We don t need no generation - a practical approach to sliding window RLNC

Transcription

1 We don t need no generation - a pratial approah to sliding window RLNC Simon Wunderlih, Frank Gabriel, Sreekrishna Pandi, Frank H.P. Fitzek Deutshe Telekom Chair of Communiation Networks, TU Dresden, Dresden, Germany simon.wunderlih@mailbox.tu-dresden.de,[frank.gabriel sreekrishna.pandi frank.fitzek]@tu-dresden.de Abstrat Random Linear Network Coding (RLNC) is a popular oding sheme to improve ommuniation over lossy hannels. For paket streaming appliations (video streaming, general IP streams), reent researh has shown that sliding window shemes an improve in-order delay properties ompared to the blok/- generation based oding. However, implementing sliding window RLNC with a limited oding window poses new hallenges in both theoretial and engineering aspets. We introdue the first pratial generation-less sliding window RLNC sheme, whih is built on existing generation based oders. Through disrete simulation and a proof of onept implementation, we show that, the in-order delay an be improved ompared to generation based shemes while retaining the reliability, omputational omplexity and overhead. Index Terms sliding window, random linear network oding (RLNC), delay I. INTRODUCTION Random Linear Network Coding (RLNC) [], [] is an inreasingly popular method to effiiently transfer data in omplex, haoti or lossy networks. It is applied not only in wireless networks [3] and satellite networks, but also in storage [] and peer-to-peer networks [5]. RLNC has relaxed requirements on speifi feedbak, network topologies and strutures, and therefore an be easily applied to improve throughput or reover errors in those appliations. Linear blok odes were introdued as a method to redue the omputational omplexity and the oeffiient overhead []. A large message is split into several hunks (also alled generations). The ode then operates only on symbols within the same hunk. This redues the omputational omplexity of enoding and deoding ompared to working on the whole message. But more signifiantly, it enables effiient, pratial implementations with urrent CPU and memory arhitetures. In reent researh, the fous of ode design has moved towards the in-order delay of oding shemes. The in-order delay is an important metri for all forms of streaming appliations. Sliding window was proposed as a sheme that is apaity ahieving and has a better throughput-delay trade-off than blok odes [7]. While the omputational omplexity of the arithmeti operations looks promising, the requirements for memory and memory management make the sheme diffiult to implement in pratie. This paper introdues a new oding sheme that loses the gap between hunked oding and sliding window network oding. We show that this sheme improves the performane for delay and jitter sensitive appliations while keeping the omputational performane benefits of hunked oding. This makes it the first omputationally feasible onvolutional RLNC sheme. It operates on a limited set of symbols to enable effiient enoding, similar to hunked oding. But instead of separating the symbols into generations, the set is maintained by inrementally adding new symbols and removing old symbols, like a sliding window. We present an algorithm to implement this sheme with minimal omputational overhead. To ahieve this, the algorithm is designed to operate without additional memory alloations. The remainder of this paper is strutured as follows. Setion II provides a review of related work on RLNC and sliding window. In Setion III we present the building bloks for the finite sliding window oding sheme. Setion IV provides a performane evaluation based on simulative results and a proof of onept implementation. Finally, we summarize our onlusions in setion V. II. RELATED WORK Most network oding researh as of today is built on the assumption of generation based blok oding, as introdued by Chou et. al. []. By prepending the oding oeffiient vetor to the symbol payload and grouping data pakets into generations (also sometimes alled hunks or bloks ), it is possible to enode, reode and deode with limited memory and proessing power, and without prior knowledge of the whole network. This approah was demonstrated to effiiently utilize the network and inrease the throughput for multiast file transfers [8]. Chunked oding was further improved by relaxing the boundaries of generations with overlapping hunks [9], and multi-generation mixing [] allows using smaller generations and mixing them to redue the omputational omplexity while keeping good error resiliene. However, these methods are diffiult to apply to streaming appliations. Chunked oding requires a transmission shedule that an reate a oded paket from any generation. But in live streaming appliations the data is ontinuously reated and not fully available from the start. Also, storing several generations at eah node inreases the memory requirements in the network, and maintaining the state for multiple generations at a time adds omplexity and delay. These tehniques fous mainly on network utilization and throughput ignoring the delay of eah paket, whih has an impat on the performane of streaming appliations /7/$3. 7 IEEE 8

2 For paket streams, it has been suggested to use a sliding window sheme to apply RLNC as a onvolutional ode. A general study of suh a sliding window ode and its properties was provided by Karzand et. al. [7]. A possible implementation for SATCOM was studied by Cloud et. al. [], and for multiple ombined paths []. This sliding window approah shows superior delay properties for streaming appliations ompared to generation based odes. However, the authors generally assume an infinite sliding window, or a sliding window of dynami size, whih is losed only by feedbak. In ontrast, we propose a finite sliding window approah of fixed size that an be effiiently implemented. It ombines the good delay properties from sliding window odes with the effiieny and reliability properties from the generation based odes. A reent work by Roa et. al. [3] suggests the use of onvolutional odes over blok odes for delay sensitive appliations, based on a omparison of Reed-Solomon and a finite sliding window RLC. In ontrast, we fous on blok and sliding window odes using RLNC. Reduing the hanges between the ompared odes makes the impat of the differenes more lear. We provide further insights into the subjet by omparing the performane of a finite sliding window with the sliding window sheme proposed in [7]. In addition, our work provides a detailed desription of effiient algorithms for enoding and deoding. III. RLNC AND CODE OPERATION For our work, we fous on optimizing network streams where pakets are transmitted in order, with a single soure and one or more destinations. We use an intra-session sheme where pakets from the same stream are ombined together to reate redundant pakets. For this senario, we propose a systemati sliding window sheme with a finite, predetermined window size. The ode sends the unoded soure pakets in order and periodially adds oded pakets to ounter erasures on the hannel. The oded pakets are reated using RLNC [], ombining only the soure pakets within the sliding window. This is similar to the onstrution in [7], but onstraint to a fixed window size. This onstraint limits the omputational omplexity, the required memory at all nodes in the network, and the overhead in eah paket. In the following setions, we give a detailed desription of the enoding and deoding proess, as well as the paket format of the oded pakets. A. Enoding Enoded pakets are reated by linearly ombining pakets over a Galois field GF (q). The original data pakets are used as soure symbols, and are represented as a vetor of elements of GF (q). In typial network appliations like Ethernet with a maximum transfer unit (MTU) of 5 bytes and a field size q = 8, an Ethernet frame is represented using m = 5 elements of GF ( 8 ). Shorter frames are padded with zeros at the end. A oded symbol is obtained by linearly ombining n s soure symbols of length m with the random oeffiient vetor. With the soure symbol matrix X with n s rows and m olumns this an be summarized with a linear equation. y = X () The resulting oded symbol y is of the same length m and ontains ombined information of all soure symbols in X. Generation based shemes group symbols into equally sized bloks of g subsequent symbols eah. Suh a blok is alled generation, and g is alled the generation size. In these generation based shemes, typially multiple oded symbols are reated in a bath. The oding step an be expressed as matrix multipliation of a oeffiient matrix C GF ( 8 ) n g with the soure symbol matrix X. The oded symbols are represented in n rows of the oded symbol matrix Y. Y = C X () In ontrast, a sliding window ode does not split the soure symbols into artifiial sets. Instead, all symbols within a onseutive sequene of soure symbols are used. This sequene is alled the sliding window. By appending new symbols to this sequene the window is opened, while losing the window removes symbols from the beginning. In previous work it was proposed to use feedbak to lose the window, exluding soure symbols whih are already known by the reeiver [7], []. As a result the size of the sliding window is not bounded. In ontrast, we use a finite sliding window of size w e. The enoder uses the last w e soure symbols to reate a oded symbol. In a systemati sliding window sheme, n s symbols are sent unoded, followed by n oded symbols. With these definitions, we observe that the systemati blok ode with generation size g an also be produed by using a sliding window RLNC with w e = n s = g. To study the performane of the finite sliding window, we fous on the three groups of oding shemes illustrated in Fig.. We define the groups by hoosing the parameters w e, n s, n as follows: ) Systemati blok odes: We set n s = w e >, n >. First, n s soure symbols are sent, followed by n oded n s n s+n symbols. The ode rate is ) Infinite sliding window: With w e the size of the enoding window is not limited. After n s soure symbols are sent, one oded symbol ombining all previous soure symbols n s n s+. is sent (n = ). The ode rate is 3) Finite sliding window: After n s soure symbols are sent, one oded symbol ombining the w e previous soure symbols n is sent (n = ). The ode rate is s n. s+ Beause the finite sliding window enoder works on a limited number of symbols it is possible to operate in a onstant spae of memory. The same is true for the generation based shemes. In ontrast, the memory requirements for the infinite sliding window are not bounded. B. Paket format A oded paket inludes the data of the oded symbol, the oeffiients and auxiliary information. For unoded (systemati) symbols, the oeffiient vetor is a unit vetor with i = 9

3 Coded paket Soure paket (a) Systemati blok ode Coded paket Soure paket (b) Infinite sliding window Coded paket Soure paket () Finite sliding window Fig. : Examples for the ompared systemati oding shemes with the ode rate R = soure symbol soure symbol soure symbol 3 soure symbol oded symbol - oded symbol - soure symbol 5 soure symbol soure symbol 7 soure symbol 8 oded symbol 5-8 oded symbol 5-8 (a) Blok ode pakets soure symbol soure symbol oded symbol - soure symbol 3 soure symbol oded symbol - soure symbol 5 soure symbol oded symbol 3- soure symbol 7 soure symbol 8 oded symbol 7-8 (b) Sliding window pakets Fig. : Paket format with example pakets aording to the parameters of Fig. for the first pakets. The blok ode sends the generation id in the first olumn while the sliding window ode sends the sequene number. The oeffiient of the latest soure symbol is highlighted in eah row. at the respetive index representing the soure symbol. We note that there are various ways to ompress the header, but we fous on the most simple form here. For a generation-based ode, the oeffiient vetor has the fixed size of g oeffiients. The oeffiient vetor an be enoded as a simple series of oeffiients as depited in Fig. a. For finite sliding window, we propose to plae the oeffiient for symbol i at position (i mod w). This is shown in Fig. b. With this format, eah oeffiient vetor has the same alignment. The oeffiient for a speifi symbol number i an always be found at the same position. This will be utilized by the deoder to effiiently build the deoding matrix. In addition to the raw oeffiient values, the deoder needs to know to whih symbol the oeffiients apply. In the generation based approah, this an be ommuniated with a generation id. For the finite sliding window sheme, the same information is arried by the number of the last soure paket. We refer to this number s as the sequene number. Given this number for a speifi oded paket, the deoder an attribute eah oeffiient to the orret soure symbol. We note that both shemes exhibit an overhead of w e elements of GF (q) plus the sequene/generation id. The oeffiient enoding for the infinite sliding window is disussed in []. C. Deoding In general, the deoding step is the inversion of the enoding. For a generation based sheme, the reeiver ollets g linear independent oeffiient vetors C with the respetive oded symbols Ȳ. The soure symbols an be reonstruted by matrix multipliation: X = C Ȳ. (3) With Gaussian elimination the linear equation system an be solved iteratively. The reeiver maintains a oeffiient matrix C t at time t. When a paket is reeived the oeffiient matrix C t+ is formed as follows. First, the new oeffiient vetor is inserted into the matrix C t, whih we an all C t. Then, Gaussian elimination is used to transform the matrix into redued row ehelon form to yield C t+. If a row in the oeffiient matrix ontains only one nonzero element, the orresponding soure symbol is deoded and an be retrieved. This algorithm an be used to deode eah generation in a generation-based sheme. Similar to the enoder, the deoder an operate on one generation at a time in onstant memory. After a generation is deoded, or a symbol from the next generation is reeived, the matrix an be leared and reused for the next generation. In ontrast, the size of the matrix is not bounded for the infinite sliding window sheme. Every soure symbol ould be required for deoding a single paket. But in general, this is also true for pakets reated from a finite sliding window enoder. There are two ases that allow to redue the size of the matrix by disarding a row: ) Suessfully deode: A deoded row an be removed if it is not required to deode any future symbols. Sine the enoding window is limited by w e, this is guaranteed to happen after at most w e rank inreases of C t. ) Reovery impossible: If the row is not deoded, it is not so easy to dismiss it after w e steps. The redued lower ehelon form an allow deoding of a row even if the leading oeffiient is not overed by the enoding window anymore. But, it is impossible to reover the symbol, if the seond nonzero oeffiient slides out of the enoding window. In this ase the symbol an be removed. However, the number of steps until this requirement is met is not bounded. Beause the design goal for our sheme is to operate in onstant memory, we propose to limit the size of a row in the oeffiient matrix by a onstant deoding window size w d. When the deoder reeives a new oded paket with sequene number s, all rows with a nonzero oeffiient at an index i s w d are removed. This now allows to safely lose the deoding window. The new oeffiient vetor an now be inserted without extending the width of the matrix beyond w d. Before we propose an effiient algorithm for the deoder, we observe that the number of rows to remove is bounded by the differene between the sequene number of the paket s p and the sequene number of the deoder s d. So basially for eah inrement in sequene number at most one row will be removed from the matrix. This is a result of the row ehelon form of the matrix.

4 C t C t+ (a) s p = s d : paket is inserted C t C t+ (b) s p > s d : inrement s d, erase rows with nonzero oeffiients C t C t+ () s p < s d : paket an only be inserted beause of the zero oeffiients Fig. 3: Visualization of the three different ases for inserting a new oded symbol (with paket sequene number s p ) into the deoder oeffiient matrix (with deoder sequene number s d ). The red line represents the index at the latest sequene number. White spaes indiate oeffiients with the value zero. Theorem. A matrix C GF (q) n m in row ehelon form with olumn vetors i, i =,..., n has at most d rows with nonzero elements in the olumns k, k =,..., d. Proof. For a number d n, let r i, i =,..., t denote the rows with nonzero elements in the olumns k, k =,..., d. Beause C is in row ehelon form, eah row r i has a leading oeffiient at index p i. Further, the redued row ehelon form requires that eah leading oeffiient is the only nonzero element in its olumn. Therefore, eah index p i is unique and there are t distint indies. Beause the leading oeffiient must be in one of the olumns k, k =,..., d and there are t distint index values p i, i =,..., t it follows that t d. For an effiient implementation, we propose to set w d = w e. In ombination with the aligned oeffiient enoding proposed in III-B, this allows to use an existing generation-based deoder as a base implementation for a finite sliding window deoder. Some modifiations have to be made to support the infinite stream of inoming pakets: when a new paket arrives the matrix has to be prepared based on the sequene number of the paket s p and the sequene number of the deoder s d. s p > s d (new paket): the paket an only be used, if all oeffiients outside of the deoder window are zero. s p < s d (old paket): the deoders sequene number is set to the pakets sequene number. This triggers the removal of rows with oeffiients at indies before s p w d. s p = s d (same age): no additional ations are required. After this preproessing, the paket an be diretly inserted into the deoding matrix. Then the deoder applies Gaussian Elimination to bring the matrix into a permuted redued lower ehelon form. The physial representation of the matrix stays in a fixed memory segment while the logial oeffiient matrix is mapped onto this memory in the same way as the oeffiients are mapped in III-B. This is illustrated in Fig. 3. We note that the desribed steps require only a onstant number of operations per sequene number inrement. Therefore, the omputational omplexity of the deoder is of the same order as a equally sized linear blok deoder. IV. SIMULATION RESULTS For an evaluation of the finite sliding window oding sheme, we implemented a time-slotted, stohasti simulator. The simulation models a sender-reeiver proess with a binary erasure hannel. We measured the paket loss and the in-order delay as performane indiators. A paket is onsidered lost if it has not been deoded when it is shifted out of the deoding window, or the next generation was started. Delivery is generally assumed in-order. Therefore, the deoding proess waits at the head of the line until the next paket is reeived or annot be reovered anymore. The inorder delay is measured as the number of time slots between the paket being first sent and its in order delivery. In the simulator, we assumed an infinite field size, whih avoids linear dependenies of the oeffiient vetors. As a result, every oded paket inreases the rank of an inomplete deoding matrix. In pratie, a suffiiently large finite field an be used to ahieve omparable results. Comparisons between the simulator and a proof of onept implementation with GF ( 8 ) onfirmed that. A more in-depth analysis on the matter an be found in [7]. We used a Gilbert-Elliot hannel [], [5] with the transition probability matrix [ ] γ γ P = () β β with γ denoting the probability to transition from the good state, where all pakets arrive, to the bad state, where all pakets are lost. The parameter β respetively denotes the probability to swith from the bad state to the good state. We derive the parameters γ and β from the steady-state γ γ+β distribution for the bad state π B = and the expeted number of onseutive paket erasures E[L] = β. With this hannel we simulated an open-loop system, where the reeiver does not send feedbak. Feedbak is not stritly neessary to make those systems work, but ould be used to lose the window (in the sliding window methods), request more redundant pakets or provide hannel information. Without feedbak the infinite sliding window does not have any mean to lose the window, and annot be implemented in pratie on memory limited mahines. We used the infinite sliding window only to ompare its performane to other shemes as a theoretial bound. Using this model, we ompared the three different oding shemes systemati blok odes, infinite sliding window and finite sliding window as introdued in Setion III. In eah run of the simulation, 8 pakets were sent and for eah onfiguration, the experiment was repeated times for improved statistial auray.

5 fsw blok fsw 3 blok 3 fsw blok fsw blok fsw 3 blok 3 fsw blok fsw blok fsw 3 blok 3 fsw blok Paket loss ɛ 3 5 Paket loss ɛ 3 5 Paket loss ɛ Code rate R Code rate R Code rate R (a) E[L] = (b) E[L] = () E[L] = 8 Fig. : Paket loss probability as a funtion of the ode rate for infinite sliding window (), various finite sliding window (fsw) and blok ode sizes in a hannel with paket loss probability π B = 5% and different expeted burst lengths E[L] In-Order Delay 5 3 fsw blok fsw 3 blok 3 fsw blok In-Order Delay 5 3 fsw blok fsw 3 blok 3 fsw blok In-Order Delay 5 3 fsw blok fsw 3 blok 3 fsw blok Code Rate Code Rate Code Rate (a) E[L] = (b) E[L] = () E[L] = 8 Fig. 5: Mean in-order delay as a funtion of the ode rate for infinite sliding window (), various finite sliding window (fsw) and blok ode sizes in a hannel with paket loss probability π B = 5% and different expeted burst lengths E[L] A. Paket Loss The results for the paket loss of the shemes for various burst lengths are illustrated in Fig.. The width of the onfidene interval for eah point is less than.% and is omitted for improved visibility. For g = w e, the finite sliding window and the systemati blok ode performed very similar, but we observe a small advantage for the finite sliding window oding sheme. With an inreasing generation size, respetively window size, the performane gets loser to the infinite sliding window. The infinite sliding window ahieved the best results for all suffiient ode rates. Only when the ode rate is higher than the theoretial apaity of the hannel (C = π B pakets per time slot) the infinite sliding window was unable to fix any losses. B. In-order Delay In Fig. 5, the simulation results for the in-order delay are summarized. The graphs show the mean values with a onfidene interval width of less than.3 time slots. The mean delay for the blok ode was mostly onstant for the given hannel properties. With low ode rates the sliding window odes ahieve lower delays. But with an inreasing ode rate the delay of the sliding window odes surpasses the blok odes. Looking at the delay alone an be misleading. Beause the delay of a lost paket annot be represented in a mean value, a low delay ould be ahieved by aepting many losses. For this reason, we also present our results in reliability CDF graphs. The lost pakets an be inluded by defining their delay as infinite. CDFs for speifi ode rates are presented in Fig.. The rates were seleted as the highest rate for whih the paket error probability of the finite sliding window of window size is below.%. The results onfirm that the finite sliding window showed similar delay properties like the infinite sliding window. The urve for the finite sliding window is ut off at a maximum delay. This effet of the limited deoding window has been disussed in more detail in [3]. Compared to the blok odes, the finite sliding window shows the lower delay. For the blok odes an inreasing generation size inreases the delay, but redues the error probability. The finite sliding window does not show this trade-off. Inreasing the window size improves the reliability without sarifiing the delay. C. Computing Performane To validate the pratiality of our approah, we evaluated the omputational effort required by the sheme. For the systemati blok ode, we used the implementation provided by the KODO library []. The finite sliding window was implemented on top of this with the modifiations desribed in III. The infinite sliding window was not evaluated here, beause of its high omputational omplexity when used without feedbak. We evaluated the blok ode and the finite sliding window within the previously desribed system model, and the time to ompletion was measured. This measurement inludes the time required by the enoder and the deoder. The

6 Cumulative Probability fsw blok fsw 3 blok 3 fsw blok Cumulative Probability fsw blok fsw 3 blok 3 fsw blok Cumulative Probability fsw blok fsw 3 blok 3 fsw blok Delay Delay Delay (a) E[L] =, ode rate.8 (b) E[L] =, ode rate.7 () E[L] = 8, ode rate.5 Fig. : CDFs of the paket in-order delay for infinite sliding window (), various finite sliding window (fsw) and blok ode sizes in a hannel with π B = 5% and different expeted burst lengths E[L] Duration 8 fsw blok fsw 3 blok 3 fsw blok Code Rate R Fig. 7: Duration for enoding and deoding for various ode rates and π B = 5%, E[L] = tests were performed on a Intel NUCi5SYH, whih features a.8ghz Core i5 proessor and 8 GB RAM. The mean duration for runs, eah with 8 pakets, is depited in Fig. 7. The width of the onfidene interval for all results was below ms and is omitted from the graphis for improved visibility. While the finite sliding window performs slightly worse than the blok ode in this metri, the additional runtime ost is not prohibitive. Also the results suggest that both shemes sale very similarly. With inreasing ode rate less pakets will be generated, whih speeds up the proess. Also with inreasing generation size, respetively window size, the omplexity of the enoding and deoding dereases. V. CONCLUSION In this work we proposed a finite sliding window RLNC as an alternative to generation based RLNC for streaming appliations. We ompared the finite sliding window RLNC to an infinite sliding window and a generation based RLNC. Our evaluation showed that the finite sliding window ombines the benefits of both other shemes. The reliability and omputational overhead is omparable to the generation based ode, but it ahieves the superior delays of the infinite sliding window sheme. We therefore argue that this sheme should supersede the use of blok based RLNC for streaming appliations like VoIP and video streaming. This initial work should be omplemented by onsidering feedbak shemes, whih is known to improve performane. We will also investigate reoding and reordering of pakets, whih is of essential importane for multi-hop and multipath networks, like mesh networks. ACKNOWLEDGEMENTS This artile is based upon work supported by Erisson, Deutshe Telekom, and the Free State of Saxony through funds from the European Commission for the Atto3D projet. REFERENCES [] T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi, and B. Leong, A random linear network oding approah to multiast, IEEE Transations on Information Theory, vol. 5, no., pp. 3 3,. [] T. Ho, R. Koetter, M. Medard, D. R. Karger, and M. Effros, The benefits of oding over routing in a randomized setting, 3. [3] P. Pahlevani, M. Hundebøll, M. V. Pedersen, D. E. Luani, H. Charaf, F. H. Fitzek, H. Bagheri, and M. Katz, Novel onepts for devie-todevie ommuniation using network oding, IEEE Communiations Magazine, vol. 5, no., pp. 3 39, Apr.. [] C. Fragouli, J.-Y. Le Boude, and J. Widmer, Network oding: an instant primer, ACM SIGCOMM Computer Commun. Rev., vol. 3, no., pp. 3 8, Jan.. [5] C. Gkantsidis and P. R. Rodriguez, Network oding for large sale ontent distribution, in Pro. IEEE Infoom, vol., 5, pp [] P. A. Chou, Y. Wu, and K. Jain, Pratial network oding, 3. [7] M. Karzand, D. J. Leith, J. Cloud, and M. Medard, Low delay random linear oding over a stream, arxiv preprint arxiv:59.7, 5. [8] P. Maymounkov, N. J. Harvey, and D. S. Lun, Methods for effiient network oding, in Pro. th Annual Allerton Conferene on Communiation, Control, and Computing,, pp [9] A. Heidarzadeh and A. H. Banihashemi, Overlapped hunked network oding, in Information Theory (ITW, Cairo), IEEE Information Theory Workshop on. IEEE,, pp. 5. [] M. Halloush and H. Radha, Network oding with multi-generation mixing, in Information Sienes and Systems, 8. CISS 8. nd Annual Conferene on. IEEE, 8, pp [] J. Cloud and M. Médard, Network oding over satom: Lessons learned, in International Conferene on Wireless and Satellite Systems. Springer, 5, pp [] J. Cloud and M. Medard, Multi-path low delay network odes, arxiv preprint arxiv:9.,. [3] V. Roa, B. Teibi, C. Burdinat, T. Tran, and C. Thienot, Blok or onvolutional al-fe odes? a performane omparison for robust lowlateny ommuniations,. [] E. N. Gilbert, Capaity of a burst-noise hannel, Bell system tehnial journal, vol. 39, no. 5, pp. 53 5, 9. [5] E. O. Elliott, Estimates of error rates for odes on burst-noise hannels, Bell system tehnial journal, vol., no. 5, pp , 93. [] M. V. Pedersen, J. Heide, and F. H. Fitzek, Kodo: An open and researh oriented network oding library, in International Conferene on Researh in Networking. Springer,, pp