Advanced coding schemes against jamming in telecommand links
Marco Baldi, Marco Bianchi, Franco Chiaraluce, Roberto Garello, Nicola Maturo, Ignacio Aguilar Sanchez, Stefano Cioni
aa r X i v : . [ c s . I T ] O c t Advanced coding schemes against jamming in telecommand links
M. Baldi ∗ , M. Bianchi ∗ , F. Chiaraluce ∗ , R. Garello † , N. Maturo ∗ , I. Aguilar Sanchez ‡ , S. Cioni ‡∗ DII, Universit`a Politecnica delle MarcheAncona, ItalyEmail: { m.baldi, m.bianchi, f.chiaraluce, n.maturo } @univpm.it † DET, Politecnico di TorinoTorino, ItalyEmail: [email protected] ‡ TEC-ETC, ESA-ESTECNoordwijk, The NetherlandsEmail: { ignacio.aguilar.sanchez, stefano.cioni } @esa.int Abstract —The aim of this paper is to study the performanceof some coding schemes recently proposed for updating theTC channel coding standard for space applications, in thepresence of jamming. Besides low-density parity-check codes,that appear as the most eligible candidates, we also considerother solutions based on parallel turbo codes and extendedBCH codes. We show that all these schemes offer very goodperformance, which approaches the theoretical limits achiev-able.
Keywords -Error correcting codes, jamming, telecommands.
I. I
NTRODUCTION S Pace missions can be impaired by intentional or uninten-tional jamming. Such a threat is particularly dangerousfor telecommands (TC), since the success of a mission maybe compromised because of the denial of signal receptionby the satellite. It is well known that to counter the jammingthreat, error correcting codes can be used jointly with directsequence spread spectrum. This topic has been investigatedin previous literature, but rarely taking into account thepeculiarities of the TC space link. As a consequence, incurrent standards or recommendations on TC space links,only weak countermeasures are included, that do not appearadequate to face the increasing skill of malicious attacks.Whilst for the spreading technique a relevant advance isbrought by the introduction of long cryptographic pseudo-noise sequences [1], the discussion is quite open as regardspossible usage of new coding techniques.As a matter of fact, the only error correcting code cur-rently included in the standards and recommendations forTC applications [2], [3] is the BCH code with dimension
Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected] work was supported in part by the ESA Contract No: 4000106268:Advanced Coding Schemes for Direct Sequence Spread Spectrum Telecom-mand Links and in part by the MIUR project “ESCAPADE” (GrantRBFR105NLC) under the “FIRB - Futuro in Ricerca 2010” fundingprogram. k = 56 and length n = 63 exploiting hard-decisiondecoding. The performance of this code in the presence ofjamming is generally not good and it is possible to verifythat significant losses appear even when an interleaver (notcurrently included in the standard) is employed. Actually,the performance of the BCH (63 , code is unsatisfactoryeven when considered on the additive white Gaussian noise(AWGN) channel. For this reason a number of new proposalshave been formulated with emphasis on binary and non-binary low-density parity-check (LDPC) codes [4], [5],[6]. Since good correction capability and short frames areneeded, the most recent proposals consider codes with rate R c = 1 / and k = 64 , or . Using non-binaryLDPC codes allows to improve on their binary counterparts[7]: as an example, a non-binary LDPC (128 , code at acodeword error rate (CER) of about − gains roughly dB over the binary LDPC (128 , code. The performanceof these codes over jamming channels has not yet beeninvestigated.In this paper we consider different types of jammingsignals, namely: pulsed jamming, continuous wave (CW)jamming and pseudo-noise (PN) jamming. We also studythe impact of jammer state information (JSI), clipping andinterleaving, under typical TC application constraints. Inorder to assess how far the performance of the consideredcodes is from the theoretical limits, we extend the conceptof Shannon’s sphere packing lower bound (SPLB). Throughour analysis, we are able to identify which are the criticalvalues of the signal-to-interference ratio (SIR) for which thecoding scheme is no more able to guarantee an acceptablelevel of protection.As further possible candidate schemes, we consider shortparallel turbo codes (PTC) and extended BCH (eBCH) codeswith soft-decision decoding. For decoding the eBCH codeswe consider the most reliable basis (MRB) algorithm, whichhas been successfully applied to these codes over the AWGNchannel. We extend its use also to the jamming channel.he organization of the paper is as follows. In Section IIwe introduce the types of jamming. In Section III we giveexamples of the performance of the BCH (63 , code overthe jamming channel. In Section IV we describe the newcoding schemes and in Section V the performance metricsadopted, including an extension of the SPLB. In SectionVI we provide some numerical examples. In Section VIIwe evaluate the impact of finite length interleavers for thesystem using PTC. Finally, Section VIII concludes the paper.II. T YPES OF JAMMING
The definition of the types of jamming we consider isshortly reminded next. We assume that the system adoptsa direct sequence-spread spectrum (DS-SS), and binaryphase shift keying (BSPK) modulation with carrier circularfrequency ω . The DS-SS is characterized by bandwidth W ss and processing gain K . Moreover, the length (period) of thespreading sequence is denoted by L . A. Pulsed jamming
A pulsed jamming signal has the following characteristics: • white Gaussian noise on the whole bandwidth W ss ; • discontinuity, with pulse active time D and period T ,which means that the pulse is active for a fraction oftime (also called duty cycle) < ρ = DT ≤ ; • power J P during the active time D , and zero for theremaining time T − D .During the active time the jamming signal has a powerspectral density which is constant over the W ss band withvalue J P , where J P = J P W ss . For proper comparison itis also useful to introduce an equivalent (with the sameenergy) Gaussian continuous jamming signal. Since thesame energy is transmitted over T instead of D , it has apower J = ρJ P . This equivalent jamming signal has a powerspectral density constant over the W ss band, with value J ,where J = JW ss = ρJ P . The error rate performance canbe expressed in the terms of the ratio E b J between the energyper bit and the equivalent one-side jamming spectral density.When an error correcting code is applied, the impact ofpulsed jamming can be mitigated through interleaving. Thisis because most of the forward error correction schemesare designed for an AWGN channel which exhibits nomemory. They do not handle bursts of errors. An inter-leaver distributes a burst of errors among many consecutivecodewords. By doing this the number of errors contained ineach codeword is limited, the code is able to correct themand the burst is neutralized. For analysis purposes, we caninitially refer to an ideal interleaver. This implies that if aburst of errors corresponds to a fraction ρ of the symbols, itsimpact after de-interleaving is modeled as a probability ρ ,for each symbol, of having a higher noise variance. Even ifan ideal interleaver cannot be implemented, it is very usefulfor analytically investigating the performance over jamming channels. Then, the performance in the presence of a realinterleaver can be determined through simulation. B. CW jamming
A CW jamming is a narrowband, continuous signal oftype j ( t ) = √ J cos ( ω j t + θ j ) . (1)Hence, it is a pure tone with: • circular frequency ω j = 2 πf j which, in general, maybe different from the signal circular frequency ω ; • initial phase θ j which, in general, may be different fromthe signal initial phase (conventionally set to ); • power J .The worst case occurs when ∆ ω = ω j − ω = 0 . Underthe hypothesis of having a large K < L , the jammingcontribution on a generic symbol can be modeled by aGaussian random variable with zero mean and variance JT b K cos ( θ j ) , where T b is the bit time duration. In the caseof CW jamming it is preferable to express the error rateperformance in terms of the SIR, SJ , where S is the signalpower. C. PN jamming
Let us denote by c ( t ) the spreading sequence used in theDS-SS system. A PN jamming is a signal of type j ( t ) = √ Jc ( t − τ ) cos ( ω t ) . (2)Hence, it is a DS-SS signal with: • circular frequency equal to ω ; • spreading sequence c ( t ) different from c ( t ) (althoughit may have the same length); • time delay τ on the spreading sequence; • power J .A common choice for the spreading sequence c ( t ) is touse a Gold code. The novel cryptographic PN sequencesproposed for TC applications [1] are very long (a suggestedlength is L = 2 − ). Once again, under the hypothesisof a large K < L , the interfering contribution on a genericsymbol can be modeled by a Gaussian random variable withzero mean and variance
JS E b K . Thus we can obtain the sameexpression as for CW jamming with θ j = 0 .For all the types of jamming the channel is also impairedby thermal noise with signal-to-noise ratio per bit E b N . Asthe two disturbances are independent one each other, whensimultaneously present, their variances can be summed.III. C URRENT STANDARD
The TC protocols for synchronization and channel codingare specified (with some differences) both in the recom-mendation [3] issued by the Consultative Committee forSpace Data Systems (CCSDS) and in the standard [2] issuedby the European Cooperation for Space Standardization(ECSS). Let us refer to the CCSDS recommendation [3]: -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] No jamming r = 1 r = 0.5 r = 0.2 Figure 1. CER performance of the BCH (63 , code over pulsed jammingchannel, for E b /J = 10 dB and no interleaver. it specifies the functions performed in the “Synchronizationand Channel Coding sublayer” in TC ground-to-space (orspace-to-space) communication links. In short, the sublayertakes transfer frames (TFs) produced by the upper sublayer(“Data Link Protocol sublayer”), elaborates them and out-puts Communications Link Transmission Units (CLTUs) thatare passed to the lower layer (‘Physical layer”) where theyare mapped into the transmitted waveform by adopting aproper modulation format. Details on the structure of the TFand CLTU can be found in [3], [2] and are here omitted forthe sake of brevity. The current CCSDS recommendation andECSS standard use a BCH (63 , code for error protectionagainst noise and interference. At the receiver side a harddecision is taken on the received symbols. The perfor-mance against pulsed jamming of the hard-decision decodedBCH (63 , code is rather poor. Examples are shown in Fig.1 for the case without the interleaver (as addressed by thecurrent standard) and in Fig. 2 for the case with the (ideal)interleaver. Performance is expressed in terms of the CER;the value of E b /J has been set equal to dB and theCER is plotted as a function of E b /N for some valuesof ρ . As expected, performance degrades for decreasing ρ ; the use of the (ideal) interleaver introduces for ρ < an improvement that however remains unsatisfactory to thepoint that reaching CER = − , that is a reference valuefor TC applications, is practically impossible for ρ = 0 . and ρ = 0 . .An example of the performance of the hard-decisiondecoded BCH (63 , code against CW jamming is shownin Fig. 3 for K = 100 , θ j = 0 and ∆ ω = 0 (worst case).The SIR value is assumed as a parameter and we see that aSIR in the order of − dB makes the system unpractical.A further reduction in the value of K would require higherSIR values; for example, by assuming K = 10 , the CER -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] No jamming r = 1 r = 0.5 r = 0.2 Figure 2. CER performance of the BCH (63 , code over pulsed jammingchannel, for E b /J = 10 dB and ideal interleaver. -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] No jamming SIR = 10 dB SIR = 0 dB SIR = -10 dB
Figure 3. CER performance of the BCH (63 , code over CW jammingchannel ( θ j = 0 and ∆ ω = 0 ), for K = 100 . target of − becomes practically unreachable just for SIR = 0 dB.For PN jamming it is possible to verify that under theGaussian approximation its impact is equivalent to that ofCW jamming with θ j = 0 and ∆ ω = 0 . In Fig. 4 we haveplotted the CER curve as a function of the SIR for differentvalues of K and E b /N = 10 dB. So this curve also appliesto the worst case CW jamming with the same SIR.IV. N EW CODING SCHEMES
The codes proposed for TC channel coding updating havelength greater ( n = 128 , and ) and rate smaller( / ) than the standard code. Also the code type has beenchanged, with the aim to introduce state-of-the-art codes.The mostly addressed candidates, in this sense, are LDPCcodes, both binary and non-binary. Recently, however, we
10 -8 -6 -4 -2 0 2 4 6 8 1010 -7 -6 -5 -4 -3 -2 -1 SIR [dB]
K = 10 K = 100 K = 1000
Figure 4. CER performance of the BCH (63 , code over PN jammingchannel, for fixed E b /N = 10 dB. have also shown that potential competitors can be PTC andeven BCH codes, if they are soft-decision decoded withmaximum likelihood (ML)-like algorithms characterized bylimited complexity. The main features of these schemes arereminded next. A. LDPC codes
A class of binary LDPC codes that is suitable for TCapplications has been proposed by the National Aeronauticsand Space Administration (NASA) and is described in [4].It is based on the adoption of three systematic short binaryLDPC codes designed using protographs with circulantmatrices. Soft-decision decoding can be realized by usingthe classic sum-product algorithm with log-likelihood ratios(LLR-SPA).Non-binary LDPC codes with the same lengths havebeen analyzed by NASA [5] and independently by theDeutsches Zentrum f¨ur Luft- und Raumfahrt (DLR) in ajoint work with the University of Bologna (UniBO) [7].In this paper we refer to the DLR-UniBO implementation.Decoding is realized by using iterative algorithms based onfast Hadamard transforms.
B. Parallel turbo codes
Parallel turbo codes are one of the coding options of theCCSDS recommendation for telemetry (TM) links [8]. TheCCSDS turbo encoder is based on the parallel concatenationof two equal 16-state systematic convolutional encoderswith polynomial description (1 , (1 + D + D + D ) / (1 + D + D )) . The interleavers are based on an algorithmicrule proposed by Berrou and described in [8]. The CCSDSturbo encoder has four possible information frame lengths: , , and bits. The nominal code ratecan be / , / , / and / . However, higher rates areobtainable by puncturing [9]. Table IS
ELECTED INTERLEAVERS FOR THE PARALLEL TURBO CODES . Input length k Interleaver Minimum distance parameters d min /A min /w min
64 DRP 10/5/23128 DRP 13/13/40256 QPP 15/1/1
Maintaining unchanged the encoder structure, we haveconsidered frame lengths shorter than those in the TMrecommendation and fixed the nominal code rate to / ,in such a way as to comply with the NASA’s choicesdiscussed above. Because of the shorter length, we cannotuse the interleavers in [8] and we must design new smallerinterleaving structures.Among a wide number of different options, we havefocused attention on: completely random, spread [10],Quadratic Permutation Polynomial (QPP) [11] and DitheredRelative Prime (DRP) [12] interleavers. Moreover, since theconstituent CCSDS convolutional codes have states, fourextra-tail bits are needed for termination; then, the turbocodeword length is n = 2( k +4) . As an example, for the caseof k = 64 , this implies to have n = 136 and an actual coderate . . In order to achieve exactly the code rate / , asit is necessary for fair comparison, we have implemented asuitable puncturing strategy.The interleaver and the puncturing pattern have beenjointly optimized, in such a way as to maximize the min-imum distance d min and minimize the codewords multi-plicity A min (i.e., the number of codewords with Hammingweight d min ); these parameters, in fact dominate the codeperformance at low error rates.The results of the design optimization are shown in TableI; besides d min and A min , also the information multiplicity w min (sum of the input weights over all the codewordswith weight d min ) is provided since, together with d min ,it determines the asymptotic bit error rate performance. Thedecoding of turbo codes is performed by iteratively applyingthe well-known BCJR algorithm [13] to the constituentencoders. C. Extended BCH codes
Although soft-decision decoding of BCH codes is gen-erally complex, in the case of short BCH codes with highrate, an exact ML soft-decision decoding is possible, throughits trellis representation (for example, based on the Viterbior the BCJR algorithms). However, having now decided touse codes with rate / and the shortest length , thesetechniques are too involved and therefore cannot be applied.Alternative solutions can be found, at least for the case of k = 64 . In fact, the eBCH (128 , code can be efficientlydecoded by using sub-optimal soft-decision decoding algo-rithms. Several options are available for this purpose, alsoexploiting LDPC-like code representations [14]. For thisode, we have focused on the MRB algorithm [15], whichhas very good performance and acceptable complexity. Itconsists of the following steps:1) Identify k = 64 most reliable received bits and obtainfrom them a vector v ∗ .2) Construct a systematic generator matrix G ∗ corre-sponding to these bits.3) Encode v ∗ by G ∗ to obtain a candidate codeword c ∗ = v ∗ G ∗ .4) Choose the order i of the algorithm.5) Consider all (or a proper subset of) test error patternsof length k and weight w ≤ i .6) For each of them: sum to v ∗ , encode by G ∗ , verifyif the likelihood is higher than that of the previouscandidate codeword and, if this is true, update thecandidate.Further details can be found in [15] and the referencestherein. The MRB algorithm can be applied also to the otherschemes (to LDPC codes, in particular) where, dependingon the order i , it can provide performance comparable to,or even better than, that offered by the iterative algorithms.V. P ERFORMANCE EVALUATION
The performances of the new coding schemes presentedin Section IV have been recently discussed and comparedover the AWGN channel [16]. We have verified they canprovide an advantage of more than dB over the currentBCH (63 , code. In Section VI we will show that similarimprovements can be achieved against jamming.For soft-decision decoding, the knowledge of the jammingstate can play a relevant role. For the case of pulsedjamming, for example, to have JSI means to know thenoise variance for each symbol. Thus, in our simulations,we have considered the case of perfect JSI, which meansthe receiver is able to identify the fraction, ρ , of symbolsaffected by jamming and the remaining fraction, − ρ , thatis only affected by thermal noise, and properly estimate theirnoise variances, which are σ ρ = N + J ρ and σ − ρ = N ,respectively. On the opposite side, we have also consideredthe case when JSI is not available, and the variance used forLLR calculation of the decoder input is always equal to theaverage value N + J .The goodness of the proposed solutions can be measuredthrough the distance of the CER curves from the SPLB.Among the various approaches available to compute theSPLB, the most suitable one is the so-called SP59 [17]. Amodified version of this bound is also available (called SP67[18]), that is able to take into account the constraint put bythe signal constellation (BPSK in the present analysis). Morerecent improvements [19], [20] are significant only for highcode rates or long codeword lengths, and these conditionsare not satisfied by the codes here of interest. Thus, in thepresent study, we consider the SP59 as the most significantSPLB. The SPLB reported in the literature refers to the AWGNchannel and needs generalization. This is easy to achievefor pulsed jamming on the condition that the pulse du-ration is a multiple of the codeword length and no in-terleaving is applied. For this purpose, let us denote bySPLB ( R c , n, E b /N ) the SP59 bound for the AWGN chan-nel. Under the hypotheses above an extended sphere packinglower bound, ESPLB, for the case of pulsed jamming canbe defined as: CER ≥ ESP LB (cid:18) R c , n, E b N , E b J , ρ (cid:19) = ρSP LB R c , n, EbN + EbJ ρ (3) + (1 − ρ ) SP LB (cid:18) R c , n, E b N (cid:19) . By setting ρ = 1 in (3), we obtain an expression whichis valid also for CW and PN jamming channels, when theGaussian approximation is applied. The ESPLB given by (3)will be considered in Section VI as a useful benchmark forthe case with JSI and without interleaving.VI. N UMERICAL EXAMPLES
Due to limited space, we focus on pulsed jamming andon codes with n = 128 and k = 64 . The analysis canbe extended to the longer codes for which, however, thecomplexity issue for the decoding algorithms adopted canbecome more critical.The performances of the new coding schemes presentedin Section IV are compared in Figs. 5-8 assuming ρ = 0 . , E b /N = 10 dB and variable E b /J , with and without an(ideal) interleaver, with and without JSI. In the latter case, tolimit the impact of the incorrect noise estimation a clippingthreshold, equal to twice the amplitude, has been applied tothe signal at the channel output. For the PTC we have usedthe optimal DRP interleaver reported in Table I. The orderused for the MRB algorithm applied to the eBCH code is i = 4 .For three of the considered scenarios the relative behaviorof the proposed schemes is very similar: the best perfor-mance is achieved by the eBCH code, while PTC and non-binary LDPC codes are very close one each other and suffera penalty with respect to the eBCH code that depends on thesimulation conditions. An interesting exception occurs forthe case with interleaving and without JSI where, becauseof the ordering mechanism that is at the basis of MRB, theeBCH code loses its leadership. The eBCH code is also veryclose to the ESPLB, where applicable (see Fig. 7). On thecontrary, the performance of the binary LDPC code is ratherpoor with a loss than can be larger than dB with respectto the best solution. These gaps are even more pronouncedthan those found over the AWGN channel [16]. -7 -6 -5 -4 -3 -2 -1 C E R E b /J [dB] Binary LDPC eBCH PTC Non-Binary LDPC
Figure 5. Performance of the new coding schemes ( n = 128 , k = 64) without interleaving and without JSI. -7 -6 -5 -4 -3 -2 -1 C E R E b /J [dB] Binary LDPC eBCH PTC Non-Binary LDPC
Figure 6. Performance of the new coding schemes ( n = 128 , k = 64) with interleaving and without JSI. VII. I
MPACT OF FINITE LENGTH INTERLEAVER
The results shown in the previous section for the systemsusing interleaving referred to the adoption of an idealinterleaver. In this section we discuss the effect of usingreal interleavers characterized by finite length. We considerthe particular case of using PTC, but a similar analysis couldbe developed for the other schemes.With reference to the short description given in SectionIII (but further details can be found in [3]), we suppose thatinterleaving is applied at CLTU level and also taking intoaccount the possible presence of partitioning. This occurswhen the TF has length M that is not a multiple of k : theTF is partitioned into N = ⌈ Mk ⌉ input blocks and, if needed,zero filling is used to complete the last block. Each blockis then encoded producing C = N n
CLTU coded bits. Forthe sake of simplicity, in this first evaluation we neglect the -7 -6 -5 -4 -3 -2 -1 C E R E b /J [dB] Binary LDPC ESPLB eBCH PTC Non-binary LDPC
Figure 7. Performance of the new coding schemes ( n = 128 , k = 64) without interleaving and with JSI. -7 -6 -5 -4 -3 -2 -1 C E R E b /J [dB] Binary LDPC eBCH PTC Non-Binary LDPC
Figure 8. Performance of the new coding schemes ( n = 128 , k = 64) with interleaving and with JSI. presence of the preamble ( bits) and the postamble (64bits) that are added for CLTU synchronization [3]. So, weassume that an interleaver is applied to the C CLTU codedbits to increase the protection against bursts; it involvesall the N codewords of the CLTU. As a simple example,we consider a square R × R row-by-column interleaverwhere R = ⌈√ C ⌉ ; the C CLTU coded bits are writtenby row and read by column (or vice versa). In this case,wishing to compare the impact of bursts of growing lengthit is preferable to refer directly to the power J P and thecorresponding ratio E b J P = SJ P K . An example of the impactof a real interleaver on the transfer frame error rate (FER)in the considered scenario is shown in Fig. 9, for E b J P = 0 dB, M = 2048 and a burst length of bits. We observethat the CLTU interleaver is very effective against burstsproduced by pulsed jamming. For long TFs (like the one -4 -3 -2 -1 F E R E b /N [dB] Without interleaving With interleaving No jamming
Figure 9. Performance of the PTC (128 , without interleaving andusing a × row-by-column interleaver, for a burst length of bits; E b /J P = 0 dB and M = 2048 . here considered) the burst is practically neutralized by theinterleaver.VIII. C ONCLUSION AND OPEN ISSUES
For the first time at our knowledge, the performance overjamming channels of new coding schemes that potentiallyreplacing the BCH (63 , code in the space TC channelcoding standard has been investigated. The adoption ofsoft-decision decoding, combined with interleaving and JSI,allows to achieve significant improvements. Whilst it isconfirmed that non-binary LDPC codes, that are consideredthe most eligible candidates, are generally very good, wehave shown that comparable performances can also beachieved by using PTC or eBCH codes. We have studiedthe case of n = 128 and k = 64 , where eBCH codes oftenoffer the best results at an acceptable complexity.This fact, however, cannot be used to draw general con-clusions. First of all, extending the sub-optimal decodingalgorithms to longer codes, while maintaining acceptablecomplexity, may be quite difficult. Additionally, the sub-optimal algorithms generally define “complete” decoders. Asit is well known, this may be a penalty for the undetectedcodeword (or frame) error rate that, in TC applications, isat least as important as the codeword (or frame) error rate.Further work is progress to assess either the complexity andthe completeness issues.R EFERENCES [1] Thales Alenia Space, “Cryptographic Pseudo-Noise Codesand Related Acquisition Techniques for Direct-SequenceSpread Spectrum Transponders - Final Report,” RPT-RFP-ESA-00013-AASI, Tech. Rep. Issue 2, Sep. 2011.[2] ECSS, “Space Data Links - Telecommand Protocols, Syn-chronization and Channel Coding,” Jul. 2008, ECSS-E-ST-50-04C. [3] CCSDS, “TC Synchronization and Channel Coding,” Sep.2010, Blue Book, CCSDS 231.0-B-2.[4] ——, “Short Blocklength LDPC Codes for TC Synchroniza-tion and Channel Coding,” Apr. 2012, Orange Book, CCSDS231.0–O–x.x.[5] E. Greenberg, “CCSDS Telecommand Sync and ChannelCoding Specification using advanced Block Codes,” CCSDSNGU WG meeting, Cleveland, OH, Tech. Rep., Oct. 2012.[6] G. Liva, E. Paolini, T. De Cola, and M. Chiani, “Codes onhigh-order fields for the CCSDS next generation uplink,” in
Proc. 2012 6th Advanced Satellite Multimedia Systems Conf.(ASMS) and 12th Signal Proc. Space Commun. Workshop(SPSC) , Wessling, Germany, Sep. 2012.[7] L. Costantini, B. Matuz, G. Liva, E. Paolini, and M. Chiani,“Non-binary protograph low-density parity-check codes forspace communications,”
Int. J. Sat. Commun. and Network-ing , vol. 30, pp. 43–51, 2012.[8] CCSDS, “TM Synchronization and Channel Coding,” Aug.2011, Blue Book, CCSDS 131.0-B-2.[9] G. P. Calzolari, E. Vassallo, F. Chiaraluce, and R. Garello,“Turbo code applications on telemetry and deep space com-munications,” in
Turbo Code Applications: a Journey from aPaper to Realization , K. Sripimanwat, Ed. Springer, 2005,ch. 13, pp. 321–344.[10] D. Divsalar and F. Pollara, “Multiple turbo codes for deep-space communications,” JPL, TDA Prog. Rep. 42-121, Tech.Rep., May 2005.[11] J. Sun and O. Y. Takeshita, “Interleavers for turbo codes usingpermutation polynomials over integer rings,,”
IEEE Trans.Inform. Theory , vol. 51, no. 1, pp. 101–119, Jan. 2005.[12] S. Crozier and P. Guinand, “High-performance low-memoryinterleaver banks for turbo-codes,” in
Proc. IEEE 54th Vehic-ular Technology Conference (VTC Fall 2001) , Atlantic City,NJ, Oct. 2001.[13] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimaldecoding of linear codes for minimizing symbol error rate,”
IEEE Trans. Inform. Theory , vol. IT-20, no. 2, pp. 284–287,Mar. 1974.[14] M. Baldi, G. Cancellieri, and F. Chiaraluce, “Iterative soft-decision decoding of binary cyclic codes based on spreadparity-check matrices,” in
Proc. SoftCOM 2007 , Split -Dubrovnik, Croatia, Sep. 2007.[15] Y. Wu and C. N. Hadjicostis, “Soft-decision decoding usingordered recodings on the most reliable basis,”
IEEE Trans.Inform. Theory , vol. 53, no. 2, pp. 829–836, Feb. 2007.[16] M. Baldi, M. Bianchi, F. Chiaraluce, R. Garello,I. Aguilar Sanchez, and S. Cioni, “Advanced channelcoding for space mission telecommand links,” in
Proc. IEEE78th Vehicular Technology Conference (VTC Fall 2013) , LasVegas, NV, Sep. 2013.[17] C. E. Shannon, “Probability of error for optimal codes ina Gaussian channel,”
Bell Syst. Tech. J. , vol. 38, no. 3, pp.611–656, May 1959.[18] C. E. Shannon, R. G. Gallager, and E. R. Berlekamp, “Lowerbounds to error probability for coding on discrete memorylesschannels,”
Info. and Control , vol. 10, pp. 65–103, 1967.[19] A. Valembois and M. P. C. Fossorier, “Sphere-packing boundsrevisited for moderate block lengths,”
IEEE Trans. Inform.Theory , vol. 50, no. 12, pp. 2998–3014, Dec. 2004.[20] G. Wiechman and I. Sason, “An improved sphere-packingbound for finite-length codes over symmetric memorylesschannels,”