1, r 1 ) S = (c, r )}. Let s introdce the clod drift b to denote the variation of nmber of clod elements in the transition from to 1 active symbols, i.e., b := c c 1. We can now express the clod and ripple cardinality after the transition respectively as c 1 = c b r 1 = r a + b. Since otpt symbols are generated independently, the random variable associated to b is binomially distribted with parameters c and P, and making se of (8), we obtain the following recrsion Pr{S 1 = (c b, r a + b ) S = (c, r )} ( ) c = P b (1 P ) c b b ( ) ( ) a 1 ( r ) r a (1) a 1

7 FRANCISCO LÁZARO ET AL.: INACTIVATION DECODING OF LT AND RAPTOR CODES: ANALYSIS AND CODE DESIGN 7 which is valid for r >, while for r =, according to (9), we have Pr{S 1 = (c b, b ) S = (c, )} ( ) c b = P (1 P ) c b. (11) b Note that the probability of S 1 = (c 1, r 1 ) can be compted recrsively via (1), (11) by initializing the decoder state as ( ) m Pr{S k = (c k, r k )} = Ω r k 1 (1 Ω 1) c k for all non-negative c k, r k sch that c k + r k = m, where m is the nmber of otpt symbols. The following proposition establishes how the nmber of inactivations can be determined sing the finite state machine. Theorem 1. Let T denote the random variable associated to the nmber of inactivations at the end of the trianglation process. The expected vale of T is given by E [T = k =1 r k c Pr{S = (c, )}. (12) Proof: Let s denote by f = (f 1, f 2,, f k ) the binary vector associated to the inactivations performed dring the trianglation process, and let F = (F 1, F 2,, F k ) denote the associated random vector. In particlar, the -th element of f, f is set to 1 if an inactivation is performed when inpt symbols are active, and it is set to if no inactivation is performed. Ths, for a given instance of inactivation decoding, the total nmber of inactivations corresponds simply to the Hamming weight of f, which we denote as w H (f). The expected nmber of inactivations can be obtained as E [T = w H (f) Pr {F = f} f ( ) = f Pr {F = f} = f Pr {F = f} f where the smmation is taken over all possible vectors f. We shall now define f \ = (f 1,, f 1, f +1, f k ), i.e., f \ denotes a vector containing all bt the -th element of f. We have E [T = f Pr {F = f} f \ f = f Pr {F = f} f f \ = f Pr {F = f } f = Pr {F = 1}. If we now observe that by definition Pr {F = 1} = Pr{S = (c, )} c we obtain (12), and the proof is complete. f Average nmber of inactivations Eqation (12) Monte Carlo simlation ɛ Fig. 4. Average nmber of inactivations vs. relative overhead ɛ for an LT code with k = 1 and with degree distribtion Ω (1) (x). Figre 4 shows the expected nmber of inactivations for a k = 1 LT code with the otpt degree distribtion Ω (1) (x) : =.98x +.459x x x x x x 4 (13) which is the one adopted by standardized Raptor codes [17, [39. The figre shows the nmber of inactivations according to (12) and reslts obtained throgh Monte Carlo simlations. In particlar, in order to obtain the simlation reslts for each vale of relative overhead 1 decoding attempts were carried ot. A tight match between the analysis and the simlation reslts can be observed. The reslts in this section are strongly based on [16, where LT codes nder inactivation decoding were analyzed. The difference is that, when considering the trianglation step of inactivation decoding, the decoding process does not stop when the decoder is at state S = (c, ). Instead, decoding can be resmed after performing an inactivation. B. Distribtion of the Nmber of Inactivations The analysis presented in Section IV-A provides the expected nmber of inactivations at the end of the trianglation process nder random inactivation decoding. In this section we extend the analysis to obtain the probability distribtion of the nmber of inactivations. To do so, we extend the finite state machine by inclding the nmber of inactive inpt symbols in the state definition, i.e., S = (C, R, T ) where T is the random variable associated to the nmber of inactivations at step (when inpt symbols are active). We proceed by deriving a recrsion to obtain Pr{S 1 = (c 1, r 1, t 1 )} as a fnction of Pr{S = (c, r, t )}. Two cases shall be considered. When the ripple is not empty (r > ) no inactivation takes place,

8 8 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS at the transition from to 1 active symbols the nmber of inactivations is nchanged (t 1 = t ). Hence, we have Pr{S 1 ( = )(c b, r a + b, t ) S = (c, r, t )} c b = P (1 P ) c b b ( r 1 a 1 ) ( 1 ) a 1 ( 1 1 ) r a. (14) If the ripple is empty (r = ) an inactivation takes place. In this case the nmber of inactivations increases by one yielding Pr{S 1 = (c b, b, t + 1) S = (c,, t )} ( ) c b = P (1 P ) c b. (15) b The probability of S 1 = (c 1, r 1, t 1 ) can be compted recrsively via (14), (15) starting with the initial condition Pr{S k = (c k, r k, t )} = ( ) m Ω r 1 r (1 Ω 1) c k for all non-negative c k, r k sch that c k + r k = m and t k =. Finally, the distribtion of the nmber of inactivations needed to complete the decoding process is given by 3 f T (t) = Pr{S = (c, r, t)}. (16) r c In Figre 5 the distribtion of the nmber of inactivations is shown, for an LT code with degree distribtion Ω (1) (x) given in (13), inpt block size k = 3 and relative overhead ɛ =.2. The chart shows the distribtion of the nmber of inactivations obtained throgh Monte Carlo simlations and by evalating (16). In order to obtain the simlation reslts for each vale of absolte overhead 1 decoding attempts were carried ot. As before, we can observe a very tight match between the analysis and the simlation reslts. V. INACTIVATION DECODING OF RAPTOR CODES The analysis introdced in Section IV holds for LT codes. We shall see next how the proposed tools can be sccessflly applied to the design of Raptor codes whose oter code has a dense parity check matrix (see [3). We proceed by illstrating the impact of the LT component of a Raptor code on the inactivation cont. We then develop a methodology for the design of Raptor codes based on the reslts of Section IV. 3 From (16) we may obtain the cmlative distribtion F T (t). The cmlative distribtion of the nmber of inactivations has practical implications. Let s assme the fontain decoder rns on a platform with limited comptational capability. For example, the decoder may be able to handle a maximm nmber of inactive symbols (recall that the complexity of inactivation decoding is cbic in the nmber of inactivations, t). Sppose the maximm nmber of inactivations that the decoder can handle is t max. The probability of decoding failre will be lower bonded by 1 F T (t max ).The probability of decoding failre is actally higher than 1 F T (t max ) since the system of eqations to be solved in the Gassian elimination (GE) step might be rank deficient. Distribtion of inactivations Eqation (16) Monte Carlo simlation t Fig. 5. Distribtion of the nmber of inactivations for an LT code with k = 3, relative overhead ɛ =.2 and degree distribtion Ω (1) (x) given in (13). A. Average Nmber of Inactivations for Raptor Codes Let s now consider a Raptor code based on the concatenation of a (dense) (h, k) oter code and an inner LT code. We denote by H P the ((h k) h) parity-check matrix of the oter code. At the inpt of the Raptor encoder, we have a vector of k inpt symbols, = ( 1, 2,..., k ). Ot of the inpt symbols, the oter encoder generates a vector of h intermediate symbols v = (v 1, v 2,..., v h ). The intermediate symbols serve as inpt to an LT encoder which prodces the codeword c = (c 1, c 2,..., c n ), where n can grow nbonded. The relation between intermediate and otpt symbols can be expressed as c = vg where G is the generator matrix of the LT code. The otpt symbols are sent over a BEC, at the otpt of which the receiver collects m = k + δ otpt symbols, denoted as y = (y 1, y 2,..., y m ). The relation between the collected otpt symbols and the intermediate symbols can be expressed as G T vt = y T. where G is a matrix that contains the m colmns of G associated to the m received otpt symbols. De to the oter code constraints, we have H P v T = T where is the length-(h k) zero vector. Let s now define the constraint matrix M of the Raptor code M := [ H T P G. ML decoding can be carried ot by solving the system M T v T = [ y T. (17) If we compare the system of eqations that need to be solved for LT and Raptor decoding, given respectively by (1) and (17), we can observed how Raptor ML decoding is very similar to

9 FRANCISCO LÁZARO ET AL.: INACTIVATION DECODING OF LT AND RAPTOR CODES: ANALYSIS AND CODE DESIGN 9 ML decoding of an LT code. The main difference lies in the fact that matrix M is formed, for the first h k colmns, by the transpose of the oter code parity-check matrix. The high density of the oter code parity-check matrix lowers the probability that the ripple contains (some of) the h k otpt symbols associated to the zero vector in (17) (this is especially evident at the early steps of the trianglation process). As a reslt, we shall expect the average nmber of inactivations to increase with h k, for a fixed overhead 4. In Figre 6 we provide the average nmber of inactivations needed to decode two Raptor codes, as a fnction of the receiver overhead δ. Both Raptor codes have the same oter code, a (63, 57) Hamming code, bt different LT degree distribtions. The first distribtion is Ω (1) (x) from (13), and the second distribtion is Average nmber of inactivations Raptor code, Ω (1) (x) LT code, Ω (1) (x) Raptor code, Ω (2) (x) LT code, Ω (2) (x) Ω (2) (x) :=.5x +.2x 2 +.4x 3 +.3x 4 +.5x 4. The Figre also shows the nmber of inactivations needed to decode the two standalone LT codes. If we compare the nmber of inactivations reqired by the Raptor and LT codes, we can see how for both degree distribtions, the nmber of additional inactivations needed for Raptor decoding with respect to LT decoding is very similar. We hence conjectre that the impact of the (dense) oter code on the inactivation cont depends mostly on the nmber of the oter code paritycheck eqations. This empirical observation provides a hint on a practical design strategy for Raptor codes: If one aims at minimizing the nmber of inactivations for a Raptor code based on a given oter code, it is sfficient to design the LT code component for a low (i.e., minimal) nmber of inactivations. Based on this consideration, an explicit design example is provided in the following sbsection. B. Example of Raptor Code Design We consider a Raptor code with a (63, 57) oter Hamming code and we assme decoding is carried ot when the absolte receiver overhead reaches δ = δ, i.e., we carry ot decoding after collecting δ otpt symbols in excess of k. Frthermore, we want to have a probability of decoding failre, P F, lower that a given vale P F. Ths, the objective of or code design will be minimizing the nmber of inactivations needed for decoding while achieving a probability of decoding failre lower than P F. Hence, the design problem consists of finding a sitable otpt degree distribtion for the inner LT code. For illstration we will introdce a series of constraints in the otpt degree distribtion. In particlar we constraint the otpt degree distribtion to have the same maximm and average otpt degree as standard R1 Raptor codes ( Ω 4.63 and d max = 4, [17). Frthermore, we constraint the otpt degree distribtion to have the same spport as the degree distribtion of R1 raptor codes, that is, only degrees 1, 2, 3, 4, 1, 11 and 4 are allowed. These design constraints allow s to perform a fair comparison between the Raptor 4 In practice, the oter codes sed are not totally dense, bt the rows of their parity check matrix are sally denser that the rows of G T. This is, for example, sally the case if the oter code is a high rate LDPC code, since the check node degree increases with the rate Fig. 6. Expected nmber of inactivations two Raptor codes sing a (63, 57) Hamming code with LT degree distribtions Ω (1) (x) and Ω (2) (x), and for two LT codes with k = 63 and degree distribtions Ω (1) (x) and Ω (2) (x). code obtained throgh optimization and a Raptor code with the same oter code and the degree distribtion from R1 Raptor codes. The design of the LT otpt degree distribtion is formlated as a nmerical optimization problem. For the nmerical optimization we sed is simlated annealing (SA) [4. More concretely, we define the objective fnction to be minimized to be the following fnction [41 η = E[T + φ ( PF ) where E[T is the nmber of inactivations needed to decode the LT code and the penalty fnction φ is defined as φ ( { ) if PF < P F PF = b ( 1 P ) F / PF otherwise being b a large positive constant 5, P F the target probability of decoding failre at δ = δ and PF the pper bond to the probability of decoding failre of the Raptor code in [33, which for binary Raptor codes has the expression 6 P F PF := δ h l=1 A o l πk+δ l where A o is the mltiplicity of codewords of weight l in the l oter code, and π l depends on the LT code otpt degree 5 In or example, b was set to 1 4. A large b factor ensres that degree distribtions which do not comply with the target probability of decoding failre are discarded. 6 The se of the pper bond on the probability of decoding failre, PF in place of the actal vale of P F in the objective fnctions stems from the need of having a fast (thogh, approximate) performance estimation to be sed within the SA recrsion. The evalation of the actal P F presents a prohibitive complexity since it has to be obtained throgh Monte Carlo simlations. Note also that the pper bond of [33 is very tight.

10 1 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS distribtion as d max π l = j=1 min(l, j) Ω j i=max(,l+j h) i even ( j ( h j ) i) l i ( h ). l In particlar, two code designs were carried ot sing the proposed optimization. In the first case we set the overhead to δ = 15 and the target probability of decoding failre to P F = 1 3, and we denote by Ω (3) the distribtion obtained from the optimization process. In the second case we chose the same overhead, δ = 15, and set the target probability of decoding failre to P F = 1 2, and we denote by Ω (4) the reslting distribtion. The degree distribtions obtained are the following PF Ω (1) (x) (simlation) Ω (1) (x) (pper bond) Ω (3) (x) (simlation) Ω (3) (x) (pper bond) Ω (4) (x) (simlation) Ω (4) (x) (pper bond) Ω (3) (x) =.347 x x x x x x x 4 Ω (4) (x) =.823 x x x x x x x 4. Monte Carlo simlations were carried ot in order to assess the performance of the two Raptor codes obtained as reslt of the optimization process. In order to have a benchmark for comparison, a third Raptor code was considered, employing the same oter code (Hamming) and the degree distribtion of standard R1 Raptor codes given in (13). Note that in all three cases we consider the same oter code, a (63, 57) Hamming code, and ths, the nmber of inpt symbols is k = 57. To derive the probability of decoding failre for each overhead vale δ simlations were rn ntil 2 errors were collected, whereas in order to obtain the average nmber of inactivations, 1 transmissions were simlated for each overhead vale δ. Figre 7 shows the probability of decoding failre P F as a fnction of δ for the three Raptor codes based on the (63, 57) oter Hamming code and inner LT codes with degree distribtions Ω (1) (x), Ω (3) (x) and Ω (4) (x). The pper bond to the probability of failre PF is also provided. It can be observed how the Raptor codes with degree distribtions Ω (3) (x) and Ω (4) (x) meet the design goal, being their probability of decoding failre at δ = 15 below 1 3 and 1 2, respectively. It can also be observed that the probability of decoding failre of the Raptor code with degree distribtion Ω (1) (x) lies between that of Ω (3) (x) and Ω (4) (x). Figre 8 shows the average nmber of inactivations as a fnction of the absolte receiver overhead for the three Raptor codes considered. It can observed that the Raptor code reqiring the least nmber of inactivations is Ω (4), followed by Ω (1), and finally the Raptor code with degree distribtion Ω (3) is the one reqiring the most inactivations, and ths has the highest decoding complexity. The reslts in Figres 7 and 8 illstrate the tradeoff existing between probability of decoding failre and nmber of inactivations (decoding complexity): In general if one desires to improve the probability of decoding failre, it is necessary to adopt LT codes with degree distribtions that lead to a larger average nmber of inactivations δ Fig. 7. Probability of decoding failre P F vs. absolte receiver overhead δ for binary Raptor codes with a (63, 57) Hamming oter code and LT degree distribtions Ω (1) (x), Ω (3) (x) and Ω (4) (x). The markers represent the reslt of simlations, while the lines represent the pper bond to the probability of decoding failre in [33. Average nmber of inactivations δ Ω (1) (x) Ω (3) (x) Ω (4) (x) Fig. 8. Nmber of inactivations vs. absolte receiver overhead δ for binary Raptor codes with a (63, 57) Hamming oter code and LT degree distribtions Ω (1) (x), Ω (3) (x) and Ω (4) (x). VI. CONCLUSIONS In this paper the decoding complexity of LT and Raptor codes nder inactivation decoding has been analyzed. Using a dynamic programming approach, recrsions for the nmber of inactivations have been derived for LT codes as a fnction of their degree distribtion. The analysis has been extended to obtain the probability distribtion of the nmber of inactivations. Frthermore, the experimental observation is made that decoding a Raptor code with a dense oter code reslts in an increase of the nmber of inactivations, compared to decoding a standalone LT code. Based on this observation, it has been

11 FRANCISCO LÁZARO ET AL.: INACTIVATION DECODING OF LT AND RAPTOR CODES: ANALYSIS AND CODE DESIGN 11 shown how the recrsive analysis of LT codes can be sed to design Raptor codes with a fine control on the nmber of inactivations vs. decoding failre probability trade-off. APPENDIX A INDEPENDENT POISSON APPROXIMATION In Section IV we have derived recrsive methods that can compte the expected nmber of inactivations and the distribtion of the nmber of inactivations reqired by inactivation decoding. The proposed recrsive methods, albeit accrate, entail a non negligible comptational complexity. In this appendix we propose an approximate recrsive method that can provide a reasonably-accrate estimation of the nmber of inactivations with a mch lower comptational brden. The development of the approximate analysis relies on the following definition. Definition 3 (Redced degree-d set). The redced degree-d set is the set of otpt symbols of redced degree d. We denote it by Z d. The cardinality of Z d is denoted by z d and its associated random variable by Z d. Obviosly, Z 1 corresponds to the ripple. Frthermore, it is easy to see how the clod C corresponds to the nion of the sets of otpt symbols of redced degree higher than 1, i.e., C = d max d=2 Z d. Moreover, since the sets Z d are disjoint, we have d max C = Z d. d=2 We aim at approximating the evoltion of the nmber of redced degree d otpt symbols, Z d, as the trianglation procedre of inactivation decoding progresses. As it was done in Section IV, in the following a temporal dimension shall be added throgh sbscript (recall that the sbscript corresponds to the nmber of active inpt symbols in the graph). At the beginning of the trianglation process we have = k, and the conter decreases by one in each step of trianglation. Trianglation ends when =. It follows that Z d, is the set of redced degree d otpt symbols when inpt symbols are still active. Moreover, Z d, and z d, are respectively the random variable associated to the nmber of redced degree d otpt symbols when inpt symbols are still active and its realization. We model the trianglation process by means of a finite state machine with state S := ( Z 1,, Z 2,,..., Z dmax,). This model is eqivalent to the one presented in Section IV-A. However, the evalation of the state evoltion is now more complex de to the large state space. Yet, the analysis can be, greatly simplified by resorting to an approximation. Before decoding starts (for = k), de to the independence of otpt symbols we have that S k follows a mltinomial distribtion, which for large nmber of otpt symbols m can be approximated as the prodct of independent Poisson distribtions. Figre 9 shows the probability distribtion of Z 1,, Z 2, and Z 3, for an LT code with a robst soliton distribtion and k = 1 obtained throgh Monte Carlo simlation, for = 1, 5 and 2. The figre also shows the crves of the Poisson distribtions which match best the experimental data in terms of minimm mean sqare error. As we can observe, the Poisson distribtion tightly matches the experimental data not only for = k, bt also for smaller vales of. Hence, we shall approximate the distribtion of the decoder state at step as a prodct of independent Poisson distribtions, d max z λ d, d, e λ d, Pr{S = z } z d,! d=1 (18) where z is the vector defined as z = ( z 1,, z 2,,... z dmax,, ), i.e., we assme that the distribtion of redced degree d otpt symbols when inpt symbols are active follows a Poisson distribtion with parameter λ d,. We remark that by introdcing this assmption, the nmber of received otpt symbols m is no longer constant bt becomes a sm of Poisson random variables. In spite of this mismatch, we will later see how a good estimate of the nmber of inactivations can be obtained resorting to this approximation. Next, we shall explain how the parameters λ d, can be determined. For this prpose let s define B d, as the random variable associated with the nmber of otpt symbols of redced degree d that become of redced degree d 1 in the transition from to 1 otpt symbols. We have Z d, 1 = Z d, + B d+1, B d,. If we take the expectation at both sides we can write E [ Z d, 1 = E [ Zd, + E [ Bd+1, E [ Bd,. (19) Let s now derive the expression of E [ B d,. We shall distingish two cases. First we shall consider otpt symbols with redced degree d 2. Since otpt symbols select their neighbors niformly at random, we have that B d,, d 2, conditioned on Z d, = z d, is binomially distribted with parameters z d, and P d,, where P d, is the probability that the degree of Y decreases to d 1 in the transition from to 1, i.e., P d, := Pr{Y Z d 1, 1 Y Z d, }. The following proposition holds. Proposition 4. The probability that a randomly chosen otpt symbol Y, with redced degree d 2 when inpt symbols are active, has redced degree d 1 when 1 inpt symbols are active is P d, = d. Proof: Before the transition, Y has exactly d neighbors among the active inpt symbols. In the transition from to 1 active symbols, 1 inpt symbol is selected at random and marked as either resolvable or inactive. The probability that

12 12 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS Z 1 Z 2 Z 3 Poisson approx. and removed from the graph. Frthermore, any other otpt symbol in the ripple being connected to inpt symbol v also leaves the ripple dring the transition. Hence, for z 1, 1 we have [ E B 1, Z 1, = z 1, 1 = ( z1, 1 ) probability (a) = k = (b) = (c) = 2 nmber of otpt symbols Fig. 9. Distribtion of Z 1,, Z 2, and Z 3, for an LT code with robst soliton distribtion and k = 1 obtained throgh Monte Carlo simlation. The pper, middle and lower figres represent respectively the distribtion for = 1, 5 and 2. The black bars represent Z 1,, the light grey bars Z 2, and the dark grey bars Z 3,. The red lines represent the best Poisson distribtion fit in terms of minimm mean sqare error. the degree of Y gets redced is simply the probability that one of its d neighbors is marked as resolvable or inactive. Ths, the expected vale of B d, is E [ B d, = E [ E [ Bd, Z d, = d E [ Z d,. (2) If we now replace (2) in (19) a recrsive expression is obtained for λ d,, d 2, λ d, 1 = λ d, + d + 1 λ d+1, d λ d, (21) ( λ d, 1 = 1 d ) λ d, + d + 1 λ d+1, where we have replaced [ E Z d, = λd, according to or Poisson distribtion assmption. We shall now consider the otpt symbols of redced degree 1. In particlar, we are interested in B 1,, the random variable associated to the otpt symbols of redced degree d = 1 that become of redced degree in the transition from to 1 active inpt symbols. Two different cases need to be considered. In the first one, the ripple is not empty, and hence there are one or more otpt symbols of redced degree 1. In this case, an otpt symbol Y is chosen at random from the ripple and its only neighbor v is marked as resolvable whereas for z 1, = we have E [ B 1, Z 1, = =. Hence, we have [ ( E B 1, = 1 1 ) Pr{Z 1, 1} + 1 = z 1, m z 1, =1 z 1, Pr{Z 1, = z 1, } ( = 1 1 ) (1 Pr{Z1, = } ) + 1 [ E Z 1, ( = 1 1 ) ( ) 1 e λ 1, + 1 λ 1,. (22) Replacing (22) in (19) a recrsive expression is obtained as, ( λ 1, 1 = 1 1 ) λ 1, + 2 ( λ 2, 1 1 ) ( ) 1 e λ 1,. (23) The decoder state probability is obtained by setting the initial condition λ d,k = m Ω d and applying the recrsions in (21) and (23). Frthermore, the expected nmber of inactivations after the k steps of trianglation, can be approximated as E [T = k Pr{Z 1, = } =1 k e λ 1,. In Figre 1 we provide again the probability distribtion of Z 1,, Z 2, and Z 3, for an LT code with robst soliton distribtion (RSD) (see [5) and k = 1 obtained throgh Monte Carlo simlation, for = 1, 5 and 2. The figre also shows the crves of the approximation to Z i, obtained sing the model in this section. We can observe how the proposed method is able to track the distribtion of Z i, very accrately for = k and = 5. However, at the end of the trianglation process a divergence appears as it can be observed for = 2 in Figre 1 (c). The sorce of this divergence cold, in large part, be attribted to the independence assmption made in (18). As the nmber of active inpt symbols decreases, the dependence among the different Z i, becomes stronger, and the independence assmption approximation falls apart. Figre 11 shows the average nmber of inactivations needed to complete decoding for a linear random fontain code (LRFC) 7 and a RSD, both with average otpt degree Ω = 12 and k = 1. The figre shows reslts obtained by Monte Carlo simlation and also the estimation of the nmber of inactivations obtained nder or Poisson approximation. A tight match between simlation reslts and the estimation can be observed. The experimental reslts indicate that, althogh =1 7 The degree distribtion of a LRFC follows a binomial distribtion with parameters k and p = 1/2 (see [42).