Binary Systematic Network Coding for Progressive Packet Decoding

Transcription

1 Binay Systematic etwok Coding fo Pogessive Packet Decoding Andew L. Jones, Ioannis Chatzigeogiou and Andea Tassi School of Computing and Communications, Lancaste Univesity, United Kingdom {a.jones2, i.chatzigeogiou, axiv:5.337v [cs.it] 4 Jan 25 Abstact We conside binay systematic netwok codes and investigate thei capability of decoding a souce message eithe in full o in pat. We cay out a pobability analysis, deive closed-fom expessions fo the decoding pobability and show that systematic netwok coding outpefoms conventional netwok coding. We also develop an algoithm based on Gaussian elimination that allows pogessive decoding of souce packets. Simulation esults show that the poposed decoding algoithm can achieve the theoetical optimal pefomance. Futhemoe, we demonstate that systematic netwok codes equipped with the poposed algoithm ae good candidates fo pogessive packet ecovey owing to thei oveall decoding delay chaacteistics. Index Tems etwok coding, Gaussian elimination, decoding pobability, ank-deficient decoding. I. ITRODUCTIO etwok coding (C, oiginally poposed in [], has the potential to significantly impove netwok eliability by mixing packets at a souce node o at intemediate netwok nodes pio to tansmission. The classical implementation of C, which is often efeed to as staightfowad C [2], andomly combines souce packets using finite field aithmetic. As the size of the field inceases, the likelihood of the tansmitted packets being linealy independent also inceases. Howeve, the decoding pocess at the eceive is computationally expensive, especially if the field size is lage. Futhemoe, staightfowad C incus a substantial decoding delay because souce packets can be ecoveed at the eceive only if the eceived netwokcoded packets ae at least as many as the souce packets. Heide et al. [3] poposed the adoption of binay systematic C, which opeates ove a finite field of only two elements, as a means of educing the decoding complexity of staightfowad C. A souce node using systematic C fist tansmits the oiginal souce packets and then boadcasts linea combinations of the souce packets. The eduction in decoding complexity at the eceive deceases enegy consumption and makes systematic C suitable fo enegyconstained devices, such as mobile phones and laptops. Lucani et al. [4] developed a Makov chain model to show that the decoding pocess of systematic C in time division duplexing channels equies consideably fewe opeations, on aveage, than that of staightfowad C. Baos et al. [5] and Pio and Rodigues [6] obseved that opting fo systematic C as opposed to staightfowad C educes decoding delay without sacificing thoughput. Theefoe, systematic netwok codes exhibit desiable chaacteistics fo multimedia boadcasting and steaming applications. Moe ecently, Saxena and Vázquez-Casto [7] discussed the advantages of systematic C fo tansmission ove satellite links. As in [3], we also conside binay systematic netwok codes and investigate thei potential in deliveing sevices, such as multimedia and steaming, which often equie the pogessive ecovey of souce packets and the gadual efinement of the souce message. Ou objective is to pove that systematic C not only exhibits a lowe decoding complexity than staightfowad C, as shown in [4], but also a bette pefomance, as obseved in [5]. Even though ou focus is on binay systematic C, we explain that ou analysis can be easily extended to finite fields of lage size. In addition, we develop a decoding algoithm and popose a famewok, which helps us study the pefomance of systematic C in tems of the pobability of ecoveing a souce message eithe in pat o in full. The est of the pape has been oganised as follows. Section II analyses the pefomance of systematic C and intoduces metics fo evaluating its capability of pogessively ecoveing souce messages. Section III poposes a modification to the Gaussian elimination algoithm that allows souce packets to be pogessively decoded. Section IV discusses the computational cost and accuacy of the poposed decoding algoithm, validates the deived theoetical expessions and contasts the pefomance of systematic C with that of benchmak tansmission schemes. The main contibutions of the pape ae summaised in Section V. II. BIARY SYSTEMATIC ETWORK CODIG Let us conside a souce node, which segments a message s = [ s i ] K i= into K souce packets and encodes them using a systematic C encode. The encode geneates and tansmits packets, which compise K systematic packets followed by K coded packets. The systematic packets ae identical to the souce packets, while the coded packets ae obtained by linealy combining souce packets. The n-th tansmitted packet, denoted by t n, can be expessed as follows s n if n K t n = K ( g n,i s i if K < n i= whee g n,i is a binay coefficient chosen unifomly at andom fom the elements of the finite field GF(2. We can also expess t n in matix notation as t n = G[ n ] s, whee G[ n ] = [ g n,i ] K i= is the coding vecto associated with t n.

2 ote that when n K, in line with the definition of binay systematic C, we set g n,i = if i = n, else g n,i =. In the emainde of this section, we investigate the theoetical pefomance of systematic C and deive analytical expessions fo the pobability of decoding the entie souce message o a faction of the souce message. We also pesent pefomance metics and benchmaks fo the evaluation of systematic C fo pogessive packet ecovey. A. Pobability of Decoding the Entie Souce Message As peviously mentioned, a souce node using systematic C tansmits packets, of which K ae systematic and the emaining K ae coded. Assume that a eceive successfully ecoves packets, of which h ae systematic and h ae coded. The coding vectos of the eceived packets ae stacked to fom the K decoding matix G. Let f K (, denote the pobability of decoding the K souce packets given that packets have been eceived. We undestand that f K (, is non-zeo only if K. The value of also detemines the smallest allowable value of h. Fo instance, if K < 2K, the K tansmitted coded packets ae fewe than the K tansmitted systematic packets; given that K packets ae eceived, the numbe of eceived systematic packets h should be at least ( K. Othewise, if 2K, the smallest value of h can be zeo. Theefoe, h is defined in the ange max (, +K h K. Having defined the paametes of the system model and thei intedependencies, we can now poceed with the deivation of an analytical expession fo f K (,. Lemma. Fo K tansmitted packets, the pobability of a eceive decoding all of the K souce packets, given that K packets have been successfully eceived, is f K (, = ( K K K + h ( K h whee h min = max (, +K. K h ( 2 +h+j j= ( (2 Poof: The decoding pobability f K (, can be decomposed into the sum of the following pobabilities f K (, = P{h=K} + K P{h<K} w K h ( h. (3 The tem P{h = K} epesents the pobability of ecoveing the K souce packets diectly fom the K successfully eceived systematic packets. This is the case when K out of the K coded packets have been successfully deliveed to the eceive along with the K systematic packets. Consideing that out of the tansmitted packets have been eceived, we can deduce that P{h = K} is given by P{h=K} = ( K K (. (4 The sum of poducts in (3 consides the pobability of ecoveing h < K systematic packets and decoding the emaining K h souce packets fom the h eceived coded packets. Moe specifically, the pobability P{h < K} of eceiving h out of the K systematic packets and h out of the K coded packets is equal to ( K h h P{h < K} = (. (5 On the othe hand, the pobability of having K h linealy independent coded packets among the h eceived ones can be obtained fom the liteatue of staightfowad C, fo example [8]. We find that w K h ( h = K h j= ( 2 ( h+j. (6 Substituting (4, (5 and (6 into (3 gives (2. This concludes the poof. Poposition. The pobability of a eceive decoding all of the K souce packets, afte the tansmission of K packets ove a channel chaacteized by a packet easue pobability p, can be expessed as follows P K ( = ( p p f K (,. (7 =K ( Poof: The poof follows fom Lemma. The conditional pobability f K (, has been weighted by the pobability of successfully eceiving out of tansmitted packets and aveaged ove all valid values of. The closed-fom expessions fo the decoding pobability of systematic netwok codes can be used to contast thei pefomance to the pefomance of staightfowad netwok codes and give ise to the following poposition. Poposition 2. Systematic netwok codes exhibit a highe pobability of decoding all of the K packets of a souce message than staightfowad netwok codes. Poof: Fo the same numbe of eceived packets, the pobability of decoding all of the K souce packets is f K (, fo systematic C and w K ( fo staightfowad C, whee w K ( = K ( j= 2 +j as pe (6. If we show that the elationship f K (, w K ( holds fo all valid values of, we can infe that the decoding pobability of systematic C is highe than that of staightfowad C. Dividing f K (, by w K ( gives f K (, w K ( = ( [ ( K K K A + ] ( K h h Bh whee, fo h = K 2 j A= 2 j and B h h = 2 j j= 2 j, fo h >. (9 j= ote that A > and B h fo all valid values of, that is, K. Theefoe, the ight-hand side of (8 can become (8

3 a lowe bound on the atio f K (, /w K ( if coefficients A and B h ae emoved. Moe specifically, we can obtain f K (, w K ( > ( K ( K h h ( if the binomial coefficient ( K K in (8 is included into the sum and the uppe limit of the sum is updated accodingly. We distinguish the following two cases fo the value of : 2K: In this case, we have h min =. Invoking a special instance of the Chu-Vandemonde identity [9, p. 4], we can educe the sum at the ight-hand side of ( to K ( K ( h h =. ( h= K <2K: As peviously explained, h min = +K. Setting h = K +h, expessing the sum in ( in tems of h, exploiting the popeties of binomial coefficients and using the widely-known Vandemonde s convolution [, p. 29] gives K ( K ( ( h h = K K h h h= +K h =. (2 = ( ( = If we combine identities ( and (2 with inequality (, we obtain f K (, /w K ( > fo all valid values of, which concludes the poof. We note that the atio f K (, /w K ( appoaches as the value of K inceases. Remak. Even though this pape is concened with binay systematic C, i.e. the elements of matix G ae selected unifomly at andom fom GF(2, the same easoning can be employed to obtain P K ( when opeations ae pefomed ove GF(q fo q 2. The pobability f K (, of decoding the entie souce message, given that packets have been eceived, can be witten as f K (, = ( K K K + h ( K h K h ( q +h+j j= (. (3 Both Popositions and 2 hold fo q 2. Substituting (3 into (7 gives the geneal expession fo P K (. B. Pobability of Decoding a Faction of the Souce Message In Section II-A, we focused on deiving the pobability of decoding the K souce packets when K packets have been tansmitted. Of equal inteest is the pobability of ecoveing at least M < K souce packets when M packets have been tansmitted. To the best of ou knowledge, a closed-fom expession fo this pobability, denoted heeafte as P K,M (, has not been obtained fo staightfowad C. Howeve, a good appoximation, which follows eadily fom Poposition, can be computed fo the case of systematic C. Coollay. The pobability of ecoveing at least M < K souce packets, when M packets have been tansmitted ove a channel with packet easue pobability p, can be appoximated by P K,M ( min =M whee min = min (K,. ( min ( p p min (4 Poof: The numbe of tansmitted systematic packets is eithe if < K, o K if K. In geneal, min (K, systematic packets ae sent ove the packet easue channel, fo any value of. If we wish to ecove at least M < min (K, souce packets and the easue pobability p is small, M o moe eceived packets will most likely be systematic and, thus, linealy independent. As a esult, the pobability of decoding at least M souce packets educes to the pobability of ecoveing at least M systematic packets, given by (4. We emak that the assumption of a low value of p is easonable when the physical laye employs eo coecting codes that impove the channel conditions as seen by highe netwok layes, whee C is usually applied. Fo example, the Long Tem Evolution Advanced (LTE-A famewok consides an easue pobability of p =. []. C. Pefomance Metics and Benchmaks In ode to assess the pefomance of systematic C and exploe its capability to pogessively decode a souce message, we will compae it with odeed uncoded (OU tansmission [2] and staightfowad C. In OU tansmission, the K souce packets ae peiodically epeated. The tansmitted packet at time step n = i + mk can be expessed as t i+mk = s i fo i =,..., K and m. We note that tansmission is uncoded in the sense that tansmitted packets ae not linea combinations of the souce packets. By contast, the n-th tansmitted packet in staightfowad C is given by t n = K i= g n,i s i fo n >, implying that all tansmitted packets ae linea combinations of the souce packets. Pobabilities P K,M ( and P K ( will be used to contast the pefomance of systematic C, staightfowad C and OU tansmission. In ode to ceate links between the two decoding pobabilities, we intoduce the following paametes: ˆP is a pedetemined taget pobability of packet ecovey that a tansmission scheme has to attain. Pobabilities P K,M ( and P K ( 2 can be set equal to ˆP in ode to detemine the numbe of tansmitted packets and 2 that ae equied fo the patial o full ecovey of the souce message, espectively. ˆ signifies the minimum numbe of tansmitted packets equied by the eceive to ecove at least M souce packets with a pobability of at least ˆP. denotes the minimum numbe of additional packets that should be tansmitted so that the eceive ecoves the K souce packets with a pobability of at least ˆP. A pefomance compaison of the investigated schemes will be caied out in Section IV. Pio to that, we discuss decoding algoithms fo C schemes and popose a decoding pocess

4 that allows pogessive decoding of souce packets in the following section. III. PROGRESSIVE DECODIG If the objective of the decoding algoithm is the ecovey of the K souce packets afte the eception of at least K tansmitted packets, Gaussian Elimination (GE could be used especially when the value of K is small. The GE algoithm tansfoms the decoding matix G into ow-echelon fom [3]. The ank of the tansfomed matix, which is equal to the ank of the oiginal decoding matix, can be obtained by inspecting the numbe of non-zeo ows within the echelon fom. If the ank is K, that is, if G is a full-ank matix, the K souce packets can be successfully ecoveed. GE and schemes based on Belief Popagation (BP [4] expeience a lage spike in computation when K tansmitted packets ae eceived. On-the-Fly Gaussian Elimination (OFGE [5] manages to mitigate the decoding delay and computational complexity of GE by invoking an optimized tiangulation pocess evey time a packet is eceived. The OFGE decode speads computation out ove each packet aival and the decoding matix G is aleady in patial tiangula fom by the time the K-th tansmitted packet is eceived. Both GE and OFGE have been designed to pefom full-ank decoding. As a esult, if the ank of G is less than K, that is, if the decoding matix is ank-deficient, some souce packets might still be decodable but GE o OFGE will not necessaily identify them. A modified vesion of OFGE, which we efe to as OFGE fo Pogessive Decoding (OFGE-PD, was pesented in [2]. Similaly to OFGE, OFGE-PD also compises a tiangulation stage and a back-substitution stage. An additional stage, called the XORing phase, enables OFGE-PD to decode souce packets fom ank-deficient decoding matices at the expense of inceased computational complexity. We evisited the oiginal GE algoithm and we amalgamated the OFGE pinciple of initiating the decoding pocess wheneve a packet is eceived. A sketch of the poposed algoithm, efeed to as Gaussian Elimination fo Pogessive Decoding (GE-PD, is pesented in Algoithm. To facilitate the desciption of GE-PD, we intoduced function Degee, which detemines the numbe of non-zeo elements in a ow vecto; function Diag, which geneates a ow vecto containing the elements of the main diagonal of a matix; function LeftmostOne, which etuns the position of the fist non-zeo enty in a ow vecto; and function Swap, which swaps two ows in a matix. The decoding matix G is initially set equal to the K K zeo matix. Recall that G[ n ] epesents the n-th ow of G, while G[ i ][ j ] denotes the enty of G in the i-th ow and j-th column (equivalent to g i,j. We note that, depending on the adopted pogamming language, the code can be futhe optimized and the execution speed of GE-PD impoved. As line 2 in Algoithm indicates, wheneve a new coding vecto R is eceived, it is updated so that any peviously decoded souce packets ae not consideed again in the decoding pocess. If the updated ow-vecto R still contains non-zeo Algoithm Gaussian Elimination fo Pogessive Decoding : Receive new K coding vecto R 2: Set enties in R that coespond to decoded packets to 3: if (Degee(R > then 4: G[ K + ] R 5: fo i = to K do 6: one in diag TRUE 7: if (G[ i ][ i ] = then 8: one in diag FALSE, j i + 9: epeat : if LeftmostOne(G[ j ] = i then : Swap(G[ i ], G[ j ] 2: one in diag TRUE 3: end if 4: j j + 5: until ( j > K + o one in diag 6: end if 7: if one in diag then 8: fo j = to (K + do 9: if ( j i and (G[ j ][ i ] = then 2: G[ j ] G[ j ] G[ i ] 2: end if 22: end fo 23: end if 24: end fo 25: G BackSubstitution(G, K 26: G Top K ows of G 27: end if Function BackSubstitution(G, K : fo i = K to step do 2: if Degee(G[ i ] = then 3: j = LeftmostOne(G[ i ] 4: fo k = to K do 5: if k i then G[ k ][ j ] 6: end fo 7: end if 8: end fo 9: etun G enties, it is appended to the bottom of the decoding matix G (lines 3-4. Lines 6-6 eaange the ows of G in an effot to tansfom it into an uppe tiangula matix. Lines 7-23 aim to tansfom G into ow-echelon fom by ensuing that each non-zeo element on the main diagonal of G is the only non-zeo element in that column. Finally, function BackSubstitution is called in line 25 to establish which souce packets ae decodable. The efficiency and accuacy of GE-PD ae investigated in the following section. IV. RESULTS AD DISCUSSIO This section compaes the poposed GE-PD with OFGE-PD, OFGE and GE in tems of computational cost and capability of pogessively ecoveing souce packets. The decoding algoithm that achieves the best accuacy but equies the least computational time is identified. It is then used to obtain simulation esults, which ae compaed to theoetical pedictions in ode to validate the deived analytical expessions fo systematic C. The pefomance of systematic C is then contasted to that of staightfowad C and OU tansmission, and the suitability of each scheme fo pogessive packet ecovey is discussed. A. Assessment of GE-PD Fig. compaes the computational cost of the consideed decoding schemes. Recall that GE-PD and OFGE-PD ae modified vesions of GE and OFGE, espectively, which have

5 Time pe Simulation (sec OFGE-PD OFGE GE-PD GE K Fig.. Computational cost of the decoding schemes fo diffeent numbes of souce packets (K =,..., 3. been adapted to ecove souce packets fom ank-deficient decoding matices, as descibed in Section III. The computational cost has been expessed in tems of the time equied fo a decode to ecove the full sequence of K souce packets when staightfowad C is applied and channel conditions ae pefect, i.e. p =. The plotted esults wee obtained on a simulation platfom equipped with an Intel Coe i7-377 pocesso and 8 GB of RAM. As expected [5], Fig. shows that OFGE yields substantial computational savings ove the conventional GE. Howeve, the inclusion of pogessive decoding capabilities in OFGE adds noticeable ovehead to the decoding pocess. We obseve that the computational cost of the esultant OFGE-PD inceases apidly fo lage values of K. On the othe hand, GE-PD is not only moe efficient than the oiginal GE but also executes faste than OFGE. Staightfowad C fo K = 2 souce packets and pefect channel conditions wee also assumed fo the pefomance assessment of the fou decoding schemes. Fig. 2 depicts the pobability of each scheme ecoveing at least half (M = o all (M = 2 of the souce packets when packets have been tansmitted. As we see, OFGE is not optimized fo ecoveing a faction of the souce message in contast to OFGE-PD, which equies a smalle numbe of tansmitted packets to ecove half of the souce message but at a highe computational cost. A fact wothy of attention is that the decoding accuacy of GE is matched by that of GE-PD, which exhibits a computational cost as low as that of OFGE. Fo this eason, the poposed GE-PD was the decoding algoithm of choice fo the simulation of the consideed C-based schemes. B. Pefomance Validation of Systematic C In ode to validate the deived analytical expessions fo the decoding pobability of systematic C, a compaison between theoetical and simulation esults was caied out. We consideed a souce message compising K = 4 packets, which ae encoded using a systematic C and tansmitted ove a packet easue channel with p = {.,.5,.3}. Fig. 3a shows that expession (4 fo P K,M ( accuately pedicts the pobability of decoding at least half of the souce message (M = 2. Similaly, expession (7 fo Decoding Pobability OFGE-PD M =.6 OFGE-PD M = 2.5 OFGE M =.4 OFGE M = 2.3 GE-PD M = GE-PD M = 2.2 GE M =. GE M = Fig. 2. Pefomance compaison of the decoding schemes fo K = 2. PK,M( Theoy (p =. Simulations (p =. Theoy (p =.5 Simulations (p =.5 Theoy (p =.3 Simulations (p = (a M = 2 PK( (b M = K = 4 Fig. 3. Pefomance validation of systematic C fo K = 4, diffeent values of p and (a patial ecovey (M = 2 o (b full ecovey (M = 4 of the souce packets. P K ( matches the simulated esults fo decoding the entie souce message (M = 4, as epoted in Fig. 3b. The excellent ageement between theoy and simulation establishes the validity of the theoetical analysis. It also demonstates that the poposed GE-PD is both efficient and accuate, consideing that the numbe of decoded souce packets matches the one pedicted by the theoetical model. C. Evaluation of Systematic C fo Pogessive Decoding Fig. 4 shows the pobability that a eceive employing systematic C ecoves at least half (M = K/2 o all (M = K of the souce packets, when packets have been tansmitted. The pefomance of systematic C is contasted with that of OU tansmission and staightfowad C, efeed to hee as SF C fo bevity. Two scenaios have been consideed; Fig. 4a depicts the pefomance of the thee tansmission schemes when K = 2, while Fig. 4b pesents plots fo the case of K = 4. In both scenaios, the packet easue pobability has been set to p =.. We obseve in Fig. 4a that OU tansmission allows the ecovey of at least half of the souce message fo a small value of. Howeve ecovey of the whole souce message equies a lage numbe of tansmitted packets. Fo example, fo a taget pobability of ˆP =.7, a system using OU tansmission

6 Decoding Pobability Decoding Pobability OU Tans., M = SF C, M = Sys. C, M = OU Tans., M = 2 SF C, M = 2 Sys. C, M = (a K = 2 OU Tans., M = 2 SF C, M = 2 Sys. C, M = 2 OU Tans., M = 4 SF C, M = 4 Sys. C, M = (b K = 4 Fig. 4. Decoding pobabilities as a function of fo p =. and (a K = 2 o (b K = 4. can etieve M = souce packets if just ˆ = packets ae tansmitted. On the othe hand, ecovey of all M = 2 souce packets equies the tansmission of at least 39 packets. In othe wods, = 39 = 28 packets need to be tansmitted, on aveage, to allow ecovey of the whole souce message, when half of the message has aleady been etieved. As we see in Fig. 4b, a lage value of K will makedly incease the value of. By contast, SF C incus a significant delay in ecoveing at least half of the souce message but only a few exta tansmitted packets ae equied to obtain the entie message. We obseve in Fig. 4a that if ˆP =.7 then ˆ = 24 packets ae needed to econstuct half of the message, while the tansmission of only = additional packet is sufficient fo the decoding of the entie message. As is appaent fom Fig. 4a and Fig. 4b, systematic C combines the best pefomance chaacteistics of both OU tansmission and SF C. We obseve that the value of ˆ fo ecoveing at least half of the souce packets is as small as that of OU tansmission, while the equied numbe of tansmitted packets fo etieving all of the souce packets is smalle than o simila to that of SF FC. The latte obsevation confims Poposition 2. Consequently, systematic C is the most appopiate of the consideed tansmission schemes fo pogessive packet decoding, as it exhibits a high pobability of eithe patially o fully decoding the souce message. V. COCLUSIOS In this pape, we consideed systematic andom linea netwok coding, obtained theoetical expessions that accuately descibe its decoding pobability and poved that systematic netwok codes exhibit a highe pobability of decoding the entiety of a souce message than staightfowad netwok coding. We also poposed Gaussian elimination fo Pogessive Decoding (GE-FD, which aims to ecove souce packets as soon as one o moe tansmitted packets ae successfully deliveed to a eceive. We demonstated that GE-PD pefoms similaly to the optimal theoetical decode in tems of decoding pobability and also exhibits low computational cost. Futhemoe, we established that the decoding delay chaacteistics of systematic netwok coding fo both patial and full ecovey of souce messages ae notably bette than those of staightfowad netwok coding. ACKOWLEDGMET This wok was conducted as pat of the R2D2 poject, which is suppoted by EPSRC unde Gant EP/L625/. REFERECES [] R. Ahlswede,. Cai, S.-Y. Li, and R. Yeung, etwok infomation flow, IEEE Tans. Inf. Theoy, vol. 46, no. 4, pp , Jul. 2. [2] S. Zhang, S. C. Liew, and P. P. Lam, Hot topic: Physical-laye netwok coding, in Poc. MobiCom, Los Angeles, USA, Sep. 26. [3] J. Heide, M. Pedesen, F. Fitzek, and T. Lasen, etwok coding fo mobile devices - Systematic binay andom ateless codes, in Poc. IEEE ICC Wokshops, Desden, Gemany, Jun. 29. [4] D. Lucani, M. Médad, and M. Stojanovic, Systematic netwok coding fo time-division duplexing, in Poc. IEEE ISIT, Austin, USA, Jun. 2. [5] J. Baos, R. Costa, D. Munaetto, and J. Widme, Effective delay contol in online netwok coding, in Poc. IEEE IFOCOM, Rio de Janeio, Bazil, Ap. 29. [6] R. Pio and A. Rodigues, Systematic netwok coding fo packet loss concealment in boadcast distibution, in Poc. ICOI, Kuala Lumpu, Malaysia, Jan. 2. [7] P. Saxena and M. Vázquez-Casto, etwok coding advantage ove MDS codes fo multimedia tansmission via easue satellite channels, in Pesonal Satellite Sevices. Spinge Intenational Publishing, 23, vol. 23, pp [8] O. Tullols-Cuces, J. Bacelo-Odinas, and M. Fioe, Exact decoding pobability unde andom linea netwok coding, IEEE Commun. Lett., vol. 5, no., pp , Jan. 2. [9] W. Koepf, Hypegeometic Summation: An Algoithmic Appoach to Summation and Special Function Identities. Vieweg Velag, 998, p. 4. [] S. Roman, The Umbal Calculus. Academic Pess, 984, p. 29. [] S. Sesia, I. Toufik, and M. Bake, LTE - The UMTS Long Tem Evolution: Fom Theoy to Pactice. John Wiley & Sons, 2. [2] A. L. Jones, I. Chatzigeogiou, and A. Tassi, Pefomance assessment of fountain-coded schemes fo pogessive packet ecovey, in Poc. CSDSP, Mancheste, UK, Jul. 24. [3] J. Eppeson, An Intoduction to umeical Methods and Analysis, 2nd ed. John Wiley & Sons, 23, ch. 7, pp [4] W. iu, Z. Xiao, M. Huang, J. Yu, and J. Hu, An algoithm with high decoding success pobability based on LT codes, in Poc. ISAPE, Guangzhou, China, ov. 2. [5] V. Bioglio, M. Gangetto, R. Gaeta, and M. Seeno, On the fly Gaussian elimination fo lt codes, IEEE Commun. Lett., vol. 3, no. 2, pp , Dec. 29.