Efficient FEC Codes for Data Loss Recovery

Transcription

1 Efficient FEC Codes for Data Loss Recovery Cheng Huang Lihao Xu Dept. of Computer Science and Engineering, Washington University in St. Louis, MO, 633 {cheng, Abstract Real-time applications are becoming more and more popular and important. To address the specific needs, such as data loss and transmission delay, of these applications, FEC (Forward Error Correction techniques based on error correcting codes are widely used. The choice of codes and their parameters are crucial in determining the efficiency of FEC, in terms of loss recovery capability, bandwidth use and computation complexity. The parity code and the Reed-Solomon code are two classes of error correcting codes widely used for FEC. The former, however, is limited by loss recovery capability while the later is limited by computation complexity. In this paper, we propose to study MDS (Maximum Distance Separable array codes as FEC codes. Especially, we study the suitability of the EVENODD code for both random and bursty data loss recovery. We also propose an efficient and versatile decoding algorithm for the EVENODD code that is suitable to correct both random and bursty losses. Our analytical and simulation results show that the EVENODD code with our decoding algorithm is more efficient and effective to combat most data loss patterns and is therefore very suitable to be used as an FEC code in a wide range of environments. I. INTRODUCTION Real-time data applications, such as streaming video/audio, video conferencing and Internet telephony, are becoming more and more popular and important. One prominent requirement of such real-time applications distinguishing them from classical bulk data applications, is that they are sensitive to data loss and transmission delay. One example is when watching a streaming video through network, users find it annoying to be forced to wait for media player re-buffering. To address the specific needs of this type of applications, FEC (Forward Error Correction techniques based on error correcting codes are widely used to protect data loss and in turn to reduce or eliminate retransmission delay, e.g., in [3]. FEC has also been proposed for bulk data transmissions, e.g.,[6], and reliable multicast, e.g.,[4]. In contrast to ARQ (Automatic Repeat request techniques, FEC reduces additional network delay by sending redundant parity data along with original message data to avoid data retransmission in presence of data loss. The cost of FEC is the need to consume extra network bandwidth for parity data. FEC also introduces extra computation load for encoding and decoding of certain error correcting codes. Thus the choice of a suitable error correcting code with certain parameters for FEC is critical in deciding FEC s effectiveness in combating data loss (data loss recovery capability, as well as FEC s efficiency in both network bandwidth use and computation complexity for encoding and decoding operations. In this paper, we study how to choose suitable error correcting codes for FEC in various environments. Both our theoretical analysis and experimental results provide useful guidelines for practical data loss recovery in a variety of applications. An (n, k error correcting code encodes k message symbols to n codeword symbols, where of course n k. A symbol here is a flexible data unit: it could be a bit, a byte, a packet or a frame. Such a code can usually tolerate a data loss of r symbols during transmission, where obviously r n k. Its coding rate, defined as k/n, decides its effective use of network bandwidth. On the other hand, r determines the code s loss recovery capability. It is easy to see the larger r is, the more capable the code is in loss recovery, but the lower its coding rate is, i.e., the less effective it consumes network bandwidth. When r = n k, the code meets the Singleton Bound, and it is MDS (Maximum Distance Separable []. An MDS code is optimal in terms of its data loss recovery capability within a given coding rate. Thus it is desirable to use an MDS code as a FEC code. The single parity code with interleaving (For simplicity, we use PARITY code hereafter. and the Reed- Solomon code[2] (RS code are two widely used FEC codes. Both of them are MDS. The PARITY code has very simple structure: a PARITY code of length n and interleaving degree m can be described as an m n array, where the last column is the (parity check column of the rest n message columns, as depicted in Figure. A PARITY code is computationally efficient in both encoding and decoding, since the only operation needed is binary addition i.e., XOR (exclusive-or. But its loss

2 2 m n Fig.. Parity Code Example. Each symbol in the shadow column of this m n array is computed by all other symbols in the same line. Thus, symbols are encoded line by line. However, they are read out column by column, which incurs an interleaving effect from the perspective of symbols in a same line. recovery capability is limited, since it has only a single parity symbol in a codeword. On the other hand, the RS code is more flexible in choosing its loss recovery capability. The encoding and decoding of a RS code, however, involve operations in finite field and thus are very computationally intensive. In this paper, we propose to study another class of error correcting codes, namely MDS array code, as FEC codes for data transmissions. MDS array codes have the MDS property, as well as efficient encoding and decoding procedures using only binary additions. They have been extensively studied for data storage applications [], [2], [8], [9], [3], [7]. Especially, we study the suitability of the EVENODD code [], an MDS array code, for both random and bursty data loss recovery. We also propose an efficient and versatile decoding algorithm for both random symbol erasure (loss and bursty erasure recovery. We also compare the data recovery capability for both random and bursty data loss of the EVENODD code, as well as computation complexity of encoding and decoding, with the PARITY code and the RS code. Both our analytical and experimental results show the EVENODD code is suitable for a FEC code in a wide range of environments. This paper is organized as follows: Section II describes the EVENODD code and an extended decoding algorithm for both random and bursty erasure recovery. The computation complexity of this extended decoding algorithm is analyzed both theoretically and experimentally. We also compare the decoding throughput to other usual components in real-time data applications, such as media player (codec and ciphers. The data loss recovery capability of the EVENODD code with the extended decoding algorithm is analyzed in Section III for random data loss and in Section IV for bursty data loss. Section V studies the impact of code parameters on FEC performance and provides general guidelines for how to choose them properly in practical applications. We conclude the paper in section VI. II. EXTENDED EVENODD DECODING (XEOD ALGORITHM A. EVENODD code and Basic EVENODD Decoding (BEOD scheme EVENODD code: The EVENODD code was initially proposed to address disk failures in disk array systems. Data from multiple disks form a two dimensional array, with one disk corresponding to one column of the array. A disk failure is equivalent to a missing column. The EVENODD code uses two redundant (parity check columns (disks in a disk array of total up to p user disks (where p is a prime number. The code ensures that all the original data is fully recoverable when any two disk fail. In this sense, it is an optimal 2-erasure correcting code, i.e., it is an (p+2, p MDS code. Besides this MDS property, EVENODD code is computational efficient in both encoding and decoding, since they only need binary exclusive-or(xor operations. 2 EVENODD encoding: Consider a (p (p+2 array, where p is a prime number, such that symbol a i,j, i p 2, j p+, is the i th symbol in the j th column. The last two columns are parity check columns. Each symbol in the p th column is calculated by taking the XOR of all the symbols in the same line to its left. The (p + th column is a little bit complex. First, an EVENODD adjuster is calculated by taking the XOR of all symbols along the main diagonal. Then, each check symbol in this column is set to be the XOR sum of all the symbols on its corresponding diagonal and the adjuster. The encoding procedure can be algebraically described as follows: a l,p = S = p a l,t t= p a p t,t t= a l,p+ = S ( p a (l t%p,t t= where S is the EVENODD adjuster. Figure 2 gives a simple example for p = 5. For more description, please refer to [].

3 3 Fig. 2. EVENODD Code Encoding (p=5. Check symbols are in shadow columns. The diagonal of O s on the right side calculates the EVENODD adjuster. Fig. 3. EVENODD Code (p=5 Decoding. The Zig-Zag decoding process starts from a 2,2 and ends at a,. 3 Basic EVENODD Decoding (BEOD scheme: EVENODD code is an optimal 2-erasure correcting code in the sense that any two missing columns in a codeword block can be fully recovered. There are four different cases regarding to the position of missing columns. Here, we focus on the most common case, where neither of the missing columns is a parity check column. (The other three cases are special cases and can be dealt with easily. Refer to []. In this case, the decoder first calculates the horizontal syndromes and the diagonal syndromes by calculating the XOR sums of all the symbols along each horizontal and diagonal lines except symbols in those two missing columns. Then a starting symbol in the missing columns can be easily found, which is guaranteed to be the only missing symbol in the corresponding diagonal line. The decoder recovers this starting symbol and then goes along horizontal line to recover the missing symbol in the other missing column. It then goes along diagonal line to the next missing symbol and horizontal line again. Upon completing this Zig-Zag process, all the symbols in two missing columns are fully recovered. In the example shown in Figure 3, the starting symbol is a 2,2 and decoding process goes from a 2,2 to a 2,, a,2, a, and ends at a,. B. EXtended EVENODD Decoding (XEOD scheme The BEOD is designed for an error (loss model which is suitable for data storage devices, such as disks or tapes, where a column is considered to be an error or erasure as long as at least one of its symbols is an error or erasure. This bursty loss model is sensible for data storage applications. When the EVENODD code is used for a disk array, the BEOD can fully recover all the original data symbols when up to two disks fail. When the EVENODD code is applied to data transmission, however, it is unlikely that loss is constrained only in two columns, i.e., symbol loss can be random in addition to bursty. BEOD scheme doesn t provide a mechanism to deal with this situation. In addition, when symbol loss occurs in a column, it doesn t necessarily mean that all the symbols in the same column are lost. It is thus not computationally efficient to calculate all the horizontal and diagonal syndromes if some of them are not actually used in decoding. However, there is no simple way to decide which syndrome is needed using the BEOD scheme. To address all these issues, we propose an extended EVENODD Decoding (XEOD algorithm for both random and bursty symbol losses. The EVENODD has been shown to be an LDPC (Low Density Parity Check code and its probabilistic error correction performance based on its parity check matrix has been studied [4]. While taking advantage of the LDPC property of the EVEN- ODD code as well, our XEOD decoding algorithm is a deterministic one that corrects erasures (symbol losses instead of errors (symbol corruptions and thus is much more computationally efficient than other probabilistic decoding algorithms based on parity check matrix. Now we described the XEOD: each codeword block is represented by a bipartite graph, with its left nodes corresponding to the message symbols (in the first p columns and the right nodes corresponding to the (parity check symbols (in the p th and (p + th columns. A left node exists in the bipartite graph only if the corresponding message symbol is lost and a right node exists only if the check symbol is not lost. For simplicity, message node and check node are used to represent left node and right node, in the rest of this section. A link (edge is setup between a message node and its corresponding check node. The degree of a node represents the number of links connected to it. It is easy to see that the degree of a message node is no larger than 2 while the degree of a check node is less or equal to (p. It takes O( operation to set up the links as the symbols are received. For decoding, XEOD starts from any check node with degree. It goes to a message node through the only link from this check node. Since this message node is the only missing one corresponding to the check node, it can be easily recovered and all the links connected to this message node are removed. Then XEOD selects another check node with degree and repeats the above process until no more nodes can be recovered. Besides the above basic loop procedure, XEOD needs to calculate the EVENODD adjuster, which is used to recover the message nodes from diagonal check nodes.

4 4 The adjuster can be calculated from two redundant columns, as in [], or any diagonal check node with degree. Experiments show that the adjuster can be calculated with very high probability when the symbol loss rate is relatively low. Thus it is not an issue to recover the adjuster. Algorithm XEOD. while (check node of degree exists 2. if (adjuster not known 3. then 4. if (check node of degree exists or all check nodes exist 5. then calculate adjuster 7. select horizontal check node of degree 8. recover corresponding message node 9. remove all links from this message node. if (adjuster known. then 2. select diag. check node of degree 3. recover corresponding message node 4. remove all links from this node 6. return C. Computation Efficiency of the XEOD Scheme Analysis and Comparison of XEOD and BEOD: The XEOD is able to recover random symbol losses as well as bursts, and thus has higher loss recovery capability than the BEOD. It is desirable to know whether the gain comes at a cost of increased computation complexity. Since the BEOD can only recover loss symbols in up to two columns, it is fair to limit the comparison to cases where symbol losses are contained in two columns. We assume that each symbol has an equal and independent loss probability q. Let i, j ( i, j (p + be the indexes of the columns of the loss symbols. For the XOR-based decoding schemes, complexity analysis can be simplified by just counting the number of XOR operations, which is the only dominating factor in the decoding process. To further simplify analysis, we don t distinguish (p and p, as p is large enough. a BEOD Complexity Analysis: It is not difficult to categorize the complexity analysis into four cases and the result is summarized in Table I, where row 2 represents the number of XOR operations needed in calculating the EVENODD adjuster, row 3 the number of XOR operations in recovering message symbols and row 4 the number of occurrences in total ( p+2 2 cases. The expect value of the number of XOR operations can be calculated as: E(XOR# = p4 + (4q + 2 p3 (2q + 2 p2 ( p+2 2 b XEOD Complexity Analysis: The XEOD computation complexity can be analyzed in the similar way and summarized in Table II, using the same notation as in Table I. The expect value of the number of XOR operations can be calculated as: E(XOR# = qp4 + ( 2 + 2q 2 q2 p 3 + ( 3 2 2q q2 p 2 ( p ( q q2 p + 2qp( q p (p ( p+2 2 The above analysis results are also verified by simulation, where the number of XOR operations are counted in a real decoding implementation. In the simulation, each symbol in a codeword block corresponds to a data packet of 5 bytes, which is a proper size for some delay sensitive network applications, such as streaming video. Both analysis and simulation results are shown in Figure 4. Figure 4(a shows a special case where all the packets in the two selected columns are lost, which is the original case BEOD designed to deal with. The XEOD requires about the same number of XOR operations as the BEOD in this case. Notably, the XEOD doesn t incur more complexity in this case. When only half of the packets in the select columns are lost, Figure 4(b shows the XEOD outperforms the BEOD in terms of computation complexity. This is reasonable since XEOD doesn t calculate unnecessary horizontal and diagonal syndromes while the BEOD does. The trend results in Figure 4 are further verified by actual time measured in simulation, which is shown in Figure 5. In summary, although XEOD incurs more logic operations than BEOD, it doesn t increase the computation complexity. It is no worse than the BEOD under any situation and outperforms the BEOD in most cases. As the symbol loss probability decreases, the XEOD has much lower computation complexity than the BEOD. Moreover, XEOD can achieve higher loss recovery capability than the BEOD. 2 Throughput of XEOD: Besides loss recovery, a real time application usually has many other components, which are computation intensive. For example, a streaming media system needs a video/audio codec to operate on compressed media data. It might also use a cipher to en/decrypt content to prevent illegitimate use. Thus, it

5 5 # of XOR operations CASE I CASE II CASE III CASE IV i p, j = p i p, j = p + i, j p i = p, j = p + calculate adjuster p 2p recover message qp 2 qp 2 2p 2 + ( p + 4pq frequency p p p 2 TABLE I BEOD COMPLEXITY ANALYSIS # of XOR operations CASE I CASE II CASE III CASE IV i p, j = p i p, j = p + i, j p i = p, j = p + i = i i = i i = i i = p calculate adjuster p 2q( q p + p 2q( q p + ( qp + 2qp ( q 2 p+ p [ q( q p ]p [ q( q p ]p 2[ ( q 2 ]p recover message qp 2 qp 2 2qp 2 frequency p p p TABLE II XEOD COMPLEXITY ANALYSIS ( p 2 is desirable to study the computation impact of an FEC on a real system by comparing its throughput to other components. In our experiments, we use an MPEG-I video sequence of size about 2 MB, excerpted from the Terminator2. The MPEG-I decoder is from SMPEG[5], which is an open source decoder and is originated from UC Berkeley. AES-28 and RC4[6] are selected because they are the fastest block cipher and stream cipher, respectively. We use Wei Dai s crypto++5.[7] package here. The experiments are performed on a 733 MHZ P3 PC running Linux Redhat 7.3. Figure 6 shows that the throughput of EVENODD encoding and XEOD is much higher than MPEG-I codec, AES-28 and RC4 en/decryption. The encoding and decoding of the PARITY code is understandably even higher. Hence, the encoding or decoding of FEC codes will not become a bottleneck in a real-time data system. III. DATA RECOVERY (I: RANDOM SYMBOL LOSS To study the efficiency of XEOD in terms of loss recovery capability, this section compares it with the PARITY code and the RS code, which are the two most widely used FEC schemes. Here the random loss model is that each symbol has an equal and independent loss probability. A. performance analysis It is not difficult to analyze the loss recovery capability of the PARITY code and the XEOD for a given random symbol loss pattern. For the PARITY code and the XEOD, we can analyze the decoding procedure by viewing it as a discrete random process and apply the approach discussed in [9], which we summarize as following: An EVENODD codeword block can be represented as a bipartite graph, with each link connecting a message node on one half plane to its check node on the other half. Links adjacent to a node of degree i are links of degree i. Let λ i be the fraction of links of degree i according to message nodes and ρ i the fraction corresponding to the check nodes. Define two polynomials λ(x = i λ ix i and ρ(x = i ρ ix i. The fraction of unrecoverable data nodes with random loss probability δ is: r(x = δ( δλ(δ + ( δx [x + ρ( λ( δλ(δ + ( δx] where x is the largest value satisfies: ρ( δλ(ρ + ( δx > x, x (, ] For a PARITY code with p message symbols, its λ(x and ρ(x can be calculated as follows: each data node participates parity calculation just once, thus λ = and λ i = for all i. Since p is the width of message block (p + the width of codeword block, and every check node has p links, ρ p = and ρ i = for all i p. Therefore, λ(x = ρ(x = x p With these two polynomials defined, we can calculate the largest feasible value of x and then compute the fraction of unrecoverable message nodes. Polynomials for the XEOD are a little bit more complex to compute. We simplify our analysis by assuming

6 6 XOR operation number simulation BEOD analysis BEOD simulation XEOD analysis XEOD code parity p (a q =.. When all symbols in selected columns are lost, both BEOD and XEOD have about the same complexity. block decoding time (ms BEOD(q=. XEOD(q=. BEOD(q=.5 XEOD(q= code parity p Fig. 5. Average Measured Block Decoding Time in Simulation. Actual time verifies the analysis results. Fig. 4. XOR operation number simulation BEOD analysis BEOD simulation XEOD analysis XEOD code parity p (b q =.5. XEOD requires less XOR operations than BEOD, when only about half symbols are lost in selected columns. XOR operation number (Analysis vs. Simulation throughput (Mbps MPEG- DECODEING AES ENCODING AES DECODING RC4 ENCODING RC4 DECODING EVENODD ENCODING XEOD PARITY ENCODING PARITY DECODING Fig. 6. Comparing Throughput of XEOD to MPEG-I Decoder and Ciphers. (Throughput of MPEG- decoder is 7.5Mbps. the EVENODD adjuster can always be calculated. This is reasonable because the adjuster can be derived from any diagonal line with its corresponding check node, as long as none of them is missing. With this assumption, check nodes are divided into three types based on their degrees and check equations, as shown in Figure 7. Therefore, we get the following polynomials (define the prime parameter of EVENODD code as p: if less than k nodes are received in a non-systematic codeword block, in general none of the message nodes can be recovered. Thus, we use systematic RS code in our analysis. Let m be the number of lost data symbols and m 2 be the number of lost check symbols, then the joint loss probability P (m, m 2 in this case is: λ(x = x ρ(x = p 2 p 2(p xp 3 + 2(p xp 2 For RS(n, k ( an (n, k RS code, we use another approach to analyze its performance. First of all, a systematic code is more efficient in terms of erasure recover capability than non-systematic code. The reason is obvious: a systematic code has the original message symbols in its codeword, thus even if less than k nodes are received, which means none of those loss nodes can be recovered, there are still some meaningful message nodes out of those received nodes. On the other hand, Fig. 7. Check nodes are divided into three categories by their degrees. All the check nodes in the (p + th column have degree (p. Only one check node in the (p + 2 th column has degree (p and all the other check nodes in this column have degree (p 2. The p th column is not in use here to achieve the same rate as PARITY code.

7 7 ( k P (m, m 2 = m δ m ( δ k m ( n k δ m2 ( δ n k m2 m 2 And the expected fraction of unreceived nodes is: r = P (m, m 2, m + m 2 > k m + m 2 m,m 2 m analysis PARITY simulation PARITY analysis EVENODD simulation EVENODD analysis RS simulation RS random loss probability B. performance results To compare the performance of the PARITY code, the XEOD and the RS code, it is desirable to use the same block size and coding rate. We shorten EVENODD code by eliminating the symbols in the last message column so that it has the same rate as the PARITY code and thus is fair for comparison. Let p be the prime parameter of the XEOD, thus the height of the code block is (p and the width is (p+. The height of the PARITY code is 2(p and the width is (p+ 2. For the RS code, only its block size matters, which is (p (p + here. The coding rate, p p+, is the same for all the three codes. We use as our performance metric, which is defined as the ratio of the number of unrecoverable message symbols over the total number of loss symbols in a codeword block. Figure 8 shows its relationship with random loss probability by both analysis results derived in previous section and simulation results for p = 9 and p = 37. For any loss rate within this range of redundancy, XEOD always outperforms PARITY code. This is reasonable because in the EVENODD code, each node participates in the calculation of two check nodes, which results in higher recovery chance. The RS has the best performance among the three, which is also expected because the RS code are MDS code at the symbol level, while the EVENODD code is MDS only at the column level. Note that there is a small gap between analysis and simulation results for the XEOD. This is due to the assumption we make that the EVENODD adjuster is always calculable to simplify analysis. This makes EVENODD code a little bit stronger and thus leads to a slightly lower unrecoverable ratio in theoretical analysis. IV. DATA RECOVERY (II: BURSTY SYMBOL LOSS A. A Realistic Data Loss Model: Bursty Loss In the previous section, we use random data loss as a transmission model, which simplifies analysis. However, in a practical wired network, packet losses are mainly due to traffic congestions which result in packets being (a p = 9 analysis PARITY simulation PARITY analysis EVENODD simulation EVENODD analysis RS simulation RS random loss probability (b p = 37 Fig. 8. Unrecoverable Ratio vs. Loss Probability. Analysis results and simulation results match quite well. XEOD shows better performance than PARITY code, although not as good as RS code. dropped by the queuing management process at routers. Thus data losses usually occur in bursts[8]. In a wireless network, data losses often occur in bursty manner as well. Hence a more realistic loss model for a practical network, taking into account the dependency between packet loss, is a two state Gilbert model[5][]. With this model, the network is either in a GOOD (G state representing a packet reaches destination, or in a BAD (B state representing a packet loss. Network state changes from state B to G with probability β and remains in state B with probably ( β. Similarly, state remains in state G with probability ( α and changes to B with probability α. It is easy to calculate that the α α+β stationary loss rate is π B = and the average length of consecutive BAD states, which can also be viewed as the average length of congestion period, is µ B = β. The value of α and β can be derived by measuring π B and µ B in a real network environment. B. Loss Recovery Performance of the FECs It is not difficult to analyze the unrecoverable rate of the RS code for a bursty loss model. We use a recursive approach here, similar to [2].

8 8 Let P s,s n (k, n be the probability of k loss packets out of n total packets, beginning from state s and ending at state s n. Therefore, we can get following recursive equations when the initial state is G: P s=g,s n=g(k, n =P s=g,s n =G(k, n ( α+ P s=g,s n =B(k, n β P s=g,s n=b(k, n =P s=g,s n =G(k, n α+ P s=g,s n =B(k, n ( β with P s=g,s k=g(k, k = and P s=g,s k=b(k, k = α ( β k. And also when the initial state is B: P s=b,s n=g(k, n =P s=b,s n =G(k, n ( α+ P s=b,s n =B(k, n β P s=b,s n=b(k, n =P s=b,s n =G(k, n α+ P s=b,s n =B(k, n ( β with P s=b,s k=g(k, k = and P s=b,s k=b(k, k = ( β k. Thus, the probability P (k, n of k loss packets out of total n packets is: P (k, n = ( π B (P s=g,s n=g(k, n + P s=g,s n=b(k, n + π B (P s=b,s n=g(k, n + P s=b,s n=b(k, n It is, however, much more difficult to get closed form representation of the unrecoverable ratio of the PARITY code or the XEOD for a bursty loss model, if possible at all. This is mainly because the recovery capability depends heavily on actual loss pattern in each codeword block, which is extremely difficult to count. Therefore, here we use simulation to measure the burst loss recovery capability of the PARITY code and the XEOD. For each simulation, we let the first, states of the Gilbert model pass to ensure our experiments always start from a steady state. Then,,, codeword blocks are generated and decoded. The unrecoverable ratio is calculated as the average over all codeword blocks. Figure 9 shows the comparison of simulation results of the PARITY code and the XEOD and analysis results of the RS code. Loss rate for each case is simulated to be of the added redundancy. It is worth pointing out that the performance of the RS code is worse than XEOD under various burst patterns when loss rate is relatively low. The explanation is that in a bursty network, packet losses tend to group closer together and with longer gap between groups than in random loss situation. Whenever there are more than (n k packet losses in a block of RS(n, k, the decoder fails to solve necessary linear equations due to too PARITY XEOD RS average burst length (a p = 9, loss rate = % PARITY XEOD RS average burst length (b p = 37, loss rate =.526% Fig. 9. Unrecoverable Ratio vs. Average Burst Length. XEOD outperforms RS code most of the time and is better than PARITY code when burst is not severe. many unknowns. Thus none of these lost packets can be recovered. In contrast, each subset of packets can do their own decoding in the PARITY code and the XEOD. This provides higher chance to recover symbol losses in some cases. Thus it is important to point out for practical bursty networks, the RS code, which are optimal for random packet loss, are not necessarily the best choice even without considering encoding and decoding throughput. Also, the performance of the PARITY code is not always poor in bursty losses. When the average burst length goes beyond a cross point with the XEOD, the PARITY code actually yields better loss recovery performance. This justifies that the PARITY code is still an effective approach for loss recovery. Moreover, this cross point exists in all our simulations with various loss rates and coding rates. It remains an interesting open problem to theoretically derive this cross point for any given bursty loss pattern. V. EFFECTS OF PARAMETER ON XEOD This section discusses the effects of parameter prime p on XEOD. Fair comparisons among different p values

9 PARITY p = 9 XEOD p = 9 RS p = 9 PARITY p = 3 XEOD p = 3 RS p = average burst length Fig. 2. Unrecoverable Ratio vs. Average Burst Length. (Between the curves of p=9 and p=3 are those of p=23, 29, which are not plotted just not to complicate the figure. Fig.. Shorten Code Example. Dark shadow is a message symbol column not in use in the shortened code. Light shadow columns are check columns. The coding rate decreases from 5 to 4 in this case PARITY p = 9 PARITY p = 3 XEOD p = 9 XEOD p = 3 RS p = 9 RS p = random loss probability Fig.. Unrecoverable Ratio vs. Loss Probability with Shortened Codes. This shows shorten does not affect random loss recovery capability of these FECs. are achieved by shortening code with larger p to having the same coding rate. Shortening is a common practice to adjust a code s coding rate without changing its loss recovery capability, by setting certain message symbols to zero and then eliminating them []. A simple shortening example is shown in Figure. The shortened code has the same coding rate as the original code, although their block size is different. A. effect of shortening for random loss Figure shows simulation results of the random loss recovery capability of the PARITY code, the XEOD and the RS code. The simulations show for the PARITY code and the XEOD, shortening does not affect their loss recovery capability, while shortening has marginal effect on the RS code. B. effect of shortening for bursty loss Figure 2 shows the performance comparison for bursty losses. For a particular type of code, its loss recovery capability increases as p increases. For the same p, the relative loss recovery capability remains the same among the PARITY code, the XEOD and the RS code: the XEOD outperforms the RS code and is better than the PARITY code for short bursts. For XEOD, larger p yields better performance. However, the side effect is that codeword block size also increases, which in general requires more buffer space usage and longer decoding delay. Hence, a general rule to decide p value is to push p to the maximum value limited by recovery buffer and delay constraints. VI. SUMMARY In this paper, we study the suitability of a few classes of error correcting codes as FECs for various network environments. In particular, we propose to use the EVENODD code, a class of MDS array code, as FEC code. We provide an efficient and versatile decoding algorithm for the EVENODD code and study its loss recovery performance for both random and bursty loss models. Our analytical and simulation results show that EVENODD code has higher loss recovery capability compared to the PARITY code and the RS code, when the data loss rate is relatively low and network burstiness is not severe. We also show our decoding algorithm XEOD for the EVENODD code is efficient in terms of computation complexity and will not be a bottleneck in real time data applications. Thus the EVENODD code shall be used as another efficient and effective FEC in data transmission applications, in addition to the PARITY code and the RS code. REFERENCES [] M. Blaum, J. Brady, J. Bruck and J. Menon, EVENODD: An Efficient Scheme for Tolerating Double Disk Failures in RAID Architectures, IEEE Trans. on Computers, 44(2, 92-22, Feb. 995.

10 [2] M. Blaum, J. Bruck, A. Vardy, MDS Array Codes with Independent Parity Symbols, IEEE Trans. on Information Theory, 42(2, , Mar [3] M. Blaum, P.G. Farrell, and H.C.A. van Tilborg, Array Codes, Chapter 22 in Handbook of Coding Theory, V.S. Pless and W.C. Huffman (Eds., Elsevier Science B.V., 998. [4] M. Blaum, J. Fan and L. Xu, Soft Decoding of Several Classes of Array Codes, Proc. of IEEE International Symposium on Information Theory, Lausanne, Switzerland, Jun. 22. [5] J.-C. Bolot, S. Fosse-Parisis, and D. Towdley, Adaptive FEC- Based Error Control for Internet Telephony, Proc. of IEEE INFOCOM, 999. [6] J. Byers, M. Luby, M. Mitzenmacher and A. Rege. A Digital Fountain Approach to Reliable Distribution of Bulk Data, Proc. of the ACM SIGCOMM 98, 56-67, Sep [7] Crypto++ Library 5., weidai/cryptlib.html [8] W. Jiang and H. Schulzrinne, Modeling of Packet Loss and Delay and Their Effects on Real-time Multimedia Service Quality, NOSSDAV, 2. [9] M. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, Efficient Erasure Correcting Codes, IEEE Trans. on Information Theory, 47(2, , Feb. 2. [] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, Amsterdam: North-Holland, 977. [] T. Nguyen and A. Zakhor, Distributed Video Streaming with Forward Error Correction, Packet Video Workshop 22, Pittsburgh PA, USA. [2] I. S. Reed and G. Solomon, Polynomial Codes over Certain Finite Fields, J. SIAM, 8(, 3-34, 96. [3] L. Rizzo, Effective Erasure Codes for Reliable Computer Communication Protocols, ACM Computer Communication Review, Apr [4] L. Rizzo and L. Vicisano, A Reliable Multicast data Distribution Protocol based on software FEC techniques (RMDP, Proc. of the Fourth IEEE HPCS 97 Workshop, Chalkidiki, Greece, Jun [5] SMPEG Library, [6] W. Stallings, Cryptography and Network Security, Principles and Practices, 3rd ed., Prentice Hall, Inc. 23. [7] L. Xu and J. Bruck, Highly Available Distributed Storage Systems, Lecture Notes in Control and Information Sciences, Vol. 249, 37-33, G. Cooperman, E. Jessen and G. Michler (Eds, Springer, Jun [8] L. Xu and J. Bruck, X-Code: MDS Array Codes with Optimal Encoding, IEEE Trans. on Information Theory, 45(, , Jan [9] L. Xu, V. Bohossian, J. Bruck and D. Wagner, Low Density MDS Codes and Factors of Complete Graphs, IEEE Trans. on Information Theory, 45(, , Nov [2] J. R. Yee and E. J. Weldon, Jr., Evaluation of the Performance of Error-Correcting Codes on a Gilbert Channel, IEEE Trans. Information Theory, 43(8, , Aug. 995.