Index Terms Adaptive modulation, Adaptive FEC, Packet Error Rate, Performance.

Transcription

1 ANALYTICAL COMPARISON OF THE PERFORMANCE OF ADAPTIVE MODULATION AND CODING IN WIRELESS NETWORK UNDER RAYLEIGH FADING 723 Sab Y.M. BANDIRI, Rafael M.S. BRAGA and Danlo H. SPADOTI Federal Unversty of Itajubá Av. BPS, Pnhernho, Itajuba - MG, Tel.: (35) bandr@unfe.edu.br, Rafael.braga@unfe.edu.br spadot@unfe.edu.br Abstract In ths paper, an analytc comparson between adaptve modulaton (AdM) and adaptve Forward Error Correcton (AdFEC) n wreless network telecommuncaton systems s performed. Through the Glbert-Ellott model's, the Packet Error Rate (PER) s calculated and both the throughput and the delay, for dfferent modulatons and error correcton capacty, have been also computed. For the throughput crteron, the results show that the AdFEC outperforms AdM for hgher values of Sgnal Nose-to- Rato (SNR). However, when the SNR has low value the adaptve modulaton performs better. In case of delay crteron, the mean tme to transmt a packet usng AdM s lower than when AdFEC s used. Index Terms Adaptve modulaton, Adaptve FEC, Packet Error Rate, Performance. I. INTRODUCTION The rapd growth n the use of nformaton networks, ncreasng the data rates, led to the necessty to mplement novel technque amng to mprove wreless networks communcaton performance. In conventonal communcaton systems, transmsson characterstcs are adjusted for the worst channel settng behavor n order to provde a lower lmt to a specfc error rate []. Adaptve Modulaton and Codng (AMC), whch ncludes both adaptve modulaton (AdM) and adaptve Forward Error Correcton (AdFEC) operatng together, have been developed to face fadng and enhance the wreless systems performance. They maxmze bandwdth effcency through selectng an optmal combnaton from the modulaton and codng scheme (MCS), where the resoluton depends on the channel state nformaton. Consequently, every modulaton and code scheme wll be lnked to a constellaton sze and codng rate respectvely [2]. AMC allows spectrally effcent transmsson n functon of tme-

2 varyng channels. The AMC technque s desgned to track channel varatons and then change the modulaton and/or codng scheme. The goal s to yeld a hgher throughput by transmttng wth hgh nformaton rates under favorable channel condtons and by reducng the nformaton rate n response of channel degradaton [2]- [5]. In case of hgh fadng, for example, the sze of the sgnal constellaton s reduced to mprove the fdelty. Thus, a more robust transmsson for effectve SNR s requred. Conversely, for low fade or hgh gan perod, the sze of the sgnal constellaton s ncreased to reach hgh throughput [6]. In AdM, the number of transmtted bts per symbol vares as a functon of the channel s characterstcs [6]. The basc dea s as followng: f the bt error rate (BER) n the channel exceeds a gven threshold, the number of transmtted bts per symbol s reduced (reducng the transmsson rate n the channel), whle keepng the transmtted average symbol energy at a constant level [6-]. Ths mproves the BER and, consequently, the packet error rate (PER). Therefore, the modulaton scheme depends on the nstantaneous BER (or SNR) n the wreless lnk. For Ad-FEC schemes case, the redundancy of the code used to control error n the wreless lnk vares accordng to ts performance. The man dea s the followng: f the BER n the channel exceeds a gven threshold, the number of bt party s ncreased []. As a consequence, the error correcton capacty of the code s mproved (reducng the probablty of a packet to contan an uncorrectable error) and the throughput s reduced. Adaptve Automatc Repeat request (ARQ) scheme has the same prncple, nevertheless, here the am s to reduce the probablty of a packet to contan undetectable error [2]. Several papers have approached AMC technques n wreless networks under fadng channels wthout consderng the relatonshp between adaptve modulaton and adaptve code [6-8]. The authors n [8] combne trells codng wth adaptve modulaton to ncrease the spectral effcency of adaptve modulaton over fadng channels. Ther results show that a 3dB codng gan relatve to uncoded adaptve modulaton for a smple 4-state trells code, and a 4dB codng gan for an 8-state trells code. Despte of these results, the authors dd not carry out the performance of adaptve modulaton and code. In [2], the authors have deployed a combned modulaton and FEC adaptve scheme to wreless mult-access Asynchronous Transfer Mode (ATM) networks. Nonetheless, he has consdered addtve whte Gaussan nose (AWGN) channel, whch does not accurately descrbe the real condton of the channel. In references [3] and [4], respectvely, the performance of adaptve modulaton and adaptve FEC n wreless network under fadng channel were separately nvestgated. But, the comparatve analyss of the performance of both technques were not been realzed. Therefore, the man objectve of ths work s to compare, adaptve technques performance when they operate together n the same condton. Throughput and delay crtera are used for analyss n the wreless network under Raylegh fadng. Thus, a more realstc stuaton of the wreless network communcaton s consdered n ths paper. 724

3 The remander of ths paper s organzed as follows: n Secton 2, the packet error rate under Raylegh fadng channel usng Glbert-Ellot models s evaluated. In Secton 3, we have compared the performance of both AMC technques (AdM and AdFEC) by usng throughput crtera. In Secton 4, we have calculated and compared the mean tme to forward a correct packet usng both technques. Fnally, the conclusons are presented n Secton II. MODEL TO COMPUTE THE PACKET ERROR RATE In a wreless communcaton system wth a Raylegh fadng channel, errors tend to occur n bursts nstead of random errors that occur n an Addtve Whte Gaussan Nose (AWGN) channel [5]- [7]. Several studes and proposals have ntroduced emprcal or approxmate methods for PER computng n a channel wth memory [5]- [7]. They concluded that such calculaton s qute complex, mprecse and cannot be generalzed to the real applcatons. To calculate or estmate the PER, all proposed methods above model the communcaton channels accordng to a Markov chan, where the SNR rato s parttoned nto a fnte number of states, that can range between two and several. The dffcultes of workng wth Markovan models are n to set the transton probabltes of states to reflect the real channel behavor. In another way, the Glbert-Ellott (GE) s one of the smplest model, whch provdes a useful dscrete model, beng that the parameters can be readly related to the statstcs of the fade [8]- [20]. The GE channel assumes that the channel can be represented by a dscrete tme Markov chan wth two states: Good state (G) and Bad state (B). Fg. llustrates a GE wth transton probabltes (The channel condtons move from bad state to good state) and (the channel condtons move from good state to bad state). As shown n Fg., each state s modeled as a Bnary Symmetrc Channel (BSC) wth bt error probabltes p g, n G state, and p b n B state. G B -p g -p b p g p 0 b p g p b -p g -p b Fg.. Glbert-Ellott Channel [6]. Consderng that the channel fades slowly under Raylegh fadng and wth respect to a bt nterval, the probablty densty functon of the SNR s gven by [2]: where,, s the average SNR and,, s the nstantaneous SNR. f e 0 ()

4 A gven threshold, ψ, has been chosen to defne the status of the channel. The model consders that the channel s sad to be n the good state when the SNR s above that threshold, ψ, and once the SNR drops below the threshold, ψ, the channel goes to a bad state. The transton probabltes n the GE channel can be calculated usng the level crossng rate and the SNR densty functon as n [8]: 726 f 2 d T (2) e f d T 2 (3) where, T s the symbol nterval and, f d, s the maxmum Doppler speed. The parameter,, s the rato between the threshold,, and the average SNR n the wreless channel. As n [8], the was set to 0., thus, a 0 db SNR below the average SNR represents the transton to the bad state and the product, T f d, s equal to 0.0. Accordng to the results obtaned n [8] and [9] these values outperform other values proposed n ther lterature. The steady state probabltes of the Markov chan llustrated n Fg. are gven by [8]: g (4) b (5) where g (ths s the probablty of the channel condton stay n good state) and b (ths s the probablty of the channel condton stay n good bad) are the steady state probabltes of GE channel beng n the good and bad states, respectvely. In GE channel, The BER (p g, p b ) assocate to each state of the Raylegh fadng for a gven modulaton scheme s computed as followng [8]: BER f p g (6) BER f d f d 0 p b (7) 0 f d d

5 n whch BER() s the bt error rate for an AWGN channel wth SNR equal to, and f() s the probablty densty functon of Raylegh fadng presented n (). The bt error rate for an AWGN channel, wth the consdered modulaton, can be determnate usng classcal equatons [2]. Wth the bt error rates calculated usng (6) and (7), the packet error rate n each state of the channel can be computed. A system wthout FEC s consdered for adaptve modulaton technque. Thus, the packet error rate n each state s the probablty of packet error n a BSC, consderng the proper BER for each state. These probabltes are gven by [8]: n 727 P p g p g (8) n P p b p b (9) where, n s the length of each packet n bts transmtted n each slot, and p g and p b are gven by (6) and (7), respectvely. However, f FEC s consdered, the probabltes of packet error n a BSC can also be easly computed by smply consderng the error correcton capacty of the FEC code. The probabltes of packet error n each state are gven by [8]: n P p p p t n g ( g ) g 0 P p n p p t n b ( b) b 0 (0) () Fnally, the packet error rate n the Raylegh channel s determnate by [6]: PER P p P p (2) g g b b where, G and B are gven by (4) and (5), respectvely, and P(p g ) and P(p B ) are gven by (8) and (9) for AdM and by (0) and () for AdFEC, respectvely, In ths proposal, the Tme Dvson Multple Access (TDMA) was consdered, where the modulaton can be defned on a frame by frame bass [2]. The modulaton schemes consdered n ths paper are: BPSK (Bnary Phase Shft Keyng), QPSK (Quaternary PSK), 8-PSK and M-QAM (M Quadrature Ampltude Modulaton) wth M = 6, 32, 64, 28 and 256 respectvely. In the QAM systems, square constellatons for M = 6, 64 and 256 and rectangular constellatons for M = 32 and 28 have been consdered. The expresson to compute the bt error rate n each consdered modulaton as a functon of Es/No can be obtaned n [2]. There are several codes to correct bursts errors n wreless networkng [22]- [25]. Nonetheless, the BCH (Bose, Chaudhur and Hocquenghem) code s consdered n ths study, due to hs hgh capacty

6 Packet Error Rate - PER to form a large class of powerful random error-correctng cyclc codes to mplement FEC scheme. Therefore, ths code s consdered n the AdFEC system. For any postve ntegers m ( m 3 ) and t ( t 2 m ), there exsts a bnary t-error correctng BCH code wth the followng parameters [22]: 728 n 2 m block length n k mt, number of party check dgts dmn 2t, mnmum dstance To have an exact parameter for comparson as n [2], the number of bt nformaton has been set to k=424 for PER computng. Fg. 2 ndcates the PER n functon of Es/No. From ths fgure, the PER decreases wth the ncrease of the Es/No, whch means that the channel tends to stay more n the good state than to the bad state. For a gven Es/No rato, the modulaton wth hghest number of ponts n the constellaton scheme has the hghest packet error rate (PER). If the Es/No rato s set constant and when channel condtons are unfavorable, t would be approprate to use modulaton wth fewer ponts n the constellaton scheme (reducng the transmsson rate) n order to mantan hgh system performance on the wreless lnk QAM 28QAM 64QAM 32QAM 6QAM Es/No Fg.2 Packet Error Rate vs Es/No III. CRITERIA OF THROUGHPUT One of the crtera used to compare the performance for both systems s based on throughput n the wreless channel. Ths approach has been consdered n the scenaro where the traffc s real-tme and retransmsson of errored packet s not mplemented. We have defned the throughput as the mean

7 number of correct packets transmtted per second n the wreless network under Raylegh fadng channel. Thus, the throughput s computed as followng []: V log M k 2 n PER log 2 M r n (3) 729 where Ms the number of ponts n the constellaton of the current modulaton, Mr s the number of ponts n the constellaton of the reference modulaton, k s the number of nformaton bts n each packet transmtted n the wreless lnk, n s the total number of bt n each packet n the wreless lnk (k plus the number of party bts, not consderng protocol headers). PER s the packet error rate defned n (2). For the adaptve modulaton scheme, error control code s not used ( n =k), and the maxmum throughput s gven by : Th log M 2 mod PER (4) log 2 M r When the system swtches from determnate modulaton to another, the average symbol energy Es/No (or transmsson power) s kept constant. The parameter E b /No (average bt energy to nose densty) changes at the swtch tme. Therefore, the performance evaluaton of the modulaton schemes s performed as a functon of the parameter Es/No (average symbol energy to nose densty rato) [23]. For adaptve FEC scheme, fxed modulaton s used as reference n the maxmum throughput defnton ( M Mr), and t can be wrtten as: Th fec k PER (5) n To compare the performance between AdM and AdFEC, the maxmum throughput of both technques s computed accordng to (4) and (5). The results are shown n Fg.3 and n Fg.4 where 256-QAM modulaton and 64-QAM modulaton are used as reference, respectvely. In Fg.3 the swtchng pont between both curves s Es/No=22.88 db wth PER=0.03. However, for Fg.4 the swtchng pont among AdM and AdFEC curves s Es/No=7.56 db wth PER= Therefore, the swtchng ponts among AdM and AdFEC depend on the reference modulaton used. An analytc comparson of the performance n Fg.3 and Fg.4 ndcates that, for lower value of Es/No (when Es/No s less than db for 256-QAM and less than 7.56 db for 64-QAM) adaptve modulaton outperforms AdFEC. The possble reason s that: the SNR s the relatonshp between Symbol Energy (Es) and Nose (N). Snce, Es s constant (consderaton made n ths paper), low SNR

8 Throughput values mean hgh nose n the channel. Thus, AdM case requres usng modulaton wth low ponts on ts constellaton format (to ensure less PER), such as: BPSK, QPSK. Consequently, the throughput ncreases, snce PER decreases. In contrast, for the AdFEC technque, the number of bt party s ncreased to guarantee the error correcton, and the code error correcton capacty s mproved. However, ncreases the bt party, hghly decreases the throughput (see Eq.5), whch leads the AdFEC throughput to be lower than the AdM. On the other hand, for hgh values of Es/No, the AdFEC outperforms AdM. The throughput of the AdFEC technque s better for hgh SNR (low nose) values, because there s no necessty to ncrement the party bts n such case, nstead, t s reduced snce the errors are low. However, for hgh values of Es/No, AdM consders modulatons wth large constellaton schemes (e.g.: 28QAM, 256QAM) that provdes a better transmsson rate, nonetheless more susceptble to errors. Ths leads the throughput of the AdFEC technque better than AdM. Thus, AdFEC s more ndcated when the channel condtons are n bad state. AdM can be used when the channel condtons are n good state snce at ths canal condton the PER s lower. Smlar concluson has been done n [2], nevertheless, n ther case, the channel consdered was memoryless and ths type of channel does not descrbe the real stuaton of wreless network communcatons behavor. Therefore, as concluson, AdFEC wll present better performance ndependently of channel model, but the crossng pont wll change accordng to the channel and the reference modulaton Adaptve Modulaton Adaptve FEC Es/No (db) Fg.3. Throughput as a functon of Es/No for the adaptve FEC and adaptve modulaton schemes (the 256-QAM s the reference modulaton).

9 Throughput Adaptve modulaton Adaptve FEC Es/No (db) Fg.4. Throughput as a functon of Es/No for the adaptve FEC and adaptve modulaton schemes (the 64-QAM s the reference modulaton). Table summarzes the results obtaned from Fg.3 and 4 and, for easly comparson, the results from Fg.4.6, Fg.4.7 from [2] were also added. From Table we apprecate the followngs cases: The swtchng occurs n hgh values of Es/No n [2], when we compare wth ths work. The value of the PER s greater n [2] than n ths paper. Throughput value of ths proposal s lower than the presented n [2]. All the above results can be explaned by the fact that, n ths work, a more realstc stuaton was consdered, once the channel s not perfect but subject to adverse condtons. Hgh packet loss generates a hgh PER and as consequently the throughput of the wreless communcaton network s reduced. TABLE I. COMPARISON OF SWITCHING POINTS, PER AND THROUGHPUT OF THIS PAPER AND [2]. Es/No (db) PER Throughput AMC +256QAM AMC + 64QAM Ref [2] + 256QAM Ref [2] + 64QAM

10 732 IV. DELAY CRITERIA In non-real data transmsson system, typcally errors n the wreless lnk are corrected by retransmsson usng Automatc Repeat request (ARQ). One errored bt s enough to result n retransmsson of the entre packet. The receptor can request a retransmsson of the entre packet untl success transmsson of all the bts of the packets. In ths system, the average tme to transmt a correct packet s a proper qualty-of-servce (QoS) parameter to evaluate the performance. The nterval of tme between the concluson of recepton of an errored packet and the begnnng of ts retransmsson, have not beng consdered. To compute the delay, we consdered a TDMA system wth, X, tme slots n a frame, wth n s bts beng transmtted n each slot. Each packet was transmtted over, Z, slots (one packet needs Z tme slots to be transmtted), and t was retransmtted untl be correctly receved (we consder that number of retransmssons was unlmted). A slow fadng channel (the duraton of fades was much greater than the duraton of packet transmsson) was also consdered. For AdFEC, the mean tme to transmt a correct packet over Z slots T fec, consderng condtons above, where the duraton of fades s much greater than the duraton of packet transmsson tme, can be computed by [4]: T fec ns Z X (6) PER B log2 M where, B s the bandwdth, and M the symbol number accordng to the modulaton consdered. The FEC codes are appled n each tme slot, thus, (0) and () should be modfed n order to calculate correctly the new packet error probabltes assocated wth each state good and bad. Therefore, the PER s the packet error rates computed by (2) wth n = Zn s. In the context of AdM system, the mean tme to transmt a packet, for a gven modulaton, whereas retransmssons are done at packet level (.e., the retransmsson request s made to the complete packet and not the fragments transmtted n each tme slot) may be calculated by the followng equaton [3]: T mod ns Z X log 2 (7) PER B M As the retransmsson s made at packet level, (8) and (9) have to be modfed also accordng to the channel state. The new PER assocated to each state s calculated wth n = Zn s n (8) and (9). For ths crteron Z has been set equal to 5 n the analyze X=0.

11 Performance Factor Wth the purpose to compare the performance of both technques (AdM and AdFEC) usng the delay crtera, a performance factor has been defned as the relatonshp between the mean tme to transmt a packet wthout error n the system wth AdM and the mean tme to transmt the same packet by usng AdFEC. The results presented n Fg.5 show that the average tme to transmt a packet usng the AdFEC technque s larger than the average tme to transmt a packet by usng AdM technque for Es/No 28.3 db. Ths could be explaned by the fact that n the AdFEC technque the spent average tme to send a packet ncludes both the transmsson tme and the spent mean tme at the recever to correct any errors durng the transmsson of the packet. Ths spent mean tme at the recever s not consdered n the AdM technque. In [2], the AdFEC has better or equal performance to the AdM for Es/No 23.6 db. The value of the performance factor where adaptve FEC performs better or equal to adaptve modulaton s.009 n [2], and ths value was got.209. Ths hgh value of the performance factor n our proposal can be explaned by the hgh value of PER, snce the channel condtons are subject to Raylegh fadng. Hence, a hybrd technque that results from the combnaton of these two technques can mprove system performance Es/No (db) Fg. 5. Performance factor between adaptve modulaton and adaptve FEC. V. CONCLUSION In ths paper, the performance of AdFEC and AdM has been compared n wreless network communcaton under Raylegh fadng. AMC technques performs n dfferent range of SNR. AdFEC performs better than AdM n most SNR range. However, for a lower value of SNR adaptve modulaton has better performance when a throughput crteron s used. Also, the mean tme to transmt a packet usng AdFEC s hgher than the tme used to transmt the same packet for AdM. Therefore, for applcatons whch requre less tme to transmt packets, adaptve modulaton technque

12 wll be approprated rather than adaptve FEC. Further, a hybrd technque (where both technques are combned) wll be a good canddate to mprove better the performance of the wreless network. 734 ACKNOWLEDGMENT THE AUTHORS WOULD LIKE TO THANK CAPES, CNPQ AND FAPEMIG. REFERENCES [] H. I. Anwar, I. Ibrahm, K.W. Hon and Y. A. Razak, Performance and Smulaton of Adaptve Modulaton Technques of the WIMAX Network va AWGN Channel, Australan Journal of Basc and Appled Scences, 8(4) Specal 204, pp [2] J. Yang, A. K. Khandan and N. Tn, Adaptve Modulaton and Codng n 3G Wreless Systems, Proceedngs on 56th Vehcular Technology Conference, VTC 2002-Fall, pp [3] J. Faezah and K. Sabra, Adaptve Modulaton for OFDM Systems, Internatonal Journal of Communcaton Networks and Informaton Securty (IJCNIS), Vol., No. 2, August 2009, pp.-8. [4] K. Shataras and S. R. Redd, Throughput and Error State Performance of AMC Scheme n 3G Wreless Systems, IOSR Journal of Computer Engneerng, Vol. 9, Issue 6, Mar-Apr. 203, pp [5] A. Zalons, N. Mlou, I. Dagres, A. Polydoros and H. Bogucka, Trends n adaptve modulaton and codng, Advances n Electroncs and Telecommuncatons, Vol., No, Aprl 200, pp [6] J. Tang and J. Wang, Adaptve modulaton for fadng channels, IEEE Asa-Pacfc Conference on Crcuts and Systems APCCAS, 2000, pp [7] I. Forkel, A. Kramlng and D. Bernhardt, On allocaton and adaptve transmsson technology n fxed wreless access networks, 4th European Personal Moble Communcatons Conference, Venna, Austra, 200. [8] A. J. Goldsmth and S. G. Chua, Adaptve coded modulaton for fadng channels, IEEE Transactons on Communcatons, 46(5), 998, pp [9] J. M. Torrance and L. Hanzo, Latency consderatons for adaptve modulaton n an nterference-free slow Raylegh fadng channel, IEEE Vehcular Technology Conference, Vol. 2, 997, pp [0] J. M. Torrance and L. Hanzo, Optmsaton of swtchng levels for adaptve modulaton n slow Raylegh fadng, Electronc Letters, Vol. 32, No 3, June, 996, pp [] J. M. C. Brto and I. S. Bonatt, Analysng the optmal threshold level for adaptve modulaton n the wreless ATM networks, Proc. of the Iasted Internatonal Conference on Wreless and Optcal Communcatons, Banf, Canada, July 2002, pp [2] J. M. C. Brto and I.V. Bonatt, A combned modulaton and FEC adaptve scheme to wreless multaccess ATM networks Internatonal Telecommuncatons Symposum, Natal/RN-Brazl, Sept: 2002, pp [3] S.Y.M. BANDIRI and J.M.C. Brto, Analyzng the optmum swtchng ponts for adaptve modulaton n wreless networks wth raylegh fadng IEEE Latncom, n the 6th Latn-Amercan Conference on Communcatons, Nov.5th to 7th 204 n Cartagena de Indas, Colomba [4] S.Y.M. Bandr; J.M.C Brto, Analyzng the Optmum Swtchng Ponts for Adaptve FEC n Wreless Networks wth Raylegh Fadng. In: Internatonal Conference on Networks, 205, Barcelona. 4th Internatonal Conference on Networks, 205. v.. p

13 [5] H. Bschl and E. Lutz, Packet Error Rate n the non-nterleaved Raylegh channel, IEEE Transactons on Communcatons, Vol. 43, No 2/3/4, February/March/Aprl 995, pp [6] R. Khall and K. Salamatan, A new analytc approach to evaluaton of packet error rate n wreless networks, Proceedng of the 3rd Communcaton Networks and Servces Research Conference, 2005, pp [7] Y. X, A. Burr, J. We and D. Grace, A general upper bound to evaluate packet error rate over quasstatc fadng channels, IEEE Transactons on Wreless Communcatons, V.0, No 5, May 20, pp [8] C. Jao, L. Schwebert and B. Xu; On modelng the packet error statstcs n bursty channels, Proceedngs on 27th Local Computer Networks - LCN, November 2002, pp [9] G. Sharma, A. Dholaka and A. Hassan, Smulaton of error trappng decoders on a fadng channel, Proc. IEEE Vehcular Technology Conference, Atlanta, USA, May 996, pp [20] A. Seddk, A. Djebbar, J.M. Rouvaen, A. Taleb-Ahmed, Performance evaluaton of BCH correctng codes on a fadng channel usng OFDM modulaton, Proceedngs of the th WSEAS Internatonal Conference on Communcatons, Crete Island, Greece, July 2007, pp [2] E. O. Ellot, Estmates of error rates for codes on burst-nose channels, Bell Syst. Tech. J., Vol 42, September 963, pp [22] B. Sklar, Dgtal Communcatons: Fundamentals and Applcatons second edton. Prentce Hall, 200. [23] A. Seddk et Al, "BCH Codng Performance Evaluaton on a Land Moble Channel Based OFDM System", Informaton Technology Journal 5, 2006, pp: [24] J. Gomes and B.K. Mshra, Double Error Correctng Long Code", Internatonal Journal of Computer Networks & Communcatons, Vol.2, No.5, September 200, pp: [25] S. Kumar and R. Gupta, Bt Error Rate Analyss of Reed-Solomon Code for Effcent Communcaton System", Internatonal Journal of Computer Applcatons, Volume 30 No.2, September 20, pp: