Long Erasure Correcting Codes: the New Frontier for Zero Loss in Space Applications? Enrico Paolini and Marco Chiani

Transcription

1 SpaceOps 2006 Conference AIAA Long Erasure Correcting Codes: the New Frontier for Zero Loss in Space Applications? Enrico Paolini and Marco Chiani D.E.I.S., WiLAB, University of Bologna, Cesena, Italy Gian Paolo Calzolari European Space Agency, D/OPS, ESOC, Darmstadt, Germany In space communications, traditional error correction/detection techniques deliver to the upper layers of the communication stack only the data units for which integrity can be guaranteed. The uncorrectable data units are then lost, and the upper layers have typically to face data units erasures. Hence, the packet erasure channel (PEC) is the most proper channel model from the point of view of the upper layers. Automatic repeat/retransmission query (ARQ) is the traditional solution implemented at the upper layers in order to face data units erasures. However, ARQ is not always possible for space or satellite communications. In such situations, forward error correction (FEC) must be used. Nowadays new techniques for FEC are available also for application at the upper layers. Long erasure correcting (LEC) codes represent a new and very promising proposal for packet erasure FEC. They are able to overcome the complexity limitations of other types of codes, while preserving very good erasure correction capability. They are currently under investigation within the CCSDS (Consultative Committee for Space Data Systems) long erasure codes Bird of Feather (LEC-BOF), where a leading role has been so far played by ESA/ESOC and NASA-JPL. In this paper, the activity of the LEC-BOF is be illustrated. More in detail, the basic ideas behind LEC codes are presented, as well as the possible codes structures, the encoding and decoding rules, some theoretical properties. Some numerical results are presented, showing the performance of LEC codes on both memory-less and burst erasure channel. I. Introduction Space communications exploit several techniques at different layers of the communication stack in order to guarantee the correct delivery of information to the receiver. As from the Consultative Committee for Space Data Systems (CCSDS) recommendations, the space link protocols can be described according to a five layers model, as depicted in Fig 1. 1 One of the tasks of the upper layers is to verify that all the data units are correctly delivered to the receiver. Traditional error correction and detection techniques only deliver the data units for which integrity can be guaranteed; on the contrary, the uncorrectable data units are lost. For this reason, the upper layers have typically to face data units erasures, and the packet erasure channel (PEC), where whole packets of bits are either correctly received or lost, is the most proper channel model from the point of view of the upper layers. Packet losses can outcome as consequence of brief outage conditions due to weather, shadowing, or loss of frame synchronization. We explicitly remark that the term packet used in this paper has a very general meaning, and is not referred to the data units of a specific protocol within the model presented in Fig. 1. In other words, a packet may represent a transfer frame, a space packet, or any data unit properly defined by the user. The expression long erasure code (LEC) packet will be sometimes used in the paper, in order to avoid confusion. Packet erasures due to the afore mentioned causes are usually correlated, and bursts of erasures can take place. This correlation is rarely considered in the channel model, and the memory-less PEC, where epaolini@deis.unibo.it mchiani@deis.unibo.it Gian.Paolo.Calzolari@esa.int 1of12 Copyright 2006 by Authors and CCSDS. Published by the, Inc., with permission.

2 Lossless Data Compression Application Layer CCSDS File Delivery Protocol (CFDP) SCPS-FP FTP Application Specific Protocols Transport Layer SCPS-TP TCP UDP SCPS-SP Network Layer Space Packet Protocol SCPS-NP IP Version 4 IP Version 6 Data Link Layer (Data Link ProtocolSublayer) TM Space Data Link Protocol TC Space Data Link Protocol AOS Space Data Link Protocol (Sync. and Channel Coding Sublayer) TM Sync. and Channel Coding TC Sync. and Channel Coding Proximity-1 Space Link Protocol Physical Layer Figure 1. RF and Modulation Systems Space link protocols model. packets are lost independently with equal erasure probability, is usually considered. This choice can be correct if a sufficiently long packet interleaving is present in the system. The packet interleaving consists in performing a packet permutation at the transmitter, before the transmission, and the inverse permutation at the receiver. This procedure can permit to properly spread the erasure bursts on the sequence of received packets. However, when the interleaving procedure (if available) only involves packets belonging to the same codeword, and the erasure burst length is not small respect to the codeword length, the memory-less approximation cannot guarantee a confident approximation of the communication system. (As explained in Section II, in this contest a codeword is represented by a set of n encoded packets, generated by a set of k information packets.) Automatic repeat/retransmission query (ARQ) is the traditional solution implemented at the upper layers in order to face data units erasures. However, ARQ is not always possible for space or satellite communications. This typically happens in deep space missions, where the long value of the round trip delay introduces drawbacks with respect to both the time needed to request and receive the lost data unit(s) as well to the increase of the memory requirements needed to ensure a very long persistence. It also happens when a feedback channel is not available, or in the satellite broadcast, where the satellite is not able to manage several retransmission requests, or when on board memory is limited and persistency of the data cannot be guaranteed. In these cases, exploiting packet oriented forward error correction (FEC) techniques is the only possibility to face packet erasures. This upper layer FEC has no impact on the traditional FEC techniques implemented at the synchronization and channel coding sublayer of data link layer (see Fig. 1). It is also worth mentioning that, when ARQ is possible, hybrid ARQ/FEC solutions can be considered. 2 In this paper, however, only the pure FEC approach is considered. Recall that a (n, k) block linear code with minimum distance d min can successfully recover with probability 1 from any pattern of d min 1 or less erasures, and that the minimum distance always satisfies the Singleton bound d min n k + 1. When bounded distance decoding is used, if the initial number of erasures is less than d min decoding is performed, otherwise a decoding failure is declared. More powerful erasure correcting algorithms exist, but none of them can successfully recover with probability 1 from a number of erasures greater than d min 1. For Reed-Solomon codes the Singleton bound is achieved with equality (maximum distance separable codes). Hence, a Reed-Solomon code can always recover from any pattern of at most 2of12

3 n k erasures. However, the performance of Reed-Solomon codes is limited due to practical limitations to the codeword length n, which are imposed by the decoding complexity. A typical value of the codeword length n used is n = 255. This limitations to the codeword length determine other drawbacks, like the need to encode a long file by using more codewords, and the impossibility to face long erasure bursts. Nowadays new techniques for forward error correction are available also for application at the upper layers. Above all, long erasure correcting (LEC) codes represent a new and very promising proposal for packet erasure FEC. This proposal has been formulated on the basis of excellent performance that can be obtained by iteratively decoded, long codes based on sparse graphs. In general, they are able to overcome the complexity limitations of maximum distance separable codes, while preserving good or very good erasure correction capability. Their linear encoding and decoding complexity enables for long codeword lengths, thus allowing to achieve extremely good performance, outperforming the performance of maximum distance separable codes of practical use. This also permits to encode long files as an unique codeword, and to face long bursts of erasures. LEC codes are currently under investigation within the CCSDS where a long erasure codes Bird of Feather (LEC-BOF) has been created to determine the potentiality and the applicability of this kind of codes. In this CCSDS group a leading role has been played by ESA/ESOC and NASA/JPL. The aim of the LEC-BOF was set to to investigate possible benefits of these codes for future CCSDS missions, to study the proper channel model and to formulate the proper requirements. In this paper, the activity of the LEC-BOF is illustrated and the basic ideas behind LEC codes are clarified. Possible codes structures, encoding and decoding rules, some theoretical properties and technical proposals presented by members during the CCSDS Meetings are also presented. A possible algorithm for improving the performance of LEC codes on burst erasure channels is also described. II. Long Erasure Correcting Codes Suppose that q packets of bits x 1, x 2,...,x q are combined together in order to generate a new packet x q+1. Suppose that packet x q+1 is generated such that if any of the packets in the set {x 1,...,x q, x q+1 } is unknown, then it can be always reconstructed if the other q packets are known. If, as usual, x 1, x 2,...,x q have the same length L (intended as number of bits), then x q+1 can be generated by a simple bitwise XOR of x 1, x 2,...,x q, with resulting length L. In order to indicate that x q+1 is generated by x 1, x 2,...,x q,the graphical notation of Fig. 2 will be adopted. An equivalent graphical notation is depicted in Fig 3, whose meaning is that the bitwise XOR of x 1,...,x q, x q+1 is equal to the all-zero packet. Suppose that these q + 1 packets are transmitted over a PEC and that all but one of them are correctly received. In such a situation the missing packet can be always reconstructed at the receiver. If x q+1 has been generated as the bitwise XOR of x 1, x 2,...,x q, then the missing packet x i can be obtained as j i x j i.e. as the bitwise XOR of the received q packets. If more than one packet is lost, there is no chance to recover from the erasure pattern. In this example, a parity constraint exists on x 1,...,x q, x q+1. Specifically, the q + 1 bits in the same position of each packet must satisfy a parity constraint, since their XOR must be 0. For this reason, the square node in Fig. 3 is said a parity-check node (or simply a check node), and the nodes x 1, x 2,...,x q, x q+1 are said variable nodes. A parity-check node enables the correction of one erased packet. A. Low-Density Parity-Check Codes and their Iterative Decoding Consider the code structure of Fig. 4, where each packet x i (i =1,...,n) to be transmitted is involved in multiple parity constraints. The code graph is supposed to be sparse, i.e. the number of edges is supposed small with respect to the product nm(maximum possible number of edges in absence of multiple connections between two nodes), and the overall code is said a low-density parity-check (LDPC) code. 3 The sequence x 1, x 2,...,x n is said the codeword, and n is the codeword length a. The codeword is generated at the transmitter by an encoding operation, consisting in the generation of the n encoded packets x 1, x 2,...,x n from k information packets u 1, u 2,...,u k. The ratio k/n is said the code rate, denoted by R. The code graph is said a bipartite graph, since no connection are allowed between two variable nodes or two check nodes. After the transmission of the n packets on the PEC, some of the packets will be known at the decoder, while some other packets will be lost. The technique for decoding the missing packets is to iteratively applying the afore described procedure. For each check node, if it is connected to only one missing a In this contest, each encoded symbol is represented by a packet of bits. 3of12

4 x 1 x 1 x 2 x 2 x q+1 x q x q x q+1 Figure 2. Generation of x q+1 from x 1, x 2,...,x q. Figure 3. Parity constraint satisfied by x 1, x 2,...,x q, x q+1. x 1 x 2 x 3 x n e c 1 c 2 c 3 c m Figure 4. LDPC code structure. packet, this missing packet can be successfully reconstructed. As explained in the above example, the erased packet can be obtained simply as the bitwise XOR of the known packets. If the number of unknown packets connected to a check node is greater or equal than 2, no packets can be recovered by that check node. As soon as receiving the last packet connected to it, each check node can be processed for recovery of missing packet(s): In principle, for some check nodes recovery will be possible, for some other check nodes it will not be. When the last check node has been considered, a new iteration on the check nodes is started. If no packet is recovered at a certain iteration l, then it will be sure that no packet will be recovered at iteration l +1. Hence, the iterative algorithm is stopped as soon as all there are no more unknown packets, or no packet is corrected at a certain iteration and unknown packets still exist. In the former case decoding is successful, in the latter one a decoding failure is declared b. However, even in this case, some packets originally missing can be successfully reconstructed. This iterative decoding algorithm for erasure correction, is characterized by decoding complexity linearly increasing with the codeword length. 4 In fact, the decoding complexity, intended as number of XOR s performed to successfully decode the erasure pattern, is proportional to the total number of edges. Since the bipartite graph is sparse by hypothesis, the decoding complexity is proportional to the number of variable nodes. We also observe that it is possible to give an equivalent description of the same decoding algorithm as an instance of the belief propagation algorithm, 5 i.e. of the traditional iterative decoding algorithm of LDPC codes used for bit-oriented error correction. B. IRA Codes The decoding procedure for LDPC codes on the PEC is extremely efficient, even in the case where the connections between variable and check nodes are chosen at random. However, this is not the case for encoding. The encoding procedure consists in generating n encoded packets from k < ninformation packets. The complexity of encoding procedure, intended as the number of operations needed to generate the encoded packets, is in general a quadratic function of n. This quadratic complexity imposes a severe constraint to the codeword length n. Thus, the possibility to exploit long codes is bound by the capability to perform the encoding operation with low complexity. This task is achieved by carefully choosing the connections between variable and check nodes in the bipartite graph. Several promising solutions exist for an efficient encoding of LDPC codes. In the following we describe b Actually, if the code is systematic, a decoding failure is declared when at the end of the decoding algorithm at least one systematic packet of bits is still erased. 4of12

5 u 1 p 1 u 1 p n k u k }{{} u k redundant packets Figure 5. code. Structure of bipartite graph for an IRA Figure 6. Structure of a Tornado code. l 2 redundant packets l 3 redundant packets u 1... u k {}}{{}}{ c l2+l 3 c 1 c l2 c l2+1 c l2+l 3+1 }{{}}{{} l 2 check nodes l 3 check nodes Figure 7. Tornado code as an instance of LDPC code. one of these solutions, namely IRA codes. 6 IRA codes have been already considered for implementation as transport layer erasure correcting codes in satellite broadcast applications. 7 IRA encoding is systematic. This means that the first k encoded packets x 1, x 2,...,x k are equal to the information packets u 1, u 2,...,u k. The last n k encoded packets p 1 = x k+1, p 2 = x k+2,...,p n k = x n are called redundant packets. In IRA encoding, a first encoded packet p 1 is generated as the bitwise XOR of a subset of the information packets. Then, other n k 1 encoded packets are generated, according to the following rule. Packet p i, i =2,...,n k, is generated as the bitwise XOR of a subset of the information packets, and of packet p i 1, as depicted in Fig. 5. In this figure, the redundant packets have been represented on the right of the check nodes: however, the meaning of the connections between the nodes remains the same as for Fig. 3. The outcoming systematic codeword is [u 1,...,u k, p 1,...,p n k ]. The resulting complexity of IRA encoder, intended as number of XOR operations, is linear in the codeword length n. Moreover, the systematic nature of the code allows to immediately use the information packets not erased by the channel. c n k C. Tornado Codes A variation with respect to the LDPC codes structure is represented by Tornado codes. 4 The structure of a Tornado code is depicted in Fig. 6. This structure consists in a cascade of a certain number of packet layers. The k packets u 1, u 2,...,u k belonging to the first layer are the information packets, while the n k packets belonging to all other layers are the redundant packets; the overall code is then systematic. A sparse bipartite graph connects each layer to the next one, where the connections have the same meaning of Fig. 2: if a packet in layer j + 1 is connected to q packets in layer j, then it is generated as the bitwise XOR of these q packets. Therefore, the encoding algorithm for Tornado codes is very simple: packets in the second layer are generated from the information packets through bitwise XOR s, packets in the third 5of12

6 layer are generated from packets in the second layer, and so on up to the packets in the last layer. Note that the complexity of this encoding rule is linear in the codeword length, independently of the structure of the connections in the cascade. The decoding algorithm of Tornado codes exploits the same simple rule, illustrated at the beginning of this section, as LDPC codes. Consider q packets in layer j connected to one packet in layer j +1. Ifoneof the q packets in layer j is unknown, while the other q 1 packets and the packet in layer j + 1 are known, then the missing packet can be reconstructed as bitwise XOR of the known packets. This decoding rule is performed in the same recursive way as for LDPC codes, for each pair of layers in the cascade. If the cascade has S layers of packets, the iterative decoding rule is first performed at layers S and S 1, then at layers S 1andS 2, and so on up to the first layer. Hence, the direction of decoding process is opposite respect to the direction of encoding. It is worth mentioning that, if the number of packets involved in layers S 1andS is sufficiently small, then it is possible to use a more powerful algorithm at the first step of decoding, instead of the iterative decoding. For instance, maximum a posteriori (MAP) decoding can be used, combined with a linear code structure with good properties in terms of minimum distance. This can produce a little improvement in the overall code performance. The structure of a Tornado code can be seen a special instance of an LDPC code. This concept is explained in Fig. 7, where a a Tornado code with three layers of nodes (one layer of information packets and two layers of redundant packets) is represented as an LDPC code. The information nodes are connected to a number of check nodes equal to the number l 2 of redundant packets in the second layer. Each of these check nodes has an edge towards a specific redundant node in the second layer. Again, the variable nodes in the second layer are connected to a number of check nodes equal to the number l 3 of redundant nodes in the third layer, and each such check node has one connection towards a specific redundant node in the third layer. The total number of check nodes is n k. This observation suggests the interpretation of Tornado codes as special LDPC codes, where constraints are imposed on the connections between variable and check nodes, in order to achieve efficient encoding. D. Protograph Codes Protograph codes 8 are a subclass of LDPC codes. The bipartite graph of a protograph code is obtained starting from a bipartite graph with a small number of edges and nodes, called the protograph. Typically, puncturing is performed to the variable nodes: this means that some of the variable nodes in the protograph are transmitted, and some other variable nodes are not transmitted. In order to obtain the final bipartite graph, a certain number of repetitions of the protograph are first generated, in order to achieve the desired codeword length n. In this phase, the bipartite graph is composed of a certain number of unconnected graphs. Next, an edge permutation operation is performed in order to make the overall graph connected. More specifically, each edge in the graph connecting a check node c and a variable node x before the permutation, will connect a repetition of c to a repetition of x after the edge permutation. The code rate of the overall bipartite graph is the same as the code rate of the protograph. Two examples of good protographs for the erasure channel are shown in Fig 8 and Fig In these protographs, the black nodes represent punctured variable nodes. E. Generalized Low-Density Parity-Check Codes c The square node in Fig. 3 represents a bitwise parity constraint on the packets x 1, x 2,...,x q+1. This simple constraint is a single parity-check code of length q +1: since its minimum distance is d min = 2 (independently of the length), it is able to correct any single erasure. Next, suppose that the same square node represents a block binary linear code of length q + 1 and minimum distance d min > 2 (actually, it represents L such codes working in parallel, where L is the packet length). In this case, the check node has a higher erasure correction capability than a single parity-check code. Several decoding algorithms can be performed at the check node. For example, bounded distance decoding can be performed: If the number e of erasures is e<d min decoding is successfully performed, otherwise a decoding failure is declared. The most powerful decoding algorithm is MAP decoding. MAP decoding is always successful when e<d min, it can be successful when d min e n k, and it is never successful when e>n k. Despite in the last case decoding is always unsuccessful, some missing packets may be still correctable. c Generalized LDPC codes have never been considered within the CCSDS LEC BOF. They are included in this paper for completeness. 6of12

7 Figure 8. A R = 1/2 protograph achieving a threshold p = Figure 9. A R = 1/2 protograph achieving a threshold p = The idea behind generalized LDPC (GLDPC) codes, sometimes called Tanner codes, 10 is to use more powerful check codes instead of single parity-check codes. Using just block linear codes with d min > 2as check nodes has been shown to produce a rate loss that makes this solution not attractive for high rate codes (code rate R = k/n > 1/2). 11 However, hybrid check structures composed of both single parity-check codes and linear block codes have been shown to be able to overcome LDPC codes in terms of asymptotic performance. 12 III. Long Erasure Correcting Codes Performance A. Asymptotic Performance The performance of the iterative decoder described in subsection II-A heavily depends on the degree distribution of variable and check nodes. The degree of a variable or check node is the number of edges connected to that node, while the variable (check) degree of an edge is the degree of the variable (check) node the edge is connected to. For instance, variable node x 1 in Fig. 4 has degree 2, check node c 1 in the same figure has degree 3, and the edge labelled by e has variable degree 2 and check degree 3. It is usual to refer to the edge-oriented degree distributions instead of the node-oriented degree distributions. This is because most of the equations concerning the asymptotic performance of long erasure codes can be easily formulated in terms of edge-oriented distributions. For a LDPC code, or for each cascading graph in a Tornado code, the fraction of edges with variable degree i is usually denoted by λ i, the fraction of edges with check degree i by ρ i, and the edge degree distributions are defined as λ(x) = i λ i x i 1 and ρ(x) = i ρ i x i 1 (x is a dummy variable). The maximum fraction of erasures that a random LDPC (i.e. an LDPC code with randomly generated bipartite graph) code can correct on a memory-less PEC can be expressed as a function of just the degree distribution pair (λ, ρ), if infinite codeword length is assumed. More specifically, for any distribution (λ, ρ), a maximum value of the channel erasure probability p exists, denoted by p = p (λ, ρ), such that for p<p arandom(λ, ρ) LDPC code makes vanishing the post decoding erasure probability, in the limit where n tends to infinity. This value is also referred to as decoding threshold. For unstructured (i.e. random) codes, the threshold has been shown to be equal to the supremum p such that inequality ρ(1 pλ(x)) > 1 x holds x (0, 1]. 13 This inequality is recognized to be a direct consequence of density evolution 14 for the memory-less erasure channel. The threshold p of a code with code rate R cannot exceed 1 R: thisisa direct consequence of the channel coding theorem. Since for an LDPC code the code rate R is expressed by R = k/n =1 ( i λ i/i)/( i ρ i/i), then the threshold can never exceed ( i λ i/i)/( i ρ i/i). The task when considering infinite length performance is to find the degree distribution pair (λ, ρ) with the best threshold p, under a number of constraints. Typical constraints are the code rate R, the minimum and maximum variable degree, the minimum and maximum (or, alternatively, the average) check degree. If v max and v min (c max and c min ) denote the maximum and minimum variable (check) degrees, then the total number of independent variables involved in the optimization problem is v max v min + c max c min 3. This is because the three constraints i λ i =1, i ρ i =1,1 R =( i λ i/i)/( i ρ i/i) must be satisfied. Several numerical techniques can be used in order to solve this optimization problem. Among them, one of the most effective is the differential evolution algorithm. 15 It has been proved that the capacity of the memory-less PEC is achievable by the iterative decoder. 4 A sequence of degree distributions (λ m,ρ m ), all with the same code rate R, is said capacity achieving of rate R when, for any ɛ>0, there exists an m such that (1 R) p m <ɛfor any m>m, where p m is the threshold of (λ m,ρ m ). Some capacity achieving distributions have been analytically evaluated. 4, 16 However, 7of12

8 these distributions are of limited practical interest because of the high error floors (see next subsection) of 17, 18 codes designed according to them. The concept of threshold, presented for unstructured LDPC codes, can be extended to IRA codes, Tornado codes, LDPC codes based on protographs and GLDPC codes as well. In each case, the asymptotic decoding threshold can be computed by exploiting either techniques based on density evolution, or techniques based on EXIT charts. 19 For protograph codes, density evolution can be directly performed on the protograph (i.e. on a relatively simple graph) in order to evaluate the asymptotic threshold of the overall code. This threshold evaluation is usually very accurate. This also implies that the problem of searching for good codes can be performed as a search for good protographs. For instance, the rate 1/2 protograph depicted in Fig. 8 has a threshold p =0.4776, while the rate 1/2 protograph depicted in Fig. 9 has a threshold p = Protographs with higher asymptotic thresholds are also known. B. Finite Length Performance The performance of a finite-length LDPC code under iterative decoding depends on particular graph structures in the bipartite graph, namely stopping sets 20 and cycles. Suppose that a subset V of the variable nodes satisfies the following condition: any check node connected to this subset is connected to it at least twice. Recall that a parity-check node in an LDPC code can correct just one erasure: then, if the starting erasure pattern generated by the channel includes V, the iterative decoder will not be able to correct any of the variable nodes in V. Subsets of variable nodes satisfying the afore mentioned constraint are called stopping sets. If decoding is unsuccessful, the set of the residual erased variable nodes at the end of decoding is the union of all the stopping sets included in the starting erasure pattern. For GLDPC codes, the concept of stopping set is replaced by the concept of generalized stopping set. 11 A cycle is defined as a close path in the bipartite graph, starting and ending on the same node, and the girth of the bipartite graph is the minimum length of a cycle. There is a close relationship between stopping sets and cycles. Specifically, it has been proved that any stopping set contains cycles. 21 Hence, constructing the bipartite graph according to girth optimizing algorithms or stopping sets removing algorithms can improve the code performance It is important to remark that more powerful (and complex) decoding algorithms like maximum likelihood can correct erasure patterns even if they contain stopping sets. Within the performance curve of a finite length long erasure code under iterative decoding, two regions can be typically distinguished. The first region is known as waterfall: for values of channel erasure probability p slightly smaller than the asymptotic threshold, a small reduction of p corresponds to high gain in terms of post decoding packet erasure probability or decoding failure probability. The second region is called error floor, and corresponds to smaller values of p. In the error floor region, small improvements in terms of performance correspond to relatively high reductions of p. This well known phenomenon is typical of iterative decoders on a wide range of channels. In general, codes with good threshold and waterfall performance have a poor error floor, and viceversa. For unstructured LDPC codes on the erasure channel, the compromise between waterfall and error floor performance can be in part justified combining a number of results. First, the error floor depends on small stopping sets. In fact, when p is small, the starting number of unknown variable nodes is usually small as well. Second, if a code has a minimum distance d min, then it must have stopping sets of size d min. 21 Third, if (λ m,ρ m ) is a capacity achieving sequence, then for sufficiently high m and sufficiently high codeword length n the minimum distance for an unstructured (i.e. random) code with distribution (λ m,ρ m ) is proportional to log(n), and hence small. 23 Thus, unstructured finite length codes with very good thresholds have poor minimum distance, and so stopping sets with small size. This generates the high error floor. Though proved for unstructured codes, this behavior is typically shared by partially structured or structured codes, as well. Finally, it has been proved that, with high probability, the minimum distance of an unstructured code with degree distribution (λ, ρ) is a linear function of n when λ (0)ρ (1) < 1, otherwise it is a logarithmic function of n. 23 This results will be recalled in Section V. IV. Optimization of Long Erasure Codes on Burst Erasure Channels In most realistic scenarios, packet erasures are correlated and they often occur in bursts. In this section an algorithm recently proposed for the optimization of long erasure codes on burst erasure channels is recalled. 24 Though the description of the algorithm is focused on LDPC codes, the basic concepts are applicable to all 8of12

9 FaR C1 JPL AR4A C2 C3 IRA,d v =5 IRA,irreg ,32 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48 Figure 10. Performance on memory-less PEC of several (2048, 1024) long erasure correcting codes (R =1/2) in terms of decoding failure rate (FaR) versus channel erasure probability p. classes of long erasure codes. By definition, the maximal guaranteed resolvable burst length of an LDPC code, denoted by L max,isthe maximal erasure burst length such that the iterative decoder is always able to successfully recover from the burst independently of its position within the codeword. For large codeword length n and proper permutation of variable nodes, it results L max p n, where p is the threshold over memory-less erasure channel. Since in most realistic environments any codeword is prevalently affected by just one erasure burst, in order to improve long erasure codes performance in such environments parameter L max should be maximized. From an equivalent point of view, concentrated stopping sets (i.e. stopping sets made up of variable nodes which are near in the graph) should be removed from the code graph, by increasing their dispersion (the maximal distance between two variable nodes belonging to the stopping set). The proposed optimization algorithm just executes variable nodes permutations, without modifying their connections towards the check nodes. It receives in input a specific LDPC code with maximum resolvable burst length L max and returns a new LDPC code with the same degree distributions and with L max >L max. If L max is the maximum resolvable length for an LDPC code, then at least one erasure burst of length L max + 1 exists which is non-resolvable. The basic observation behind the algorithm is the following: If this non resolvable burst begins on variable node x j (see Fig. 4), then variable nodes x j and x j+l max must belong to the maximal stopping set comprised in the burst. The optimization algorithm is then based on the following rule: If burst length L 1 is non-resolvable due to a burst beginning on encoded packet x j,lookfor a variable node x i in the set {x 1,...,x j 1, x j+l1, x n } such that permuting x j (or x j+l1 1) andx i makes the erasure burst length L 1 resolvable. Despite its extreme simplicity, the afore described optimization algorithm has been shown to be surprisingly effective. In many cases, it is able to generate an LDPC code with L max quite close to the asymptotic value p n. Usually, the generated LDPC code has an L max quite higher than the L max of the input code. For instance, if applied to an irregular LDPC code with codeword length n = 2000 and asymptotic threshold p = (p n 911), it has been shown to be able to produce an LDPC code with L max = Other features of the algorithm are its flexibility (i.e. applicable to any code rate and codeword length), and the possibility to be used within pure FEC schemes, Type I / Type II ARQ protocols, and parity on demand schemes. p 9of12

10 FaR C1 optim ized C Figure 11. Performance on the CLBuEC with burst length L = 810 packets of code C1 (2048, 1024) and its optimized version in terms of decoding failure rate (FaR) versus parameter b. Figure 12. b 1 b b }{{} Good state L Bad states Constant length burst erasure channel (CLBuEC) model. V. Simulation Results In this section some performance curves for long erasure correcting codes are illustrated and discussed. Consider first Fig. 10, where the performance of several long erasure codes is shown on the memory-less PEC, in terms of decoding failure rate (FaR) versus the channel erasure probability p. Eachcodehasacode rate R =1/2 and information block length k = 1024 packets (n = 2048 packets). The code denoted by C1 is random LDPC code whose distribution has been obtained by running differential evolution subject to the constraint to have a minimum variable nodes degree equal to 3, a maximum variable degree equal to 30 and a maximum check degree equal to 14. The condition λ 2 = 0 implies good property in terms of minimum distance and error floor (λ (0)ρ (1) = 0), but does not permit to obtain capacity approaching distributions. This code does not exhibit error floor up to FaR = 10 6, and its asymptotic threshold p = , though not excellent is rather good. However, the waterfall performance of this code is very poor. The asymptotic threshold does not confidently describe the waterfall performance of code C1 at finite length n = Consider now the JPL AR4A protograph code. 25 It is characterized by a worse asymptotic threshold p 0.44 than code C1. However, it overcomes code C1 since, for codeword length n = 2048, it exhibits a smaller gap between finite length and asymptotic performance, as well as very good properties in terms of error floor. Next, consider the random LDPC codes denoted by C2 and C3. Their distributions have been obtained by running differential evolution with maximum variable degree equal to 30, minimum variable degree equal to 2, and with an upper bound on parameter λ (0)ρ (1). More specifically, λ (0)ρ (1) < 0.3 for C2 and λ (0)ρ (1) < 0.4 for C3. The threshold for code C2 is p = , while the threshold for C3 is p = of 12

11 They both outperform the AR4A code in terms of asymptotic threshold, waterfall performance and don t exhibit error floor as well, up to FaR < Efficient encoding for the codes C2 and C3 can be performed with linear complexity by exploiting a generalization of the concept of IRA code, namely GeIRA. 26 The IRA code in the same figure is characterized by a uniform degree d v = 5 for the systematic nodes, and a threshold p = Its performance is quite similar to the performance of the AR4A protograph code, with a little gain for FaR< It is possible to generate IRA codes with the same codeword length and code rate, and with an optimized distribution. For instance, the irregular IRA code in Fig. 10 has a threshold p = However, it presents a poor error floor that makes it not attractive in applications that require failures rates less than Next, consider two examples of application of the optimization algorithm on burst erasure channels. More specifically, consider the application of the algorithm to the afore presented codes C1 and irregular IRA. Code C1 has a maximum resolvable length L max = 788, quite lower than the asymptotic limit. From the performance curve of this code on the memory-less channel (Fig. 10) we observe that the decoding failure rate is about for p = /2048. Thus, the probability that a randomly chosen erasure pattern of length 788 includes a stopping set is relatively small. Hence, the number of burst positions of length 788 including stopping sets is small, and we can expect that L max will be improved. On the contrary, the decoding failure rate at p =0.395 for the same code is about Reasoning in the same way, there is a relatively high number of burst positions of length including stopping sets. We expect that the algorithm can improve L max just a little more than this value. Actually, the optimized code is characterized by a L max = 822. The decoding failure rates of both C1 and the generated optimized code are plotted in Fig. 11, on a constant length burst erasure channel (CLBuEC) with burst length L = 810 packets. This channel (see Fig 12) has a good state with erasure probability p G =0,andL bad states each with erasure probability p B = 1. The channel moves from the good state to the bad states with transition probability b, with generation of a burst of packets erasures of length L. After the generation of the last erasure in the burst, the channel returns in the good state. We observe a gain of about one order of magnitude in term of b at FaR= Considering the irregular IRA code, we found a maximum resolvable length L max = 766 packets. This value has been improved by the algorithm up to L max = 912, close to p n. VI. Conclusion In this paper long erasure correcting codes for packet oriented FEC have been presented. These codes are currently under investigation within the CCSDS (Consultative Committee for Space Data Systems) Long Erasure Codes Bird of Feather (LEC-BOF). The possible codes structures, the encoding and decoding algorithms, and some theoretical properties about asymptotic and finite length performance have been discussed. Code structures exist which permit to efficiently perform the encoding operation. Furthermore, their low complexity iterative decoding algorithm, which can asymptotically achieve the erasure channel capacity and offers good finite length performance, allows for long codeword lengths (up to thousands of packets). Long erasure correcting codes can be in principle implemented at different layers in the protocol stack illustrated in Fig. 1, depending on the transmission protocol and on the specific application. Then, the LEC packet can assume different meanings in different contexts, like transfer frame, (constant length) space packet, UDP packet or any data unit properly defined by the user. Thus, they represent an attractive solution for packet oriented FEC when ARQ is not available or within hybrid FEC / ARQ schemes, when long files need to be transmitted or when the channel can produce relatively long erasure bursts. References 1 CCSDS, Overview of space link protocols, Green Book G-1, June Shambayati, S., Jones, C., and Divsalar, D., Maximizing throughput for satellite communication in a hybrid FEC/ARQ scheme using LDPC codes, IEEE MILCOM 2005, Oct Gallager, R., Low-density parity-check codes, Cambridge, Massachussets: M.I.T. Press, Luby, M., Mitzenmacher, M., Shokrollahi, M., and Spielman, D., Efficient erasure correcting codes, IEEE Trans. Inform. Theory, Vol. 47, Feb. 2001, pp Pearl, J., Probabilistic reasoning in intelligent systems: Network of plausible inference, San Mateo, CA: Morgan Kaufmann, Jin, H., Khandekar, A., and McEliece, R., Irregular repeat-accumulate codes, Int. Symp. on Turbo codes and Related Topics, Sept of 12

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