Blind identification of code word length for non-binary error-correcting codes in noisy transmission

Transcription

1 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 DOI /s RESEARCH Blnd dentfcaton of code word length for non-bnary error-correctng codes n nosy transmsson Yasamne Zrell 1,2, Roland Gauter 1,2*,ErcRannou 1,3,MélaneMarazn 1,2 and Emanuel Rado 1,2 Open Access Abstract In cogntve rado context, the parameters of codng schemes are unnown at the recever. The desgn of an ntellgent recever s then essental to blndly dentfy these parameters from the receved data. The blnd dentfcaton of code word length has already been extensvely studed n the case of bnary error-correctng codes. Here, we are nterested n non-bnary codes where a nosy transmsson envronment s consdered. To deal wth the blnd dentfcaton problem of code word length, we propose a technque based on the Gauss-Jordan elmnaton n GFq Galos feld, wth q = 2 m,wherems the number of bts per symbol. Ths proposed technque s based on the nformaton provded by the arthmetc mean of the number of zeros n each column of these matrces. The robustness of our technque s studed for dfferent code parameters and over dfferent Galos felds. Keywords: Cogntve rado; Blnd dentfcaton; Non-bnary error-correctng codes; Galos feld Introducton Error-correctng codes are frequently used n modern dgtal transmsson systems n order to mprove the communcaton qualty. These codes are desgned to acheve a good mmunty aganst channel mparments by ntroducng redundancy n the nformatve data. Due to the complexty of both encodng and especally decodng procedures, the maorty of research and practcal mplementatons of real-tme embedded systems were often restrcted to encoders manpulatng bnary data,.e., elements of the Galos feld GF2. Over the last decade, low-densty party chec LDPC codes and turbo codes over GF2 have attracted consderable nterest of many researchers due to ther excellent error correcton capablty. They have been generalzed to fnte felds GFq [1,2], where q = 2 m, and are among the most wdely used errorcorrectng codes n wreless communcaton standards. It *Correspondence: roland.gauter@unv-brest.fr Equal contrbutors 1 Unversté Européenne de Bretagne, 5 Boulevard Laënnec, Rennes, France 2 Unversté de Brest; CNRS, UMR 6285 Lab-STICC, 6 avenue Vctor Le Gorgeu, Brest, France Full lst of author nformaton s avalable at the end of the artcle has been shown n [1] that non-bnary LDPC codes perform generally better than bnary LDPC codes and turbo codes. However, the maor drawbac of these codes s ther decodng complexty for a large Galos feld order q [3,4]. Low complexty decodng algorthms have recently been proposed [5,6], thus allowng the use of non-bnary LDPC codes n practcal mplementatons. Our man research nterests are focused on non-bnary error-correctng codes n order to blndly dentfy ther parameters. Ths topc s a part of a non-cooperatve context le a mltary ntercepton or cogntve rado applcatons. In ths case, the recever has no nowledge about the parameters used to encode the nformaton at the transmtter. The soluton s to desgn an ntellgent recever whch s able to blndly dentfy the encoder parameters from the only nowledge of the receved data stream. Ths blnd dentfcaton functon of the recever permts to ncrease the data rate transmsson, snce t wll be unnecessary to transmt supplementary nformaton about the encoder parameters wth the useful data. Such ntellgent recever s able to adapt automatcally tself to the development of new hgh-performance codng schemes and the fast evoluton of new communcaton standards wthout equpment change. In ths wor, we are only nterested 2015 Zrell et al.; lcensee Sprnger. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly credted.

2 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 2 of 16 n blndly dentfyng the code word length of lnear nonbnary bloc codes. In the case of the ntercepton, ths parameter can not be transmtted. Lewse, f we want to change the encoder or get out of the lst of possble choce of encoders, the code word length s not transmtted. In ths context, the publshed research results have been restrcted so far to the blnd recognton of the code word length of bnary codes. To the best of our nowledge, ths paper ntroduces, for the frst tme, an approach to blndly dentfy the code word length of non-bnary codes n nosy condtons. In ths wor, the am s to blndly dentfy the code word length from the only nowledge of receved data. The authors n [7] proposed a technque of dentfcaton of non-bnary LDPC parameters, but the dentfcaton s not blnd because t s based on usng a predefned canddate set of encoders whch s nown by both the transmtter and the recever. Furthermore, ths technque only wors wth LDPC codes unle our proposed technque, whch s general and sutable for all bloc codes. In our paper, the proposed blnd dentfcaton technque s based on a generalzaton of an exstng method used for bnary codes. The prncple of ths generalzaton wll be explaned n ths paper wthout specfyng n detals ts detecton performances. So, we present here state-of-the-art technques to dentfy the code word length of bnary lnear bloc codes. The dea of these technques s to fnd a bass of a dual code composed of party chec relatons. For ths purpose, an approach based on fndng code words of small Hammng weght [8,9] was mproved by Valembos [10] by usng statstcal hypothess tests and recently by Cluzeau [11,12] and Côte [13]. A second approach based on lnear algebra theory was ntroduced n [14] for noseless channel. Ths approach permts to recover the length of code words by studyng behavors of the ran of matrces composed of receved bts. However, the ran crteron was exploted wthout provdng an algebrac and theoretcal ustfcaton of such behavor. In [15], the use of ths crteron was ustfed. In [16], the ran crteron approach was generalzed to convolutonal codes over GFq, where q > 2, assumng a noseless transmsson, but t was shown that ths generalzed technque can be also performed to non-bnary lnear bloc codes. In nosy transmssons, a technque based on the Gauss elmnaton n GF2 was appled n [17-19] to matrces composed of nosy receved bts n order to fnd the number of almost dependent columns permttng the dentfcaton of the code word length n the case of bnary error-correctng codes. Indeed, an almost dependent column of a matrx composed of nosy receved symbols corresponds to a column whch may be a lnear combnaton of some precedng columns wthout thepresenceoferroneoussymbolsandwhchleadstoa column that contans more zero elements after the Gauss elmnaton. Compared to prevous wors, we demonstrate here that t s possble to generalze the blnd dentfcaton technque proposed n [17,18] to non-bnary bloc codes provded that the Galos feld parameters the cardnalty and the prmtve polynomal are nown by the recever. To dentfy the prmtve polynomal, an algorthm of dentfcaton was proposed n [20]. To acheve our purpose, t s necessary to dentfy the number of almost dependent columns n the matrces composed of nosy symbols of GFq by studyng the probablty of detecton of these columns, denoted as P. In fact, the computaton of P s essental n order to determne an optmal detecton threshold. Assumng a transmsson over q-ary symmetrc channel wth an error probablty p e,thetechnques based on fndng a base of a dual code [18,19] for bnary codes requre the nowledge of p e, where a hard decson demodulaton s consdered. For ths reason, we propose here an approach whch s more robust because t allows us the blnd dentfcaton of the code word length of nonbnary and bnary bloc codes wthout usng the error probablty p e. Ths approach s based on analyzng behavors of the arthmetc mean of the number of zeros n the columns of the matrces constructed by the Gauss elmnaton n GFq. In ths paper, the proposed method s a general method that should be appled to all nonbnary bloc codes even though most examples of codes gven here are non-bnary LDPC codes. For ths reason, the propertes of LDPC codes are not exploted by our method. Ths paper s organzed as follows. In the Techncal bacground secton, we present the encodng process of non-bnary error-correctng codes. Then, the prncple of the blnd dentfcaton of code parameters n the noseless case s descrbed. The channel model used n ths study s also defned and ustfed n ths secton. In the Blnd dentfcaton of code word length n the nosy case secton, the blnd dentfcaton method of the code word length n nosy envronment s descrbed. A comparson n terms of error probablty and detecton performances s shown n the Analyss and performances secton. Fnally, some conclusons are drawn n the Conclusons secton and planned future wor s ponted out. Techncal bacground Non-bnary error-correctng codes The use of an effcent codng system n the transmtter as error-correctng codes s essental n order to fght dsturbances present on the transmsson channel. For a long tme, cyclc codes such as BCH codes [21,22] and Reed- Solomon codes [23] have been the most commonly used as codes based on fnte felds snce they are characterzed by large mnmum dstances for a hard decson decodng. The non-bnary LDPC codes descrbed by a sparse party chec matrx wth elements n GFq havebeen

3 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 3 of 16 developed by Davey and MacKay n 1998 [1]. Sgnfcant wors on the desgn and the decodng complexty reducton of these codes have shown that they have a great potental to replace Reed-Solomon codes n some applcatons of communcaton, such as space communcatons [24], and storage systems [25,26]. In ths paper, we focus on the blnd dentfcaton of code word length for the non-bnary bloc codes, but ths proposed method can also be appled to convolutonal codes and concatenated codes. Let us present the encodng process of these codes over GFq. Actually, the prncple of a transmsson chan s to send dgtal nformaton from a source to one or more recevers. The nformaton yelded by the source s bnary data {0, 1} =GF2. Each bloc of m nformaton bts are combned to generate a symbol of GFq. Then, the generated non-bnary nformaton, denoted as d,sencodedby one of the bloc codes over GFq lstedabove.formost bloc error-correctng codes, a code word, denoted as c, composed of non-bnary symbols s obtaned by the multplcatonofthenformatond and a non-bnary generator matrx G: c = d G 1 In the case of LDPC codes, the encodng process needs the use of the party chec matrx, whch s always sparse compared to the other codes. In most of the standards, such as long-term evoluton LTE standard [27], the encodng s performed n a systematc form n order to facltate the decodng process wthout degradng performances of the error correcton. For ths reason, n the case of bloc codes, the requred parameters to perform the decodng operaton are the number of nputs, denoted as, thecodewordlength, denoted as n, and a party chec matrx, denoted as H. Indeed, the matrx H wll be used by the decoder to detect or/and to correct the errors. The recovered nformaton wll be the frst symbols of the recovered code word due to the systematc form used n the encodng. Our am n ths research wor s to blndly dentfy the parameter n from non-bnary receved symbols whch are affected by nosy transmssons. In the noseless context, we have already demonstrated n [16] that we can dentfy ths parameter wth the only nowledge of the receved data, provded that the Galos feld parameters are nown. The prncple of blnd dentfcaton of the code parameter n n the noseless case s recalled n the followng subsecton. [28] to dentfy the parameters of convolutonal codes over GFq, where q = 2 m.wehaveshownthatour method for the noseless case can be appled to bloc codes. Ths method reshapes row-wse the receved symbols, denoted as r, under a matrx form, denoted as R l,of sze M l. Indeed, R l s flled by receved symbols from the top left corner to the bottom rght as llustrated n Fgure 1. The number of columns l vares between 1 and l max and the number of rows M whch depends on l s gven by the nteger part L l where L s the length of a receved symbol stream. Then, the ran over GFq s calculated for each matrx R l.whenallmatrcesr l have full ran, t s mpossble to detect the exstence of a code. Nevertheless, the redundancy ntroduced by the code leads to ran defcences n some matrces R l.henceforth,theranbehavors of R l allowustodetectthecodeandtodentfytsparameters, n partcular the code word length. As demonstrated n [15] and studed n [16], there are two possble ran behavors accordng to the number of columns l. Ifl s a multple of n.e., l = α n, α N, the rans of the matrces R l are proportonal to the code rate /n.e., ranr l = l /n. Otherwse.e., l = α n, R l have full ran.e., ranr l = l. Thus, the value of the ran defcency depends on code parameters and n. Indeed, only two consecutve ran defcences are necessary to determne all code parameters. The code word length n can be determned by the dfference between two values of l correspondng to two consecutve ran defcences of R l. As shown n [16], the ran method gves good results n a noseless envronment. A theoretcal and algebrac study of the behavor of the ran crteron, as well as partcular cases whch can occur for specfc parameters of Prncple of blnd dentfcaton method of code word length n the noseless case In ths part, we assume that the channel ntroduces no error.in[16],wehaveadaptedthemethodproposedn Fgure 1 Example of a matrx R l. Example of a matrx constructon from the receved symbols.

4 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 4 of 16 codes, were presented n [15]. It was demonstrated that most matrces R l have full ran when l s not a multple of n, except for some partcular cases whch depend on codes generator matrx. In a nosy envronment, the ran method can not be used, snce all the matrces R l have full ran n ths case. Non-bnary channel In order to evaluate our blnd dentfcaton algorthm, we assume that the encoded sequences are transmtted through a q-ary non-bnary, for q = 2 m > 2 symmetrc channel QSC whch s the smplest channel. However, our proposed algorthms can wor for every type of channel provded that the error probablty p e computed at the output of the demodulator s nown. Indeed, we consder that the blocs of the transmsson chan, the modulator, the transmsson channel, and the demodulator can be modeled by a non-bnary channel, where a hard decson demodulaton s consdered. In a cogntve rado context, a multpath fadng channel s used. Ths realstc channel leads to burst errors whch can be corrected by usng an nterleaver and error-correctng codes. In ths context, the errors at the output of a denterleaver at the recever sdecanbemodeledbyaqscwhenadecodngprocess wth hard decson wll be used. The problem of a blnd dentfcaton of the nterleaver perod, as well as a blnd synchronzaton wth the nterleaver blocs was handled n [14,18]. Let us defne the q-ary symmetrc channel whch s the generalzaton of the bnary symmetrc channel BSC. In fact, t s a dscrete memoryless channel wth an error probablty p e andcomposedofnon-bnarynputsand non-bnary outputs belongng to the GFq, where q = 2 m. Thesymbolsatthenputofthechannelarendependent and dstrbuted unformly wth a probablty equal to 1/q. Asymbolδ GFq at the channel nput s receved ncorrectly wth a probablty p e /q 1 [29]. In other words, t s replaced at the recever by a dfferent symbol β of GFq. The probablty of correctly recevng a symbol s equal to 1 p e. The QSC channel s characterzed by the condtonal probabltes: p r = β r = δ = p e q 1, δ = β 2 p r = δ r = δ = 1 p e where the transmtted symbol s denoted r,.e., r = c,for {1,, L}, and the nosy receved symbol s denoted r such that r = r + e wth e the transmsson error ntroduced n the symbol r. An example of a non-bnary symmetrc channel for q = 2 2 s depcted n Fgure 2. In the followng secton, we present the blnd dentfcaton method of the parameter n n a nosy framewor. Blnd dentfcaton of code word length n the nosy case In ths part, we present the mplementaton method whch allows us to dentfy the code word length of a non-bnary code n a nosy envronment. Ths method s based on the concept of fndng the ran-defcent matrces among R l, l {1,..., l max }, correspondng to matrces havng at least one almost dependent column. Indeed, the matrces R l are reshaped n the same way as R l usng the nosy receved symbols r. In [19], a method devoted to determne these matrces n the case of bnary codes was presented. However, ths method requres the nowledge of the error probablty p e. In order to avod ths constrant, we propose a method based on usng the arthmetc mean crteron n order to detect the ran-defcent matrces whch have some almost dependent columns wthout the need of the error probablty p e. Prncple In a noseless case, the ran crteron s used to fnd the maxmum number of lnearly ndependent columns n the matrces R l. Ths allows us to derve the number of lnearly dependent columns n R l columns whch are lnear combnatons of other columns. The fnte-feld Gauss Fgure 2 Non-bnary symmetrc channel. An example of a non-bnary symmetrc channel for q = 2 2.

5 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 5 of 16 elmnaton method [30] has to be used to elmnate those lnear-dependent columns to zero. In nosy transmssons, all matrces R l have full ran. A matrx R l can be expressed accordng to R l by: R l = R l + E l 3 where E l s the error matrx of sze M l constructed n the same way as R l usng the errors nduced by the channel. Therefore, the dependence of the columns s dsturbed by the presence of errors n some receved symbols. In such context, the authors n [17,18] proposed to loo for the number of almost dependent columns n the matrces composed of nosy receved bts by usng the Gauss elmnaton over GF2. Inspred by ths dea, t s suffcent, n the case of non-bnary error correctng codes, to apply the fnte-feld Gauss elmnaton n GFq to R l n order to obtan a new matrx T l of sze M l. Ths algorthm gves also at output a matrx of sze l l, denoted à l, that descrbes the combnaton operatons performed to the columns of the matrx R l n order to obtan the transformaton matrx T l. A recall of the fnte-feld Gauss elmnaton over GFq spresentednalgorthm1.to descrbe ths algorthm, we denote I l the dentty matrx of sze l l, x l the -th column of a gven matrx X l and x l a coeffcent of a matrx X l placed n the -th column and n the -th row. By means of ths algorthm, the lnear-dependent columns n the matrx wll be elmnated to zeros. The whole matrx s consdered n our proposed method nstead of only the lower part of the matrx R l as mentoned n [17]. It would be more accurate than assumng that errors do not occur n the upper part of the matrx, buttsnottherealcase. We can note that the fnte-feld Gauss elmnaton over GFq can be defned by a lnear applcaton gven by: R l à l = T l 4 In noseless transmssons, the number of dependent columns n R l,forl = α n, α N, corresponds to the number of the zero columns n the matrx T l whch s the result of the transformaton of R l by the fnte-feld Gauss elmnaton n GFqR l A l = T l. The matrx form of T l s descrbed n Fgure 3. In fact, the dmenson dentfcaton of a vector space generated by a code C s equvalent to fndng the Algorthm 1 The fnte-feld Gauss elmnaton over GFq. Requre: R l Ensure: T l and à l Intalzaton: T l R l and à l I l for = 1tol do f the -th element of the -th column t l Permute the -th column t l = 0 then wth the frst column t l > thathasanon-zeroonts-th element Permute the column ã l wth ã l end f Multply the columns t l and ã l to have t l = 1 for = + 1tol do by v = 1/ t l n order Let b = t l. Apply the followng operaton to the columns t l t l = t l and t l b t l n order to have t l = 0: Apply the followng operatons to the columns of the matrx A l : end for end for ã l = ã l b ã l Fgure 3 Matrx form of T l. An example of a matrx form of T l for l = α n.

6 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 6 of 16 dmenson of a vector space generated by ts dual code C. For any vector h belongng to C and for any code word r of C, the relaton between both s defned by r h T = 0. In noseless condtons, the matrx R n,forl = n, whchs composed of M code words of length n,should satsfy: R n h T = 0 5 We can note that h belongs to the ernel of R n,denoted as er R n.so,wehavec er R n. Snce the dependent columns n R l multpled by the columns a l permt to have the zero columns n the matrx T l,thecorrespondng columns a n wll belong to er R n n whch the dual code C s contaned. Therefore, fndng the dependent columns n R l s equvalent to fndng the columns a l whch belong to the dual code C. Due to the presence of errors nduced by the channel n R l,forl = α n, thecolumnsof T l correspondng to the almost dependent columns n R l wll contan some nonzero symbols. Assumng that the frst l rows and the pvots of the matrx T l do not contan transmsson errors, usng 3 and 4 allows us to wrte the matrx T l as: T l = T l + E l A l 6 parameters M and P, denoted as BM, P.The parameter P corresponds to the probablty that a coeffcent t l of the column t l s equal to 0.e., P = Pr [ t l = 0 ã l C ]. It s possble to lmt the two behavors of the varable B l by computng an optmal threshold ˆη opt such that: { If Bl > ˆη opt then ã l C If B l ˆη opt then ã l 7 / C where ˆη opt = M q η opt s a real n the nterval [ 0, M]. The optmal threshold η opt s able to mnmze the probablty of wrong detecton of a column ã l C,denoted as P wd, whch corresponds to the sum of the false alarm probablty, denoted as P fa, and the probablty of not detectng a theoretcal dependent column, denoted as P nd. The optmal threshold s determned by: η opt = arg mn η P wd = arg mn η P nd + P fa = arg mn η 1 + M M = Mq η +1 [q M q 1 M P 1 P M ] 8 In ths case, a vector h s a party chec relaton.e., h C wth hgh probablty f the relaton R l h T has a low Hammng weght [11]. However, the opposte s not necessarly true. We can conclude that ã l belongs to C f the correspondng t l = R l ã l has a small Hammng weght. In GFq, the Hammng weght of a vector s the number of non-zero elements n ths vector. So, our am s to determne the columns t l whch have a hgh number of zeros. The dea s to study the number of zeros n the columns of the T l n order to detect the almost dependent columns n R l. Behavors of the number of zeros n the columns of T l Let B l be the number of zeros n the -th column of T l, t l.hence,thevarableb l has two behavors dependng on whether the column ã l belongs to the dual code C or not. Ths varable wll be studed as a functon of ã l assumng that the bts that represent an element of the GFq, where q = 2 m, are unformly dstrbuted and ndependent from each other. If the column ã l does not belong to the dual code C, the varable B l,forall [[ 1, l]], wll follow a bnomal dstrbuton of parameters M and 1/q wth a mean equal to M/q, denoted as BM,1/q. If the column ã l belongs to the dual code C,the varable B l wll follow a bnomal dstrbuton wth The normal dstrbuton can be used to approxmate the bnomal probabltes of B l when M s large: If ã l C : B l N μ 0, σ0 2 If ã l / C : B l N μ 1, σ where N μ 0, σ0 2 s the normal dstrbuton of parameters μ 0 = M P and σ0 2 = M P 1 P and N μ 1, σ1 2 corresponds to the normal dstrbuton of parameters μ 1 = M/q and σ1 2 = M q 1/q2. Henceforth, the optmal value of the threshold ˆη mnmzng the probablty of wrong detecton P wd can be computed by: ˆη opt = arg mn 1 φ ˆη ˆη μ1 σ 1 + φ ˆη μ0 σ where φx s the cumulatve densty functon of the standard normal dstrbuton: φx = 1 x e t2 2 dt 12 2 π

7 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 7 of 16 We can note that the optmal threshold ˆη opt depends on the parameters: M, q, andp. So, n order to delmt the two behavors of the varable B l, t s necessary to compute the probablty P. Computaton of the probablty P In the case of bnary codes, the probablty P has been calculated n [11]. But, t has never been studed n the general case of codes over GF2 m. In fact, the computaton of the parameter P s essental n order to detect the almost dependent columns n R l by delmtng the two behavors of the varable B l. Our am s to nvestgate ths probablty n the case of non-bnary codes. In the followng, the theoretcal study of P s presented. For l = n and a poston of a column ã l contaned n C, a coeffcent t l of the column t l can be obtaned, usng 6, by: t l = t l + n =1 a l e l = n =1 a l e l 13 where t l = 0 n the case of noseless transmssons as explaned prevously. Indeed, the sum n =1 a l e l s null n ths case because e l = 0, {1,, n}, and {1,, M}. However, n the case of nosy transmssons, the coeffcents e l GFq corresponds to the errors ntroduced by the nosy channel n the symbols r l GFq n order to generate the nosy symbols r l GFq. Our am s to determne P the probablty of detectng a zero coeffcent n the column t l correspondng to havng n =1 a l [ n P = Pr =1 a l e l = 0 e l = 0: ] 14 Let N l be the mnmum number of lnear combnatons of columns requred to obtan t l.thsnumbercorresponds also to the Hammng weght of the column ã l. Then, there could be postons among N l where e l = 0. Thus, P can be defned as the probablty of havng s=1 a l e l = 0suchthats s the number of postons among N l where e l = 0: [ N l P = Pr [X = 0] + Pr X = s, s=1 s =1 a l e l = 0 ] 15 where X s a random varable of the erroneous postons number among N l. Indeed,weshownAppendxthat the probablty P of havng t l = 0canbedetermned by: P = 1 + q 1 1 p e q q 1 q N l 16 In the case of GF2.e., q = 2, ths probablty can be wrtten as: P = p e N l 2 17 Ths expresson corresponds to that used n [11]. In Fgure 4, we represent the wrong detecton probablty P wd as a functon of ˆη/M and p e assumng q = 2 3, w ã l = 20 and M = 2, 000. For each value of p e,the optmal threshold ˆη opt correspondng to a root of 11 s computed. From Fgure 4, we can deduce that the threshold nterval satsfyng P wd 0 decreases when the value of p e ncreases. We can conclude that studyng the behavors of B l n order to dentfy n s based on the calculaton of the optmal threshold ˆη opt. However, ths threshold depends on the value of the error probablty p e whch s unnown for the recever. So, the need to estmate ths parameter s a blocng step n the almost dependent columns method and also leads to a lac of robustness. In order to address these problems, we propose a new teratve method based on the arthmetc mean of the varable B l whch do not depend on p e and where the teratve process permts to mprove the detecton probablty. New teratve method based on the arthmetc mean of the varable B l In ths part, the proposed method based on the arthmetc mean of the number of zeros n the columns of the matrx T l s descrbed. We recall that the Gauss elmnaton descrbed n Algorthm 1 should be appled n order to obtan T l. We show here that the dentfcaton of the parameter n by our proposed method does not depend on the error probablty p e. In ths method, n order to mprove the detecton probablty of n, an teraton process s ntroduced. We consder the dea of the teratve process proposed n [18,19]. The prncple of ths process s to perform random permutatons on the rows of the matrx R l n order to obtan a new vrtual realzaton of the receved data. These permutatons permt to ncrease the probablty to obtan non-erroneous pvots durng the Gauss elmnaton.

8 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 8 of 16 Fgure 4 Probablty P wd accordng to ˆη/M and p e. The probablty of wrong detecton of almost dependent columns P wd accordng to ˆη/M and p e s represented for q = 2 3, M = 2, 000 and w ã l = 20. The arthmetc mean of the varables B l, [[ 1, l]], denoted E l s defned by: l=1 B l E l = 18 l Property 1. If X 1, X 2,, X m are ndependent random varables respectvely followng: N μ 1, σ1 2, N μ2, σ2 2,..., N μm, σm 2 the mean defned by X 1+X 2 + +X m m follows: μ1 + μ 2 + +μ m N, σ σ σ m 2 m m 2 19 We recall that the varable B l whch s the number of zeros n the -th column of the matrx T l has two possble behavors dependng on l: If l = α n,forα N, the varable B l follows a normal dstrbuton N μ 1, σ1 2 for all columns of T l. In ths case, usng the property 1, the mean E l wll follow: E l N μ 1, σ l We can note that the mean E l wll be close to M/q. If l = α n,forα N: Ifthe -th column s an almost dependent column, the varable B l wll follow the normal dstrbuton of parameters N μ 0, σ 2 0. If the -th column s not an almost dependent column, the varable B l wll follow the normal dstrbuton of parameters N μ 1, σ 2 1. Thereby, the mean E l s gven by: E l N Ql μ0 + l μ 1 l, Ql σ l σ 2 1 l 2 where Ql s the number of almost dependent columns n the matrx R l such that: Ql = Card { [[ 0, l]], B l > ˆη opt } where Cardx s the cardnal functon whch returns the set sze. l = l Ql s the number of ndependent columns n the same matrx. In the noseless envronment, the mean E l s stable at: E l = M q n + q n 23

9 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 9 of 16 We note two behavors of E l wth respect to l = α n or l = α n: If l = α n then E l M q If l = α n then E l > M 24 q The gap between these behavors allows us to fnd the matrces whch have the number of columns l = α n. Let J be a set of l-values where the gap E l M q > 0: J = {l = 1,, l max E l Mq } > 0 25 Thereby, the dentfed length of the code words wll be such that: ñ = modedffj 26 wherethefunctons dffx and modex aredefnedby: Functon dffx: the output of ths functon s a vector of sze s 1 and t corresponds to the dfference between two consecutve elements of the vector x = x1 x2 xs : Algorthm 2 The algorthm based on the arthmetc mean calculaton Requre: r, M, q and maxmum number of teratons t max Ensure: Identfed code word length ñ Intalze the number of teratons t = 1 Intalze the stop crteron end t = 0 whle end t = 0 do for l = 1tol max do Buld matrx R l of sze M l R l T l = R l à l for = 1tol do Count B l end for Compute E l end for Create the set J 25 f J s empty then f t < t max then t = t + 1 Permute randomly the rows of R l else end t = 1 end f else Determne ñ 26 end t = 1 end f end whle dffx = x2 x1 xs xs 1 27 Functon modex: ths operaton provdes the value whch has the hghest occurrence n the vector x. The proposed teratve method of the code word length dentfcaton s summarzed n the Algorthm 2. Example 1. Let us consder the Reed-Solomon code, denoted RS15, 11, overgf2 4 whch s defned by: n = 15 and = 11. The mean E l normalzed by M, whchs set to 1,000, s represented n Fgures 5 and 6. In Fgure 5, a zero probablty of error.e., p e = 0 s consdered. For Fgure 5 Gap between the mean E l /M and 1/q of RS15, 11 over GF2 4 forp e = 0. The mean E l normalzed by M = 1, 000 s represented n the case of p e = 0.

10 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 10 of 16 Fgure 6 Gap between the mean E l /M and 1/q of RS15, 11 over GF2 4 forp e = The mean E l normalzed by M = 1, 000 s represented n the case of p e = l = α n, we can verfy that the mean E l normalzed by M s stable at 1/q = For l = α n,themeane l meets 23: 1 q n + = q n So, the matrces of sze l = α n have peas for E l M 1 q = 0.25 > 0. In Fgure 6, the gap E l M 1 q s represented wth respect to l when p e = 0.01 for one teraton of our algorthm. Accordng to 25, the set J s shown n Table 1. Henceforth, usng 26, the dentfed length of the code words s ñ = 15. Analyss and performances The am of our proposed algorthm s to blndly dentfy the length of non-bnary code words n nosy envronment. Ths purpose can be reached wth an average complexty equal to OM l 3 max t max. Indeed,theproposed algorthm performs l max 1 t max processes of Gaussan elmnatons whch have an average complexty equal to OM l 2,wherel = 2 l max.so,theaverage complexty s such that: O M t max l max l 2 t=1 l=2 = OM lmax 3 t max 28 Table 1 Szes of the matrces R l for E l M 1 q > 0 l J dffj The szes of matrces for whch El M 1 q > 0, J,andthesetdffJ are gven for p e = 0.01 nthecaseofrs15, 11 over GF2 4. In order to analyze the performances of our blnd dentfcaton method, the probablty of correct detecton ofthecodewordlengthn s chosen as a performance crteron. In the smulatons, our method s appled to the non-bnary LDPC codes whch became canddate for future communcaton systems. For each smulaton, 2,000 Monte Carlo trals are run where the data symbols are randomly chosen at each tral. In ths part, we focus on: the gan of the teraton process on the detecton probablty of n the performance comparson n the case of dfferent channels the mpact of ncreasng the Galos feld dmenson q on the detecton probabltes of n the mpact of ncreasng the code word length n on the detecton probabltes for a gven q Gan of the teratve process In our smulatons, we consder a LDPC n = 6, = 3 over GF4. Fgure 7 shows the probablty of detectng n accordng to p e for one, three, fve, and ten teratons. We can see that the gan between the frst and the tenth teraton s sgnfcantly mportant. Indeed, for p e = 0.07, wth one teraton, the detecton probablty s equal to 0.76 and t becomes equal to 0.99 after 10 teratons. We can deduce that the teratve process mproves sgnfcantly the detecton performances of the blnd dentfcaton method based on the mean calculaton. Performance comparson n the case of dfferent channels Let us llustrate the detecton obtaned by the proposed method for a LDPC n = 16, = 8 over GF8 when an AWGN channel the frst channel and a multpath Raylegh channel assocated to an AWGN channel the

11 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 11 of Detecton Probablty of n t=1 t=3 t=5 t= p e Fgure 7 The detecton probablty of the method based on the mean calculaton for LDPC n = 6, = 3nGF4. For LDPC n = 6, = 3 n GF4, the probablty of detectng n s depcted compared wth the error probablty p e for one, three, fve, and ten teratons. second channel are consdered. In order to compensate and reduce the nter-symbol nterference ISI caused by the multpath propagaton, a lnear mean square error MSE equalzer of length 20 was used. We evaluate the performances of our method when the QAM or PAM modulaton of order 8 8-QAM and 8- PAM s used to transmt the symbols coded by LDPC n = 16, = 8 over GF8. In Fgures 8 and 9, a comparson of performances of our blnd dentfcaton method usng 8-PAM or 8-QAM modulatons n the case of an AWGN channel and a multpath channel wth path number L path = 4andt max = 1spresented.InFgure8,a comparson of the detecton performances of our method n the case of AWGN channel s depcted. We can see that the proposed method for 8-QAM modulaton gves better performances than for 8-PAM modulaton when SNR< PAM, AWGN 8 QAM, AWGN 0.8 Detecton Probablty of n SNRdB Fgure 8 The detecton probablty of the method based on the mean calculaton for LDPC n = 16, = 8 n the case of AWGN channel. For LDPC n = 16, = 8 over GF8, the detecton probabltes of the method based on the mean calculaton n the case of AWGN channel are depcted when a 8-PAM and 8-QAM modulatons are used.

12 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 12 of Detecton Probablty of n PAM, multpath Raylegh channel 8 QAM, multpath Raylegh channel SNR db Fgure 9 The detecton probablty of the method based on the mean calculaton n the case of multpath Raylegh channel. For LDPC n = 16, = 8 over GF8, the detecton probabltes of our method are depcted n the case of multpath Raylegh channel wth L path = 4when 8-PAM and 8-QAM modulatons are used. 18 db. The gan between both s equal to 5 db. However, for SNR > 18 db, the performances are smlar and the detecton probablty s equal to 1. To obtan the detecton probabltes presented n Fgure 9, the modulated symbols by 8-PAM or 8-QAM modulatons are transmtted n a quas-statc Raylegh fadng multpath channel wth path number L path = 4, then the receved symbols are treated by the lnear MSE equalzer of length 20. We can observe that, n the case of 8-QAM, our proposed method provdes better performances than for 8-PAM. A gan equal to 5 db s exhbted. We have chosen to evaluate our proposed methods n the worst case of 8-PAM modulaton because our am was to show that our method has the best performances even n the case of the PAM modulaton. In the followng, the performance study of the mpact of n and q on the proposed method s presented. Impact of ncreasng q Let us consder a LDPC n = 6, = 3, constructed n the Galos feld GFq, where q = 4, 8, 16. The matrces R l are reshaped from L = 30, 000 receved symbols wth l = 2,,30and M = 1, 000. For each value of q, the method based on the mean calculaton s appled to blndly dentfy the code word length of LDPC n = 6, = 3 over GFq whent max = 1. Fgure 10 depcts the probablty of detectng the correct n by our blnd dentfcaton method accordng to the error probablty p e n the cases of GF4, GF8, and GF16. Ths fgure shows that the curve behavor s nearly smlar for all q = 4, 8, 16. We can deduce that the method based on the mean calculaton s slghtly senstve to the ncrease of the Galos feld dmenson q. Impact of ncreasng n To evaluate the detecton performances of our blnd dentfcaton method, the mpact of ncreasng the code word length should be studed. In our smulatons, we consder two LDPC codes over GF8, a LDPC n = 6, = 3 and aldpcn = 16, = 8. The matrces R l are reshaped from L = 64, 000 receved symbols wth l = 2,,64 and M = 1, 000. For each code, the method based on the mean calculaton s appled to blndly dentfy the code word length n when t max = 1. Fgure 11 shows the detecton probabltes of n by the method based on the mean calculaton. We can note that the ncrease of the code word length leads to lower detecton performances wth our proposed method. Indeed, for p e = 0.01, the detecton probablty of the method of the mean calculaton s constant and equal to 1 n the case of the two codes. For p e = 0.02, the detecton probablty decreases from 0.99 to In order to show that our method wors n the case of codes of a reasonable code word length, we computed the detecton probablty of the Reed-Solomon code RS n = 31, = 25 over GF32 whch corresponds to an equvalent code over GF2 of length m n = 5 31 = 155. For an error probablty p e = 0.01 and 1,000 trals

13 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 13 of Detecton Probablty of n GF4 GF8 GF p e Fgure 10 Impact of the Galos feld dmenson q on the detecton probablty of n by the proposed method consderng t max = 1. For LDPC n = 6, = 3, the probablty of detectng the correct n by the method based on the arthmetc mean computaton s depcted accordng to the error probablty p e n the cases of GF4, GF8, and GF16. of Monte Carlo, we obtaned a detecton probablty of 0.87 for t max = 50. Ths probablty can be mproved by ncreasng the number of teraton of our algorthm. For t max = 100, we obtaned a detecton probablty of Conclusons In ths paper, we have ntroduced an algorthm devoted to the blnd dentfcaton of the code word length for a nonbnary code n a nosy transmsson envronment. Usng ths algorthm, the code word length can be dentfed by Detecton Probablty of n n=6 n= p e Fgure 11 Impact of ncreasng n on the detecton probablty for LDPC codes. Of szes n = 6andn = 16 by usng the proposed method consderng t max = 1. For LDPC n = 6, = 3 and LDPC n = 16, = 8 over GF8, the probablty of detectng the correct n by the method based on the arthmetc mean calculaton s depcted accordng to the error probablty p e.

14 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 14 of 16 calculatng the arthmetc mean of the number of zeros that occur n the columns of the matrx obtaned by the Gauss elmnaton. We have shown that the proposed algorthm s robust because t does not requre the estmaton of error probablty, s nsenstve to the hgh order of Galos feld, and has the best detecton performances for the most of modulaton types. Furthermore, ths method provdes better performances of detecton when an teratve process s consdered n order to ncrease the probablty to obtan non-erroneous pvots durng the Gauss elmnaton. Our future wor wll focus on dentfyng the remander of the non-bnary code parameters as well as a party chec matrx, permttng to mplement a generc decoder n a nosy envronment. Furthermore, a method based on usng soft nformaton that allows us to mprove the performances of the blnd dentfcaton algorthms wll be publshed soon [31]. Appendx Proof of Equaton 8 We defne H 0 and H 1 by: H 0 f ã l C and H 1 f ã l / C 29 The two behavors of B l are lmted n 7. The am of ths appendx s to demonstrate 8. In order to determne the probabltes of P fa and P nd,weshouldstudythe behavors of the varable B l accordng to the hypotheses H 0 and H 1 : Under the hypothess H 0 : the varable B l follows a bnomal dstrbuton BM, P. So, the probablty that B l s greater than M q η s as follows: [ Pr B l > M ] q η H 0 = M = M q η +1 M P 1 P M 30 Under the hypothess H 1 : the varable B l follows a bnomal dstrbuton BM,1/q. So, the probablty that B l s less than or equal to M q η s as follows: [ Pr B l M ] q η H 1 = M q η =0 M q 1M q M 31 Usng these two probabltes, we wll calculate the false alarm probablty P fa, the probablty of not detectng a theoretcal dependent column P nd and the probablty of detecton P det. Calculaton of the false alarm probablty P fa : ths probablty corresponds to decde that a column ã l belongs to a dual code C even thought n realty t does not belong. Ths probablty can be determned by: [ P fa = Pr B l > M ] q η H 1 = M = M q η +1 M q 1M q M 32 Calculaton of the probablty of not detectng a theoretcal dependent column P nd : ths probablty corresponds to decde that a column ã l does not belong to C even thought n realty t belongs. Ths probablty can be determned by: [ P nd = Pr B l M ] q η H 0 = M q η =0 M P 1 P M 33 Calculaton of the probablty of detecton P det : ths probablty s defned by: [ P det = 1 P nd = Pr B l > M ] q η H 0 = M = M q η M P 1 P M Usng 32 and 34, the optmal threshold can be determned by: η opt = arg mn η P wd = arg mn η P nd + P fa = arg mn η 1 + P fa P det M = arg mn η 1 + M = Mq η +1 [q M q 1 M P 1 P M ] 35 Proof of the equaton 16 The probablty P s ntally expressed by 15. We denote P 1 s = Pr [X = s] and P 2 s = Pr

15 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 15 of 16 [ s=1 ] a l e l = 0 such that these two probabltes are ndependent. Henceforth, 15 becomes: N l P = P P 1 s P 2 s 36 s=1 Assumng that the errors are ndependent from each other and unformly dstrbuted n GFq\{0},thevarable X follows a bnomal dstrbuton wth parameters N l and p e. Thereby, the probablty P 1 s s determned by: N l P 1 s = p e s 1 p e N l s 37 s The probablty P 2 s s the probablty of havng s=1 a l e l = 0whereel GFq\{0}. We demonstrate by the mathematcal nducton that the probablty P 2 s can be expressed by: P 2 s = 1 P 2s 1 38 q 1 We have P 2 0 = 1 because there are no erroneous postons. In the case of a sngle erroneous poston, we have P 2 1 = 0. However, consderng the example of GF2 2, the probablty P 2 s = 2 can be obtaned by the matrx M whose the ndexes of rows and columns correspond to non-zero elements of ths feld. The coeffcents of ths matrx correspond to the sum over GF2 2 ofthendexes of a row and a column. 032 M = If we have e l 2 a l 1, e l 1 GF2 2 2,and a l 2, GF2 2 2, the probablty of havng a l 1 e l 1 + al 2 e l 2 = 0wllbeP 22 = 3/9 = 1/3. The computed probablty verfes 38. We assume that 38 s verfed for s, and we demonstrate t for s+1. If we have s+1 =1 al e l = 0, we wll have e l s+1 = 1 s a l s+1 =1 a l e l that belongs to GFq wth a probablty equal to 1/q 1. Therefore, the probablty P 2 s + 1 s determned by: s P 2 s + 1 = Pr e l s+1 GFq, a l e l = 0 =1 s = Pr e l s+1 GFq Pr a l e l = 0 = P 2s q 1 = 1 P 2s q 1 =1 40 In order to smplfy the expresson of P 2 s, achange of varable s done by consderng ϕs = q 1 s 1 P 2 s. WhenP 2 s s replaced by ϕs, the expresson 38 becomes: ϕs + ϕs 1 = q 1 s 2 41 Denotng ρs = 1 s ϕs, the expresson 41 can be wrtten as: s 1 ρs = ρ1 + 1 q 42 =0 but, the sum s 1 =0 1 q s a geometrc sequence of common rato 1 q. So,tcanbewrttenas: s 1 1 q = 1 1s 1 q 1 s 1 q =0 43 The computaton of ρ1 gves ρ1 = 0. Therefore, usng 43 and 41, the smplfed expresson of P 2 s s wrtten as: P 2 s = 1s + q 1 s 1 q q 1 s 1 44 Usng 37 and 44, the overall probablty P s gven by: P = 1 q N l =0 N l =0 N l pe 1 p e Nl + q 1 N l pe 1 p e l N q 1 45 In order to smplfy ths equaton, the Newton s bnomal formula can be appled: Z + Y N l = N l =0 N l Z Y N l Thus, the probablty of havng an element of the -th column of T l equal to 0 s determned by: P = 1 + q 1 1 p e q q 1 q N l 46

16 Zrell et al. EURASIP Journal on Wreless Communcatons and Networng :43 Page 16 of 16 Competng nterests The authors declare that they have no competng nterests. Author detals 1 Unversté Européenne de Bretagne, 5 Boulevard Laënnec, Rennes, France. 2 Unversté de Brest; CNRS, UMR 6285 Lab-STICC, 6 avenue Vctor Le Gorgeu, Brest, France. 3 Unversté de Brest; CNRS UMR 6205, Laboratore de Mathématques Bretagne Atlantque, 6 avenue Vctor Le Gorgeu, Brest, France. Receved: 12 February 2014 Accepted: 11 February 2015 References 1. MC Davey, D MacKay, Low-densty party-chec codes over GFq. IEEE Commun. Lett. 2, JA Brffa, HG Schaathun, n 5th Internatonal Symposum on Turbo Codes and Related Topcs. Non-bnary turbo codes and applcatons IEEE Lausanne, D Declercq, M Fossorer, Decodng algorthms for nonbnary LDPC codes over GFq. IEEE Trans. Commun. 554, L Barnault, D Declercq, n Proceedngs ITW. 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Dual code method for blnd dentfcaton of convolutonal encoder for cogntve rado recever desgn IEEE, Honolulu, HI, EM Moro, Algebrac geometry modelng n nformaton theory. World Scentfc, Sngapore, E Anderson, Z Ba, C Bschof, S Blacford, J Demmel, J Dongarra, JD Croz, A Greenbaum, S Hammarlng, A McKenney, D Sorensen, LAPACK user s gude. SIAM, Phladelpha, Y Zrell, Identfcaton aveugle de codes correcteurs d erreurs basés sur des grands corps de Galos et recherche d algorthmes de type décson souple pour les codes convolutfs. PhD thess, Unversté de Brest, France, 2013 Submt your manuscrpt to a ournal and beneft from: 7 Convenent onlne submsson 7 Rgorous peer revew 7 Immedate publcaton on acceptance 7 Open access: artcles freely avalable onlne 7 Hgh vsblty wthn the feld 7 Retanng the copyrght to your artcle Submt your next manuscrpt at 7 sprngeropen.com