Linear network coding - Knowledge (XXG)

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uniform distribution from a Galois field. If the field size is sufficiently large, the probability that the receiver(s) will obtain linearly independent combinations (and therefore obtain innovative information) approaches 1. It should however be noted that, although random linear network coding has excellent throughput performance, if a receiver obtains an insufficient number of packets, it is extremely unlikely that they can recover any of the original packets. This can be addressed by sending additional random linear combinations until the receiver obtains the appropriate number of packets.

1975:), then the expected number of linearly dependent packets per generation is practically zero. Since it is the last packets the major contributor to the overhead due to linear dependencies, there are RLNC-based protocols such as tunable sparse network coding that exploit this knowledge. These protocols introduce sparsity (zero-elements) in the coding coefficients at the beginning of the transmission to reduce the decoding complexity, and reduce the sparsity at the end of the transmission to reduce the overhead due to linear dependencies. 1132:

over traditional block codes is that it allows the recombination of previously coded packets into new and valid coded packets. This process is usually called recoding. After a recoding operation, the size of the appended coding coefficients does not change. Since all the operations are linear, the state of the recoded packet can be preserved by applying the same operations of addition and multiplication to the payload and the appended coding coefficients. In the following example, we will illustrate this process.

1124:). In the binary field, each element is one bit long, while in the binary-8, it is one byte long. Since the packet size is usually larger than the field size, each packet is seen as a set of elements from the Galois field (usually referred to as symbols) appended together. The packets have a fixed amount of symbols (Galois field elements), and since all the operations are performed over Galois fields, then the size of the packets does not change with subsequent linear combinations. 1744: 1704: 987: 5247: 2023:

together with the coded packets, a source can achieve bundling without much planning just by sending coded packets through all its network interfaces. The destination can decode the information once enough coded packets arrive, irrespectively of the network interface. A video demonstrating the channel bundling capabilities of RLNC is available at.

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traffic in terms of delay, packet loss, and jitter over wireless networks prone to packet erasures can be improved with RLNC. In the case of audio streaming over wireless mesh networks, the packet delivery ratio, latency, and jitter performance of the network can be significantly increased when using

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coding coefficients. The size of each coefficient is the number of bits needed to represent one element of the Galois field. In practice, most network coding applications use a generation size of no more than 32 packets per generation and Galois fields of 256 elements (binary-8). With these numbers,

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To preserve the statelessness of the code, the coding coefficients used to generate the coded packets are appended to the packets transmitted over the network. Therefore, each node in the network can see what coefficients were used to generate each coded packet. One novelty of linear network coding

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packets form a block, usually called a generation. The number of original packets combined and recombined together is the generation size. The second parameter is the packet size. Usually, the size of the original packets is fixed. In the case of unequally-sized packets, these can be zero-padded if

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The sources and the intermediate nodes can combine any subset of the original and previously coded packets performing linear operations. To form a coded packet in RLNC, the original and previously coded packets are multiplied by randomly chosen coefficients and added together. Since each packet is

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Random linear network coding (RLNC) is a simple yet powerful encoding scheme, which in broadcast transmission schemes allows close to optimal throughput using a decentralized algorithm. Nodes transmit random linear combinations of the packets they receive, with coefficients chosen randomly, with a

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It is usually thought that the transmission overhead of network coding is high due to the need to append the coding coefficients to each coded packet. In reality, this overhead is negligible in most applications. The overhead due to coding coefficients can be computed as follows. Each packet has

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If only routing were allowed, then the central link would be only able to carry A or B, but not both. Supposing we send A through the center; then the left destination would receive A twice and not know B at all. Sending B poses a similar problem for the right destination. We say that routing is

998:. Two source nodes (at the top of the picture) have information A and B that must be transmitted to the two destination nodes (at the bottom). Each destination node wants to know both A and B. Each edge can carry only a single value (we can think of an edge transmitting a bit in each time slot). 1754:

Since the coding coefficients are chosen randomly in RLNC, there is a chance that some transmitted coded packets are not beneficial to the destination because they are formed using a linearly dependent combination of packets. However, this overhead is negligible in most applications. The linear

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Channel bundling: Due to the statelessness characteristics of RLNC, it can be used to efficiently perform channel bundling, i.e., the transmission of information through multiple network interfaces. Since the coded packets are randomly generated, and the state of the code traverses the network

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Coding and recoding process in linear network coding. Each packet is seen as a set of elements from a Galois field. Therefore, multiplying and adding two packets means multiplying each of its symbols by a coding coefficient chosen from the Galois field and then adding the two packets together,

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Using a simple code, as shown, A and B can be transmitted to both destinations simultaneously by sending the sum of the symbols through the two relay nodes – encoding A and B using the formula "A+B". The left destination receives A and A + B, and can calculate B by subtracting the two values.

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Any destination node must collect enough linearly independent coded packets to be able to reconstruct the original data. Each coded packet can be understood as a linear equation where the coefficients are known since they are appended to the packet. In these equations, each of the original

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There are three key parameters in RLNC. The first one is the generation size. In RLNC, the original data transmitted over the network is divided into packets. The source and intermediate nodes in the network can combine and recombine the set of original and coded packets. The original

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RLNC instead of packet forwarding-based protocols such as simplified multicast forwarding and partial dominant pruning. The performance improvements of network coding for video conferencing are not only theoretical. In 2016, the authors of built a real-world testbed of 15 wireless

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instructions enabled, the decoding goodput of network coding was 750 MB/s for a generation size of 16 packets and 250 MB/s for a generation size of 64 packets. Furthermore, today's algorithms can be vastly parallelizable, increasing the encoding and decoding goodput even further.

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devices to evaluate the feasibility of network-coding-based video conference systems. Their results showed large improvements in packet delivery ratio and overall user experience, especially over poor quality links compared to multicasting technologies based on packet

4134: 2103:.: Linear network coding can improve the network efficiency of information-centric networking solutions by exploiting the multi-source multi-cast nature of such systems. It has been shown, that RLNC can be integrated into distributed content delivery networks such as 1815:

degrees of freedom, and then almost any coded packet will be useful and innovative. In fact, only the zero-packet (only zeroes in the coding coefficients) will be non-innovative. The probability of generating the zero-packet is equal to the probability of each of the

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wireless networks can benefit from network coding. Researchers showed that network coding and its channel bundling capabilities improved the performance of SD-WANs with a large number of nodes with multiple cellular connections. Nowadays, companies such as

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Noor-A-Rahim, Md; Liu, Zilong; Lee, Haeyoung; Khyam, M. Omar; He, Jianhua; Pesch, Dirk; Moessner, Klaus; Saad, Walid; Poor, H. Vincent (2022-05-01). "6G for Vehicle-to-Everything (V2X) Communications: Enabling Technologies, Challenges, and Opportunities".

1949:), the expected number of linearly dependent packets per generation is of 1.6 extra packets. If our generation size if of 32 or 64 packets, this represents an overhead of 5% or 2.5%, respectively. If we use the binary-8 field ( 966:

was proposed to find such paths in polynomial time. Then, Edmonds proved in the paper "Edge-Disjoint Branchings" the upper bound in the broadcast scenario is also achievable, and proposed a polynomial time algorithm.

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just an appended set of Galois field elements, the operations of multiplication and addition are performed symbol-wise over each of the individual symbols of the packets, as shown in the picture from the example.

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Torres Vega, Maria; Liaskos, Christos; Abadal, Sergi; Papapetrou, Evangelos; Jain, Akshay; Mouhouche, Belkacem; Kalem, Gökhan; Ergüt, Salih; Mach, Marian; Sabol, Tomas; Cabellos-Aparicio, Albert (October 2020).

1331:, which means that they are constructed by a linear combination of zero times one of the packets plus one time the other packet. Any coded packet would have appended other coefficients. In our example, packet 1606:

Linear network coding is still a relatively new subject. However, the topic has been vastly researched over the last twenty years. Nevertheless, there are still some misconceptions that are no longer valid:

1053:. The third key parameter is the Galois field used. In practice, the most commonly used Galois fields are binary extension fields. And the most commonly used sizes for the Galois fields are the binary field 1755:

dependencies depend on the Galois fields' size and are practically independent of the generation size used. We can illustrate this with the following example. Let us assume we are using a Galois field of

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Ali Farzamnia, Sharifah K. Syed-Yusof, Norsheila Fisa "Multicasting Multiple Description Coding Using p-Cycle Network Coding", KSII Transactions on Internet and Information Systems, Vol 7, No 12, 2013.

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Similarly, the right destination will receive B and A + B, and will also be able to determine both A and B. Therefore, with network coding, it takes only three time slots and improves the throughput.

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and a generation size of 16 packets. At the beginning of the transmission, the linear dependencies are minimal. It is the last packet of the transmission that is more likely to be linearly dependent.

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Over the years, multiple researchers and companies have integrated network coding solutions into their applications. We can list some of the applications of network coding in different areas:

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ideas. Ahlswede et al. proved that it can be achieved if additional computing tasks (incoming packets are combined into one or several outgoing packets) can be done in the intermediate nodes.

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insufficient because no routing scheme can transmit both A and B to both destinations simultaneously. Meanwhile, it takes four time slots in total for both destination nodes to know A and B.

1923:. We can use this knowledge to calculate the expected number of linearly dependent packets per generation. In the worst-case scenario, when the Galois field used contains only two elements ( 1399:. Since network coding can be applied at any layter of the communication protocol, these packets can have a header from the other layers, which is ignored in the network coding operations. 4205:. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering. Vol. 256. Cham: Springer International Publishing. pp. 68–77. 1870:. With each successive innovative transmission, it can be shown that the exponent of the probability of a non innovative packet is reduced by one. When the destination has received 748: 75:

is enough to achieve the upper bound in multicast problems with one source. However linear coding is not sufficient in general; even for more general versions of linearity such as

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INFOCOM 2008. 27th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 13–18 April 2008, Phoenix, AZ, USA

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Network coding decoders have been improved over the years. Nowadays, the algorithms are highly efficient and parallelizable. In 2016, with Intel Core i5 processors with

1122: 644: 585: 526: 487: 204: 1267:) appended. The appended coefficients can take any value from the Galois field. However, an original, uncoded data packet would have appended the coding coefficients 1265: 942: 863: 690: 1973: 1737: 1595: 1546: 1497: 1468: 1083: 553: 319: 234: 5314: 4132:, Buga, Wladyslaw Jan & Trent, Tracy Raymond, "Systems and methods for prioritizing wireless communication of aircraft", issued 2013-03-19 3738: 2547: 1947: 1894: 3583:

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Wireless broadcast: RLNC can reduce the number of packet transmission for a single-hop wireless multicast network, and hence improve network bandwidth

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In the figure, we can see an example of two packets linearly combined into a new coded packet. In the example, we have two packets, namely packet

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innovative packets (i.e., it needs only one more packet to fully decode the data). Then the probability of a non innovative packet is of

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bytes of appended overhead. If each packet is 1500 bytes long (i.e. the Ethernet MTU), then 32 bytes represent an overhead of only 2%.

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Sink nodes receive these network coded messages, and collect them in a matrix. The original messages can be recovered by performing

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are employing RLNC-based solutions due to their advantages in low latency, small footprint on computing devices, and low overhead.

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coding coefficient to be equal to the zero-element of the Galois field. I.e., the probability of a non-innovative packet is of

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sink nodes. Each node generates new packets which are linear combinations of past received packets by multiplying them by

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they are shorter or split into multiple packets if they are longer. In practice, the packet size can be the size of the

2586: 83:. Finding optimal coding solutions for general network problems with arbitrary demands is a hard problem, which can be 5394: 3536:"Instantly decodable network coding–assisted data dissemination for prioritized services in vehicular ad hoc networks" 2042: 5277: 5273: 5257: 3660:. Lecture Notes in Computer Science. Vol. 6164. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 76–86. 2895:

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The expected number of linearly dependent packets per generation is practically independent of the generation size.

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standard to improve the performance of video delivery over 5G systems. In 2018, a demo presented at the

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is a program in which intermediate nodes transmit data from source nodes to sink nodes by means of

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Any receiver can then recover the source vectors using Gaussian elimination on the vectors in its

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scenario is more complicated, and in fact, such an upper bound can't be reached using traditional

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technologies to improve video quality against packet loss due to congestion at the core network.

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packets is the unknown. To solve the linear system of equations, the destination needs at least

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Protection against network attacks such as snooping, eavesdropping, replay, or data corruption.

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on the matrix. In reduced row echelon form, decoded packets correspond to the rows of the form

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packets and combine for transmission. This process may be used to attain the maximum possible

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Now, lets assume that the network node wants to produce a new coded packet combining packet

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Expected linearly dependent packets at different stages of transmission for a Galois field

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The butterfly network is often used to illustrate how linear network coding can outperform

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Precision Agriculture in Latin America. 3737:: CS1 maint: location missing publisher ( 1176:linearly independent equations (packets). 5296:Learn how and when to remove this message 5184: 4945: 4888: 4843: 4832:IEEE Transactions on Vehicular Technology 4782:Bhadra, S.; Shakkottai, S. (April 2006). 4603: 4456: 4366: 4263: 3952: 3934: 3795: 3777: 3665: 3613: 3588: 3551: 3497:. Paris, France: IEEE. pp. 441–446. 3473: 3435: 3294: 3275:Journal of Network and Systems Management 3228: 3139: 3129: 2439: 2365: 2318: 2193:Homomorphic signatures for network coding 1954: 1928: 1903: 1901: 1875: 1852: 1843: 1841: 1821: 1800: 1780: 1760: 1712: 1669: 1651: 1630: 1581: 1575: 1554: 1532: 1526: 1505: 1483: 1477: 1454: 1448: 1427: 1407: 1380: 1367: 1358: 1337: 1304: 1272: 1248: 1242: 1221: 1199: 1161: 1141: 1105: 1090: 1058: 1029: 914: 890: 870: 835: 815: 795: 775: 755: 710: 709: 707: 664: 658: 630: 625: 619: 598: 592: 571: 566: 560: 539: 533: 509: 494: 473: 468: 462: 438: 425: 420: 410: 399: 386: 380: 356: 345: 335: 326: 305: 299: 269: 245: 220: 214: 187: 172: 144: 124: 104: 4754:"NetworkCoding - batman-adv - Open Mesh" 4096:Computers and Electronics in Agriculture 1742: 1702: 1183: 71:It has been proven that, theoretically, 5220:https://doi.org/10.1145/1111322.1111337 2428:IEEE Transactions on Information Theory 2307:IEEE Transactions on Information Theory 2272:IEEE Transactions on Information Theory 2255:. 15th ACM-SIAM SODA. pp. 142–150. 2209: 4825: 4823: 4414:. Taipei, Taiwan: IEEE. pp. 1–6. 3730: 3161: 3159: 2251:Rasala Lehman, A.; Lehman, E. (2004). 743:{\displaystyle {\mathcal {G}}=(V,E,C)} 5375:http://scholarcommons.sc.edu/etd/230/ 5126:IEEE Transactions on Mobile Computing 3174:. Las Vegas, NV: IEEE. pp. 1–2. 3107: 3105: 3037: 3035: 3010: 3008: 2983: 2981: 2890: 2888: 2811: 2809: 2763: 2761: 2715: 2713: 2006:Software-defined wide area networks ( 119:are involved in moving the data from 7: 4250:. Montreal, QC: IEEE. pp. 1–6. 2057:Remote vehicle driving applications. 1752:Overhead due to linear dependencies: 5352:http://www.icsi.berkeley.edu/~luby/ 4472:Larsson, P.; Johansson, N. (2006). 810:gives the capacity of each link of 365:{\displaystyle \{M_{i}\}_{i=1}^{S}} 4420:10.1109/GLOBECOM42002.2020.9322478 1863:{\displaystyle {\frac {1}{q^{M}}}} 1775:elements and a generation size of 1611:Decoding computational complexity: 25: 4203:Smart Grid and Internet of Things 2084:In-flight entertainment networks. 770:is the set of nodes or vertices, 5245: 4899:10.1109/LCOMM.2013.031313.121994 4764:from the original on 12 May 2021 2686:10.1109/CSNDSP16145.2010.5580304 4934:IEEE Transactions on Multimedia 4522:securenetworkcoding.wikidot.com 3217:IEEE Transactions on Multimedia 3050:. Sevilla: IEEE. pp. 1–5. 2555:IEEE Internet of Things Journal 1693:{\displaystyle M*log_{2}(s)=32} 489:are coefficients selected from 5364:Introduction to Network Coding 4677:10.1109/VTCSpring.2013.6692495 2954:10.1109/ANTS47819.2019.9117981 2909:10.1109/IC3I46837.2019.9055565 2466:, D. R. Karger and M. Effros, 2097:Information-centric networking 1916:{\displaystyle {\frac {1}{q}}} 1726: 1720: 1681: 1675: 1386: 1360: 1318: 1306: 1286: 1274: 1111: 1098: 1072: 1066: 970:However, the situation in the 931: 919: 852: 840: 737: 719: 699:A network is represented by a 679: 673: 515: 502: 287:{\displaystyle InDeg(p_{k})=S} 275: 262: 193: 180: 1: 5315:A network coding bibliography 5216:Computer Communication Review 2353:Israel Journal of Mathematics 607:{\displaystyle k^{\text{th}}} 555:along with the coefficients, 4550:. IEEE. pp. 1409–1417. 4311:10.1007/978-3-030-37629-1_48 4108:10.1016/j.compag.2007.01.019 3639:Dammak, Marwa (2018-11-20). 2774:2011 IFIP Wireless Days (WD) 2734:10.1109/NUICONE.2012.6493203 2054:remote support and training. 1549:, and each symbol of packet 1085:and the so-called binary-8 ( 1019:Operation and key parameters 1010:Random Linear Network Coding 5044:10.1109/GLOCOM.2015.7417590 4877:IEEE Communications Letters 4486:10.1109/VETECS.2006.1683207 4211:10.1007/978-3-030-05928-6_7 3833:10.1109/PIMRCW.2010.5670413 3667:10.1007/978-3-642-13971-0_8 3615:10.1007/978-3-642-13971-0_8 2834:10.1016/j.adhoc.2016.03.002 2617:10.1109/NETCOD.2014.6892129 2161:Spatial buffer multiplexing 2081:Agricultural sensor fields. 27:Computer Networking Program 5426: 5086:10.1109/SPAWC.2015.7227122 4385:10.1109/JSYST.2018.2862913 4256:10.1109/PIMRC.2017.8292785 4166:10.1109/ICCCN.2017.8038415 3878:10.1109/ECMSM.2017.7945879 3287:10.1007/s10922-020-09545-w 2407:(or more) received packets 209:More formally, each node, 5186:10.1049/iet-com.2014.0323 3503:10.1109/ICCW.2017.7962697 3437:10.1007/s13174-010-0006-7 3389:10.1007/s11276-016-1294-z 3180:10.1109/CCNC.2018.8319320 2861:10.1109/ICME.2009.5202786 2567:10.1109/JIOT.2017.2703813 2511:10.1109/WCNC.2016.7565066 2483:10.1109/ISIT.2003.1228459 2376:10.1007/s11856-022-2345-z 2198:Triangular network coding 2116:automatic repeat requests 2035:Consumer Electronics Show 1117:{\displaystyle GF(2^{8})} 1047:maximum transmission unit 639:{\displaystyle g_{k}^{i}} 580:{\displaystyle g_{k}^{i}} 521:{\displaystyle GF(2^{s})} 482:{\displaystyle g_{k}^{i}} 199:{\displaystyle GF(2^{s})} 5350:Digital Fountain Codes, 5231:10.3837/tiis.2013.12.009 5138:10.1109/TMC.2016.2642190 4999:10.1109/ICC.2015.7248631 4956:10.1109/TMM.2013.2285518 4796:10.1109/INFOCOM.2006.266 4556:10.1109/INFOCOM.2008.199 3553:10.1177/1550147719842137 3239:10.1109/TMM.2013.2241415 3056:10.1109/ITW.2013.6691231 2329:10.1109/TIT.2023.3247570 2168:Distributed file sharing 2112:forward error correction 2069:Healthcare applications. 964:Ford–Fulkerson algorithm 960:max-flow min-cut theorem 907:max-flow min-cut theorem 5410:Wireless sensor network 5344:Random Network Coding, 5310:Network Coding Homepage 4854:10.1109/TVT.2008.927729 4722:10.1109/INFCOM.2007.124 4636:10.1145/1159913.1159942 4059:10.1145/3291078.3291098 3990:10.1145/3365871.3365903 2782:10.1109/WD.2011.6098167 2284:10.1109/TIT.2006.874434 2188:Secret sharing protocol 958:scenario, known as the 5356:Coding-Aware Routing, 5338:Avalanche Filesystem, 4993:. pp. 2072–2077. 4716:. pp. 1028–1036. 3546:(4): 155014771984213. 3340:10.1109/SMC-IT.2017.29 2855:. pp. 1492–1495. 1969: 1943: 1917: 1890: 1864: 1830: 1809: 1789: 1769: 1748: 1740: 1733: 1694: 1639: 1622:Transmission Overhead: 1591: 1563: 1542: 1514: 1493: 1464: 1436: 1416: 1393: 1346: 1325: 1293: 1261: 1260:{\displaystyle C_{ij}} 1230: 1208: 1190: 1170: 1150: 1118: 1079: 1038: 991: 938: 937:{\displaystyle T(s,t)} 899: 879: 859: 858:{\displaystyle T(s,t)} 824: 804: 784: 764: 744: 686: 685:{\displaystyle e_{i}=} 640: 608: 581: 549: 522: 483: 448: 415: 366: 315: 294:, generates a message 288: 230: 200: 153: 133: 113: 5362:MIT offers a course: 2101:named data networking 2048:Remote collaboration. 1970: 1968:{\displaystyle q=256} 1944: 1918: 1891: 1865: 1831: 1810: 1790: 1770: 1746: 1734: 1732:{\displaystyle GF(2)} 1706: 1695: 1640: 1592: 1590:{\displaystyle d_{2}} 1564: 1543: 1541:{\displaystyle d_{1}} 1515: 1494: 1492:{\displaystyle d_{2}} 1465: 1463:{\displaystyle d_{1}} 1437: 1417: 1394: 1347: 1326: 1294: 1262: 1231: 1209: 1187: 1171: 1151: 1119: 1080: 1078:{\displaystyle GF(2)} 1039: 989: 982:The Butterfly Network 939: 900: 880: 860: 825: 805: 785: 765: 745: 687: 641: 609: 582: 550: 548:{\displaystyle X_{k}} 523: 484: 449: 395: 367: 316: 314:{\displaystyle X_{k}} 289: 231: 229:{\displaystyle p_{k}} 201: 154: 134: 114: 95:Encoding and decoding 36:linear network coding 5266:improve this article 5080:. pp. 670–674. 4355:IEEE Systems Journal 3827:. pp. 134–138. 2151:Coding-aware routing 2133:Avalanche filesystem 2010:): Large industrial 1953: 1927: 1900: 1874: 1840: 1820: 1799: 1779: 1759: 1711: 1650: 1629: 1574: 1553: 1525: 1504: 1476: 1447: 1426: 1406: 1357: 1336: 1303: 1271: 1241: 1220: 1198: 1160: 1140: 1089: 1057: 1028: 913: 889: 869: 834: 814: 794: 774: 754: 706: 657: 651:Gaussian elimination 618: 591: 559: 532: 493: 461: 379: 325: 298: 244: 213: 171: 167:, typically of size 143: 123: 103: 77:convolutional coding 5405:Network performance 5278:footnote references 4377:2019ISysJ..13.1376B 3927:2019Senso..19.1673P 3770:2014Senso..14.4806K 3141:10.3390/info9040072 2462:T. Ho, R. Koetter, 2138:Distributed storage 2078:Satellite networks. 1942:{\displaystyle q=2} 1889:{\displaystyle M-1} 635: 576: 478: 430: 361: 40:linear combinations 32:computer networking 5395:Information theory 5173:IET Communications 3779:10.3390/s140304806 3701:. Boca Raton, FL. 2473:2017-10-31 at the 1965: 1939: 1913: 1886: 1860: 1826: 1805: 1785: 1765: 1749: 1741: 1729: 1690: 1646:each packet needs 1635: 1587: 1559: 1538: 1510: 1489: 1460: 1432: 1412: 1389: 1342: 1321: 1289: 1257: 1226: 1204: 1191: 1166: 1146: 1114: 1075: 1034: 992: 990:Butterfly Network. 934: 895: 875: 855: 820: 800: 780: 760: 740: 682: 636: 621: 604: 577: 562: 545: 518: 479: 464: 444: 416: 362: 341: 311: 284: 226: 196: 149: 129: 109: 81:filter-bank coding 53:of a network take 5306: 5305: 5298: 5132:(10): 2697–2711. 5095:978-1-4799-1931-4 5053:978-1-4799-5952-5 5008:978-1-4673-6432-4 4790:. pp. 1–12. 4758:www.open-mesh.org 4731:978-1-4244-1047-7 4686:978-1-4673-6337-2 4645:978-1-59593-308-9 4621:"XORs in the air" 4429:978-1-7281-8298-8 4320:978-3-030-37628-4 4275:978-1-5386-3529-2 4220:978-3-030-05928-6 4175:978-1-5090-2991-4 4068:978-1-4503-6577-2 3999:978-1-4503-7207-7 3936:10.3390/s19071673 3887:978-1-5090-5582-1 3842:978-1-4244-9117-9 3716:978-1-4665-8207-1 3707:10.1201/b17124-13 3677:978-3-642-13970-3 3625:978-3-642-13971-0 3512:978-1-5090-1525-2 3377:Wireless Networks 3349:978-1-5386-3462-2 3189:978-1-5386-4790-5 3065:978-1-4799-1321-3 2963:978-1-7281-3715-5 2918:978-1-7281-5529-6 2870:978-1-4244-4290-4 2791:978-1-4577-2028-4 2743:978-1-4673-1719-1 2695:978-1-4244-8858-2 2626:978-1-4799-6217-4 2520:978-1-4673-9814-5 2450:10.1109/18.850663 2052:Augmented reality 1911: 1858: 1829:{\displaystyle M} 1808:{\displaystyle M} 1788:{\displaystyle M} 1768:{\displaystyle q} 1638:{\displaystyle M} 1562:{\displaystyle e} 1513:{\displaystyle f} 1435:{\displaystyle e} 1415:{\displaystyle f} 1345:{\displaystyle f} 1229:{\displaystyle e} 1207:{\displaystyle f} 1169:{\displaystyle M} 1149:{\displaystyle M} 1037:{\displaystyle M} 898:{\displaystyle t} 878:{\displaystyle s} 823:{\displaystyle E} 803:{\displaystyle C} 783:{\displaystyle E} 763:{\displaystyle V} 601: 457:Where the values 152:{\displaystyle K} 132:{\displaystyle S} 112:{\displaystyle P} 16:(Redirected from 5417: 5301: 5294: 5290: 5287: 5281: 5249: 5248: 5241: 5207: 5206: 5188: 5164: 5158: 5157: 5123: 5114: 5108: 5107: 5073: 5064: 5058: 5057: 5038:. pp. 1–6. 5027: 5021: 5020: 4982: 4976: 4975: 4949: 4925: 4919: 4918: 4892: 4872: 4866: 4865: 4847: 4827: 4818: 4817: 4779: 4773: 4772: 4770: 4769: 4750: 4744: 4743: 4705: 4699: 4698: 4671:. pp. 1–6. 4664: 4658: 4657: 4625: 4616: 4610: 4609: 4607: 4595: 4589: 4588: 4586: 4584: 4575: 4566: 4560: 4559: 4539: 4533: 4532: 4530: 4528: 4514: 4508: 4507: 4474:"Multi-User ARQ" 4469: 4463: 4462: 4460: 4448: 4442: 4441: 4403: 4397: 4396: 4370: 4361:(2): 1376–1385. 4350: 4339: 4338: 4336: 4335: 4294: 4288: 4287: 4267: 4239: 4233: 4232: 4194: 4188: 4187: 4145: 4139: 4138: 4137: 4133: 4126: 4120: 4119: 4087: 4081: 4080: 4042: 4036: 4035: 4033: 4032: 4018: 4012: 4011: 3973: 3967: 3966: 3956: 3938: 3906: 3900: 3899: 3861: 3855: 3854: 3816: 3810: 3809: 3799: 3781: 3764:(3): 4806–4830. 3749: 3743: 3742: 3736: 3728: 3688: 3682: 3681: 3669: 3653: 3647: 3646: 3636: 3630: 3629: 3617: 3601: 3595: 3594: 3592: 3580: 3574: 3573: 3555: 3531: 3525: 3524: 3486: 3480: 3479: 3477: 3464: 3458: 3457: 3439: 3415: 3409: 3408: 3383:(8): 2395–2414. 3368: 3362: 3361: 3323: 3317: 3316: 3298: 3265: 3259: 3258: 3232: 3223:(5): 1195–1212. 3208: 3202: 3201: 3163: 3154: 3153: 3143: 3133: 3109: 3100: 3099: 3097: 3096: 3084: 3078: 3077: 3039: 3030: 3029: 3027: 3026: 3012: 3003: 3002: 3000: 2999: 2993:www.linkedin.com 2985: 2976: 2975: 2948:. pp. 1–6. 2937: 2931: 2930: 2892: 2883: 2882: 2844: 2838: 2837: 2813: 2804: 2803: 2776:. pp. 1–5. 2765: 2756: 2755: 2728:. pp. 1–5. 2717: 2708: 2707: 2669: 2663: 2662: 2660: 2659: 2645: 2639: 2638: 2611:. pp. 1–6. 2600: 2594: 2593: 2591: 2585:. Archived from 2552: 2543: 2537: 2536: 2531:. Archived from 2505:. pp. 1–6. 2500: 2491: 2485: 2460: 2454: 2453: 2443: 2434:(4): 1204–1216. 2424:Ahlswede, Rudolf 2420: 2411: 2409: 2394: 2388: 2387: 2369: 2347: 2341: 2340: 2322: 2302: 2296: 2295: 2278:(6): 2386–2397. 2263: 2257: 2256: 2248: 2242: 2227: 2221: 2214: 2175:Device-to-device 1974: 1972: 1971: 1966: 1948: 1946: 1945: 1940: 1922: 1920: 1919: 1914: 1912: 1904: 1895: 1893: 1892: 1887: 1869: 1867: 1866: 1861: 1859: 1857: 1856: 1844: 1835: 1833: 1832: 1827: 1814: 1812: 1811: 1806: 1794: 1792: 1791: 1786: 1774: 1772: 1771: 1766: 1738: 1736: 1735: 1730: 1699: 1697: 1696: 1691: 1674: 1673: 1644: 1642: 1641: 1636: 1596: 1594: 1593: 1588: 1586: 1585: 1568: 1566: 1565: 1560: 1547: 1545: 1544: 1539: 1537: 1536: 1519: 1517: 1516: 1511: 1498: 1496: 1495: 1490: 1488: 1487: 1469: 1467: 1466: 1461: 1459: 1458: 1441: 1439: 1438: 1433: 1421: 1419: 1418: 1413: 1398: 1396: 1395: 1392:{\displaystyle } 1390: 1385: 1384: 1372: 1371: 1351: 1349: 1348: 1343: 1330: 1328: 1327: 1324:{\displaystyle } 1322: 1298: 1296: 1295: 1292:{\displaystyle } 1290: 1266: 1264: 1263: 1258: 1256: 1255: 1235: 1233: 1232: 1227: 1213: 1211: 1210: 1205: 1175: 1173: 1172: 1167: 1155: 1153: 1152: 1147: 1123: 1121: 1120: 1115: 1110: 1109: 1084: 1082: 1081: 1076: 1043: 1041: 1040: 1035: 943: 941: 940: 935: 904: 902: 901: 896: 884: 882: 881: 876: 864: 862: 861: 856: 829: 827: 826: 821: 809: 807: 806: 801: 789: 787: 786: 781: 769: 767: 766: 761: 749: 747: 746: 741: 715: 714: 691: 689: 688: 683: 669: 668: 645: 643: 642: 637: 634: 629: 613: 611: 610: 605: 603: 602: 599: 586: 584: 583: 578: 575: 570: 554: 552: 551: 546: 544: 543: 527: 525: 524: 519: 514: 513: 488: 486: 485: 480: 477: 472: 453: 451: 450: 445: 443: 442: 429: 424: 414: 409: 391: 390: 372:by the formula: 371: 369: 368: 363: 360: 355: 340: 339: 320: 318: 317: 312: 310: 309: 293: 291: 290: 285: 274: 273: 235: 233: 232: 227: 225: 224: 205: 203: 202: 197: 192: 191: 158: 156: 155: 150: 139:source nodes to 138: 136: 135: 130: 118: 116: 115: 110: 21: 5425: 5424: 5420: 5419: 5418: 5416: 5415: 5414: 5380: 5379: 5302: 5291: 5285: 5282: 5263: 5254:This article's 5250: 5246: 5239: 5211: 5210: 5166: 5165: 5161: 5121: 5116: 5115: 5111: 5096: 5071: 5066: 5065: 5061: 5054: 5029: 5028: 5024: 5009: 4984: 4983: 4979: 4927: 4926: 4922: 4874: 4873: 4869: 4845:10.1.1.321.1962 4829: 4828: 4821: 4806: 4781: 4780: 4776: 4767: 4765: 4752: 4751: 4747: 4732: 4707: 4706: 4702: 4687: 4666: 4665: 4661: 4646: 4623: 4618: 4617: 4613: 4597: 4596: 4592: 4582: 4580: 4573: 4568: 4567: 4563: 4541: 4540: 4536: 4526: 4524: 4516: 4515: 4511: 4496: 4471: 4470: 4466: 4450: 4449: 4445: 4430: 4405: 4404: 4400: 4352: 4351: 4342: 4333: 4331: 4321: 4296: 4295: 4291: 4276: 4241: 4240: 4236: 4221: 4196: 4195: 4191: 4176: 4147: 4146: 4142: 4135: 4128: 4127: 4123: 4089: 4088: 4084: 4069: 4044: 4043: 4039: 4030: 4028: 4020: 4019: 4015: 4000: 3975: 3974: 3970: 3908: 3907: 3903: 3888: 3863: 3862: 3858: 3843: 3818: 3817: 3813: 3751: 3750: 3746: 3729: 3717: 3690: 3689: 3685: 3678: 3655: 3654: 3650: 3638: 3637: 3633: 3626: 3603: 3602: 3598: 3582: 3581: 3577: 3533: 3532: 3528: 3513: 3488: 3487: 3483: 3466: 3465: 3461: 3417: 3416: 3412: 3370: 3369: 3365: 3350: 3325: 3324: 3320: 3267: 3266: 3262: 3210: 3209: 3205: 3190: 3165: 3164: 3157: 3111: 3110: 3103: 3094: 3092: 3086: 3085: 3081: 3066: 3041: 3040: 3033: 3024: 3022: 3014: 3013: 3006: 2997: 2995: 2987: 2986: 2979: 2964: 2939: 2938: 2934: 2919: 2894: 2893: 2886: 2871: 2846: 2845: 2841: 2822:Ad Hoc Networks 2815: 2814: 2807: 2792: 2767: 2766: 2759: 2744: 2719: 2718: 2711: 2696: 2671: 2670: 2666: 2657: 2655: 2647: 2646: 2642: 2627: 2602: 2601: 2597: 2589: 2550: 2545: 2544: 2540: 2521: 2498: 2493: 2492: 2488: 2475:Wayback Machine 2461: 2457: 2441:10.1.1.722.1409 2422: 2421: 2414: 2396: 2395: 2391: 2349: 2348: 2344: 2304: 2303: 2299: 2265: 2264: 2260: 2250: 2249: 2245: 2228: 2224: 2215: 2211: 2206: 2184: 2110:Alternative to 2093:infrastructure. 1981: 1951: 1950: 1925: 1924: 1898: 1897: 1872: 1871: 1848: 1838: 1837: 1818: 1817: 1797: 1796: 1777: 1776: 1757: 1756: 1709: 1708: 1665: 1648: 1647: 1627: 1626: 1604: 1577: 1572: 1571: 1551: 1550: 1528: 1523: 1522: 1502: 1501: 1479: 1474: 1473: 1450: 1445: 1444: 1424: 1423: 1404: 1403: 1376: 1363: 1355: 1354: 1334: 1333: 1301: 1300: 1269: 1268: 1244: 1239: 1238: 1218: 1217: 1196: 1195: 1182: 1158: 1157: 1138: 1137: 1101: 1087: 1086: 1055: 1054: 1026: 1025: 1021: 1012: 984: 911: 910: 887: 886: 867: 866: 832: 831: 812: 811: 792: 791: 772: 771: 752: 751: 704: 703: 697: 660: 655: 654: 616: 615: 594: 589: 588: 557: 556: 535: 530: 529: 505: 491: 490: 459: 458: 434: 382: 377: 376: 331: 323: 322: 301: 296: 295: 265: 242: 241: 216: 211: 210: 183: 169: 168: 141: 140: 121: 120: 101: 100: 97: 28: 23: 22: 15: 12: 11: 5: 5423: 5421: 5413: 5412: 5407: 5402: 5397: 5392: 5382: 5381: 5378: 5377: 5371: 5366: 5360: 5354: 5348: 5342: 5336: 5329: 5323: 5317: 5312: 5304: 5303: 5258:external links 5253: 5251: 5244: 5238: 5237:External links 5235: 5234: 5233: 5222: 5209: 5208: 5179:(3): 367–374. 5159: 5109: 5094: 5059: 5052: 5022: 5007: 4977: 4940:(2): 521–532. 4920: 4883:(5): 944–947. 4867: 4838:(2): 914–925. 4819: 4804: 4774: 4745: 4730: 4700: 4685: 4659: 4644: 4611: 4590: 4561: 4534: 4509: 4494: 4464: 4443: 4428: 4398: 4340: 4319: 4289: 4274: 4234: 4219: 4189: 4174: 4150:Cheng, Xiuzhen 4140: 4121: 4082: 4067: 4037: 4013: 3998: 3968: 3901: 3886: 3856: 3841: 3811: 3744: 3715: 3683: 3676: 3648: 3631: 3624: 3596: 3575: 3526: 3511: 3481: 3459: 3410: 3363: 3348: 3318: 3281:(4): 796–826. 3260: 3203: 3188: 3155: 3101: 3079: 3064: 3031: 3004: 2977: 2962: 2932: 2917: 2884: 2869: 2839: 2805: 2790: 2757: 2742: 2709: 2694: 2664: 2653:www.comsoc.org 2640: 2625: 2595: 2592:on 8 Apr 2022. 2561:(4): 917–933. 2538: 2535:on 2022-04-08. 2519: 2486: 2455: 2412: 2389: 2342: 2297: 2258: 2243: 2229:R. Dougherty, 2222: 2208: 2207: 2205: 2202: 2201: 2200: 2195: 2190: 2183: 2180: 2179: 2178: 2172: 2169: 2166: 2163: 2157: 2139: 2136: 2135:from Microsoft 2129: 2126: 2124:Multi-user ARQ 2108: 2094: 2088: 2085: 2082: 2079: 2076: 2070: 2067: 2064: 2061:Connected cars 2058: 2055: 2049: 2046: 2024: 2020: 2004: 1991: 1980: 1977: 1964: 1961: 1958: 1938: 1935: 1932: 1910: 1907: 1885: 1882: 1879: 1855: 1851: 1847: 1825: 1804: 1784: 1764: 1728: 1725: 1722: 1719: 1716: 1689: 1686: 1683: 1680: 1677: 1672: 1668: 1664: 1661: 1658: 1655: 1634: 1603: 1602:Misconceptions 1600: 1584: 1580: 1558: 1535: 1531: 1509: 1486: 1482: 1457: 1453: 1431: 1411: 1388: 1383: 1379: 1375: 1370: 1366: 1362: 1341: 1320: 1317: 1314: 1311: 1308: 1288: 1285: 1282: 1279: 1276: 1254: 1251: 1247: 1225: 1203: 1181: 1178: 1165: 1145: 1113: 1108: 1104: 1100: 1097: 1094: 1074: 1071: 1068: 1065: 1062: 1051:Ethernet frame 1033: 1020: 1017: 1011: 1008: 983: 980: 933: 930: 927: 924: 921: 918: 894: 874: 854: 851: 848: 845: 842: 839: 819: 799: 779: 759: 739: 736: 733: 730: 727: 724: 721: 718: 713: 701:directed graph 696: 693: 681: 678: 675: 672: 667: 663: 633: 628: 624: 597: 587:, used in the 574: 569: 565: 542: 538: 517: 512: 508: 504: 501: 498: 476: 471: 467: 455: 454: 441: 437: 433: 428: 423: 419: 413: 408: 405: 402: 398: 394: 389: 385: 359: 354: 351: 348: 344: 338: 334: 330: 308: 304: 283: 280: 277: 272: 268: 264: 261: 258: 255: 252: 249: 223: 219: 195: 190: 186: 182: 179: 176: 163:chosen from a 148: 128: 108: 96: 93: 26: 24: 18:Network coding 14: 13: 10: 9: 6: 4: 3: 2: 5422: 5411: 5408: 5406: 5403: 5401: 5400:Finite fields 5398: 5396: 5393: 5391: 5390:Coding theory 5388: 5387: 5385: 5376: 5372: 5370: 5367: 5365: 5361: 5359: 5355: 5353: 5349: 5347: 5343: 5341: 5337: 5334: 5330: 5328: 5324: 5322: 5318: 5316: 5313: 5311: 5308: 5307: 5300: 5297: 5289: 5279: 5275: 5274:inappropriate 5271: 5267: 5261: 5259: 5252: 5243: 5242: 5236: 5232: 5228: 5223: 5221: 5217: 5213: 5212: 5204: 5200: 5196: 5192: 5187: 5182: 5178: 5174: 5170: 5163: 5160: 5155: 5151: 5147: 5143: 5139: 5135: 5131: 5127: 5120: 5113: 5110: 5105: 5101: 5097: 5091: 5087: 5083: 5079: 5078: 5070: 5063: 5060: 5055: 5049: 5045: 5041: 5037: 5033: 5026: 5023: 5018: 5014: 5010: 5004: 5000: 4996: 4992: 4988: 4981: 4978: 4973: 4969: 4965: 4961: 4957: 4953: 4948: 4943: 4939: 4935: 4931: 4924: 4921: 4916: 4912: 4908: 4904: 4900: 4896: 4891: 4886: 4882: 4878: 4871: 4868: 4863: 4859: 4855: 4851: 4846: 4841: 4837: 4833: 4826: 4824: 4820: 4815: 4811: 4807: 4805:1-4244-0221-2 4801: 4797: 4793: 4789: 4785: 4778: 4775: 4763: 4759: 4755: 4749: 4746: 4741: 4737: 4733: 4727: 4723: 4719: 4715: 4711: 4704: 4701: 4696: 4692: 4688: 4682: 4678: 4674: 4670: 4663: 4660: 4655: 4651: 4647: 4641: 4637: 4633: 4629: 4622: 4615: 4612: 4606: 4601: 4594: 4591: 4579: 4572: 4565: 4562: 4557: 4553: 4549: 4545: 4538: 4535: 4523: 4519: 4513: 4510: 4505: 4501: 4497: 4495:0-7803-9392-9 4491: 4487: 4483: 4479: 4475: 4468: 4465: 4459: 4454: 4447: 4444: 4439: 4435: 4431: 4425: 4421: 4417: 4413: 4409: 4402: 4399: 4394: 4390: 4386: 4382: 4378: 4374: 4369: 4364: 4360: 4356: 4349: 4347: 4345: 4341: 4330: 4326: 4322: 4316: 4312: 4308: 4304: 4300: 4293: 4290: 4285: 4281: 4277: 4271: 4266: 4261: 4257: 4253: 4249: 4245: 4238: 4235: 4230: 4226: 4222: 4216: 4212: 4208: 4204: 4200: 4193: 4190: 4185: 4181: 4177: 4171: 4167: 4163: 4159: 4155: 4152:(July 2017). 4151: 4144: 4141: 4131: 4125: 4122: 4117: 4113: 4109: 4105: 4101: 4097: 4093: 4086: 4083: 4078: 4074: 4070: 4064: 4060: 4056: 4052: 4048: 4041: 4038: 4027: 4023: 4017: 4014: 4009: 4005: 4001: 3995: 3991: 3987: 3983: 3979: 3972: 3969: 3964: 3960: 3955: 3950: 3946: 3942: 3937: 3932: 3928: 3924: 3920: 3916: 3912: 3905: 3902: 3897: 3893: 3889: 3883: 3879: 3875: 3871: 3867: 3860: 3857: 3852: 3848: 3844: 3838: 3834: 3830: 3826: 3822: 3815: 3812: 3807: 3803: 3798: 3793: 3789: 3785: 3780: 3775: 3771: 3767: 3763: 3759: 3755: 3748: 3745: 3740: 3734: 3726: 3722: 3718: 3712: 3708: 3704: 3700: 3699: 3694: 3687: 3684: 3679: 3673: 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