Diversity Enhancement of Coded Spread Spectrum Video Watermarking

12 Pages·2002·0.182 MB·English

by  Brandao, Queluz, Rodrigues

WirelessPersonalCommunications 23: 93–104,2002.
©2002KluwerAcademicPublishers. PrintedintheNetherlands.
Diversity Enhancement of Coded
Spread Spectrum Video Watermarking
T.BRANDÃO,M.P.QUELUZ∗ andA.RODRIGUES
IT/IST,TechnicalUniversityofLisbon,Av.RoviscoPais,1049-001Lisboa,Portugal
∗
E-mail:[email protected]
Abstract. Thispaperanalysestheeffectofsignalcombinationtechniquesinvideowatermarkdetection.Aspatial
spreadspectrumbasedwatermarkingtechiqueisusedasembeddingmethod,incombinationwithcommonerror
correctioncodes(BCH,Reed-Solomonwithmultilevelsignaling,binaryconvolutionalcodeswithViterbidecod-
ing).Besidesananalytical evaluationof thebiterrorrate,theeffectivenessofthechannel coding anddiversity
techniquesisalsoassessedexperimentallyunderMPEG-2videocompression.
Keywords:watermarking,spread-spectrum,signalcombinationtechniques.
1. Introduction
With the evolution of 2G and the upcoming introduction of 3G networks, service providers
will be able to expand the number of services and to support m-commerce over mobile net-
works.Thiswillallowtheexchangeofsuchdigitalcontentasdigitalmusic,streamingvideo,
e-books, but it has become the major concern for content providers due to the possibility of
high quality copying in the digital domain. In order to prevent this problem it is necessary
to develop digital rights management systems that are able to introduce copy control, media
identification and tracing mechanisms. These goals can be achieved by using watermarking
technology.Thistechnologyisparticularlyadaptedtobeusedinmobilesystemssincemobile
terminals provide reliable information about the identity of the user, that can be used for
watermarkinganddataidentification purposes [1].
Mostproposed watermarking methods use aspread spectrum approach [2]: anarrowband
signal (the watermark information) has to be transmitted via a wideband channel that is
subject to noise and distortion (the multimedia host data, e.g., video). Under this approach,
digitalwatermarkingcanbetreatedasacommunicationproblem.Someauthors[3–5]haveal-
readyshownthaterrorprotection techniques mightbeusedadvantageously inwatermarking.
This paper considers a spatial spread spectrum based watermarking technique as embedding
methodincombinationwithcommonerrorcorrectioncodes(BCH,Reed–Solomonwithmul-
tilevel signaling, binary convolutional codes with Viterbi decoding) applied to a set of video
sequences.Inordertoimprovetheresults,theuseofdiversitytechniques,inconjunctionwith
channelcoding, isproposed. Thisconcept, wellknownfromdigitalcommunicationtheory,is
implemented by simultaneously considering a group of consecutive frames at the extraction
procedure.Analyticalexpressionsandboundsforthebiterrorrtearecomparedwithempirical
results.Theeffectivenessofthechannelcodinganddiversitytechniquesisalsoassessedunder
MPEG-2videocompression.
94 T.Brandãoetal.
Figure1. Watermarkembedding/extractionschemes:(a)Embedding;(b)Extraction.
2. Watermark Embedding
The watermark embedding system is depicted in Figure 1(a). The mark consists on a N bit
b
sequence B = {b ,b ,...,b } and isembedded inthe luminance component ofthe image.
1 2 Nb
Before embedding, the binary sequence B is mapped to a symbol sequence B , with length
s
N . If M levels are used to perform multilevel signaling, then N = N /log M resulting in
s s b 2
M differentsymbolsA ,...,A ,eachoneconveying l = log M bits.
1 M 2
The channel encoder performs error correction encoding over the symbol sequence B .
s
Encoding isperformed using either binary block codes (BCHcodes) orbinary convolutional
codes, if M = 2, and non-binary block codes (RScodes), if M > 2. Ifthe selected code has
coderatecr,theencoded symbolsequence, B ,willhaveN symbols,withN = N /cr.
cs c c s
For M > 2 and following the approach proposed in [6], the symbol sequence B is
cs
modulated using M bi-orthogonal sequences with zero mean and unitary variance, which
assigns a modulating sequence s , i ∈ {1,...,N }, to symbol i. The use of M bi-orthogonal
i c
sequences requires M/2 orthogonal sequences, which are used to modulate symbols A to
1
AM/2. The remaining symbols, AM/2+1 to AM, are modulated using the antipodal sequences
of the defined M/2 orthogonal sequences. ForM = 2, two antipodal sequences are used for
modulating thetwodifferentsymbols.
The modulating sequences, s , are then sent to a scrambler that maps each sequence to a
i
sub-setofpixelpositions.Themappingsarenon-overlappingandpseudo-randomlygenerated,
being secret key dependent, and the inverse operation is only possible if the key is known.
Symbols (m,n)designatestheelementofthesequences thatwasmappedtoimageposition
i i
(m,n).
DiversityEnhancementofCodedSpreadSpectrum VideoWatermarking 95
After the spatial assignment, the values s (m,n) are further weighted by a local factor
i
α(m,n), the purpose of which is to adapt the embedding to the human visual system. The
watermark w is then defined as the superposition of all modulated, scrambled and weighted
sequences s ,as:
i
(cid:1)Nc
w(m,n)= α(m,n)s (m,n). (1)
i
i=1
Tocomplete theprocessofwatermarkembedding, thewatermarkw isaddedtotheimage
luminancecomponentX,resultinginawatermarkluminancecomponentY.Theprocedureis
extended to video sequences following a frame-based approach – the embedding separately
processes eachframeofthevideostreamandthesamemarkisembeddedineachframe.
3. Watermark Extraction
Retrievingthewatermarkwithoutanyknowledgeoftheoriginalimagecanbeachievedusing
the system depicted in Figure 1(b). To reduce major components of the image signal itself a
receiverfilter(filterF)isused.Asshownin[7],itmaysignificantlyimprovetheperformance
of the watermark extraction system. After this pre-processing step is completed, the filtered
imagepassesthroughtheunscrambler,whichperformstheinverseoperationofthescrambler
definedinprevioussection. Usingthesamekeyoftheembedding, theunscrambler generates
theimagepositions corresponding toeachembeddedsymbol.
ThedemodulatorconsistsinM/2linearcorrelatorswherethereceivedsignaliscorrelated
witheach orthogonal sequence. Thecorrelation exhibiting the largest absolute value leads to
thechoiceoftwopossiblesymbols:symbolA oritsantipodalpair.Thesignofthecorrelation
x
completes this selection: if it is positive, then symbol A is selected, otherwise the antipodal
x
symbol is selected. To complete the watermark extraction algorithm, the received symbol
ˆ
sequence is decoded and mapped to a binary sequence. The resulting binary sequence, B, is
the received watermark. Forvideo sequences, the extraction algorithm operates over a group
ofJ (diversity window)consecutive frames(seeSection5).
4. PerformanceintheAbsenceofCompression
4.1. UNCODED CASE
AssumingthatfilterF guaranteesavalidgaussianchannelapproach(F shouldbeawhitening
filter)anddefiningµandσ astheexpectedvalueandthestandarddeviationofthecorrelators’
output,respectively, anapproximation forthesevauescanbewrittenas[6]:
DVH
µ ≈ E[α(m,n)] (2)
N
c
(cid:2)
DVH
σ ≈ (E[Yˆ2(m,n)]+E[α2(m,n)]), (3)
N F
c
where D is the density of watermark embedding (the ratio between marked image locations
ˆ
and total number of image locations), Y is the marked and filtered luminance component
F
96 T.Brandãoetal.
of the image at the unscrambler output, N is the number of embedded symbols, H,V are,
c
respectively, thehorizontal andverticalimagedimensions andE[.]denotes expectedvalue.
For uncoded M-ary bi-orthogonal signaling, the symbol error probability P , is given by
M
[8]:
1 (cid:3) +∞ v2 (cid:4) 1 (cid:5) µ(cid:6)(cid:7)M2−1
P = 1− √ e ·erf √ v+ dv, (4)
M
2π µ 2 2 σ
σ
andthebiterrorprobability P isbounded by:
b
P
M < P ≤ P . (5)
b M
2
In the binary case, P matches the upper bound (P = Q(µ/σ) = P , with M = 2) and
b b M
withincreasing numberoflevels,thisprobability approaches thelowerbound(large M leads
toP ≈ P /2).
b M
4.2. BINARY AND NON-BINARY BLOCK CODES
Letus consider the case in which binary antipodal signaling (M = 2) is used in conjunction
with a linear (n,k) binary block code with minimum distance d = 2t +1, where t is the
min
maximum number of errors corrected by the code, and a bit-by-bit hard decision. Assuming
thatthebiterrorsoccurindependently,theprobabilityofadecodedmessagebit-errorisupper-
boundedby[9]:
(cid:4) (cid:7)
(cid:1)n
1 n
P ≤ min(i +t,n) Pi(1−P )n−i, (6)
db n i b b
i=t+1
where P stands for channel bit error rate. This probability is given by (4) using M = 2 and
b
µ/σ computedforN = N n/kembeddedbits.Thetermmin(i+t,n)in(6)guarantees that
c b
impossibleoccurrences ofmorethannbiterrorspercodewordarenotconsidered.
For non-binary (N,K) linear block codes, with symbol-by-symbol hard decision, and
considering P as the probability of symbol error in the channel (as defined in Equation
M
(4),butnowcomputedforN = N N/(Klog M)inserted symbols), weget:
c b 2
(cid:4) (cid:7)
2l−1 1 (cid:1)N N
P ≤ · min(i +t,N) Pi (1−P )N−i. (7)
db 2l −1 N i M M
i=t+1
4.3. CONVOLUTIONAL CODES
In this paper, a soft decision Viterbi algorithm was used for encoder/decoder implemen-
tation. The code rate cr and the constraint length L usually characterize a convolutional
code. The minimum free distance d is also an important parameter in the definition of the
f
code performance. It can be defined as the resulting minimum distance when the constraint
lengthapproachesinfinity.Itscalculationinvolvesthegeneratingfunctionoftheconvolutional
code, T(D,N). If transmission errors occur with equal probability and independently, the
probability ofadecoded messagebit-error isupper-bounded by[9]:
(cid:9)
(cid:5)(cid:8) µ(cid:6) dfµ2 ∂T(D,N)(cid:9)(cid:9)N = 1
Pdb < Q df σ e 2σ2 ∂N (cid:9)D = e−2µσ22 . (8)
DiversityEnhancementofCodedSpreadSpectrum VideoWatermarking 97
5. Diversity Techniques
Videowatermarkcanbeseenasamulti-channel system,sincethesamemarkisembeddedin
eachframe,andeachframecanbeconsideredasanindependentchannel.Inthissense,results
from diversity theory can be aplied, and watermark extraction may be improved considering
simultaneouslyagroupofJ (diversitywindow)consecutiveframes.Threestrategiesofsignal
combinationhavebeenstudiedandimplemented. Theyarediscussedinthefollowing,forthe
caseM = 2.
5.1. MAJORITY LOGIC (ML)
In this technique, the coded symbols (or the message symbols if channel coding is not used)
are independently extracted for each frame and the final symbol sequence is obtained by
simple majority counting over the retrieved symbols. The resulting N symbol sequence is
c
thenappliedtothechanneldecoder.
Let us consider that the bit error probability of the coded watermark, P , is the same for
b
everyframe.Then,assumingindependencebetweenerrorsinconsecutiveframes,thebiterror
probability P obtained afterapplying MLoveragroupofJ framesisgivenby:
bf
(cid:1)J
P = CJPi(1−P )J−i. (9)
bf i b b
i=J+1
2
IfP (cid:16)1,Equation(9)canbesimplifiedconsidering onlythefirsttermofthesum:
b
J+1
P ≈ DJ P 2 . (10)
bf J+1 b
2
From Equation (10), we can conclude that in most cases P (cid:16) P , which proves the
bf b
interestofusingtimediversitybasedonmajoritylogic.
5.2. EQUAL GAIN COMBINING (EGC)
In this technique, for each coded symbol, the correlation output obtained in each frame
is summed-up along the group of frames. The resulting correlation value is used for the
transmittedsymboldecision.
Considering that r is the correlator output for one symbol in frame i, the resulting
i
correlation value,r(cid:17),forthesamesymbol,aftercombining J frames,isgivenby:
(cid:1)J
r(cid:17) = r . (11)
i
i=1
Assumingoncemoreagaussianmodelforthechannel,thesignalattheoutputofthesignal
combinerwillhaveanormaldistributionwithmeanµ(cid:17)andvarianceσ(cid:17)2.Thus,theprobability
densityfunction ofr(cid:17),p(r(cid:17)),willbegivenby:
p(r(cid:17)) = √ 1 e−(r(cid:17)2−σµ(cid:17)2(cid:17))2, (12)
2πσ(cid:17)
98 T.Brandãoetal.
where µ(cid:17) and σ(cid:17)2 are obtained as a function of the correlator’s statistics for each frame i
according to:
(cid:1)J (cid:1)J
µ(cid:17) = µ , σ(cid:17)2 = σ2. (13)
i i
i=1 i=1
Hence,thebiterrorprobability atthesignalcombining output, P ,isgivenby:
bf
 
(cid:1)J
 
(cid:4) (cid:7)  µ 
i
Pbf = Q µσ(cid:17)(cid:17) = Q(cid:13)(cid:14)(cid:14)i=(cid:1)1J  . (14)
(cid:15) 
σ2
i
i=1
Assuming that thestatistics (meanand variance) arethe sameforallframes (which isnot
exactlytrueforMPEG-2compressed frames),i.e.:
µ = µ = ··· = µ = µ (15)
1 2 J
σ2 = σ2 = ··· = σ2 = σ2, (16)
1 2 J
thebiterrorprobability, P ,aftercombining thereceivedsignaloverJ frames,isgivenby:
bf
(cid:5)√ (cid:6)
µ
P = Q J . (17)
bf
σ
Asfrom(4)P = Q(µ/σ),wewillhaveingeneralP (cid:16)P .
b bf b
5.3. MAXIMAL GAIN COMBINING (MGC)
In this strategy, the J signals are weighted proportionally to their estimated µ/σ2 ratio and
thensummed.Undertheassumptionofindependentchannelswithwhiteandadditivegaussian
noise,itmaximizesthesignal-to-noiseratioatthecombiningoutput[8].Thecorrelationvalue
ofthecombinedsignal, foronegivensymbol,isthengivenby:
(cid:1)J µˆ
r(cid:17) = ir , (18)
σˆ2 i
i=1 i
whereµˆ andσˆ are,respectively,theestimatedmeanandstandarddeviationofthecorrelators
i i
outputatframei.Themaximumlikelihood estimatesoftheseparametersaregivenby[10]:
(cid:13)
(cid:14)
µˆ = 1 (cid:1)Nc rjbj, σˆ = (cid:14)(cid:15) 1 (cid:1)Nc (rjbj −µˆ )2, i = 1,2,...,J , (19)
i N i i N i i
c j=1 c j=1
wherethevaluesbj ∈{−1,1},j = 1,...,N arethebinary1 antipodalsymbolrepresentation
c
of the embedded mark and where rj is the correlator output for symbol j in frame i. As the
i
receiver does not a priori know the embedded symbols, the bj values required in (19) are
1 InthedevelopmentsofthissectionweareonlyconsideringtheM =2case.
DiversityEnhancementofCodedSpreadSpectrum VideoWatermarking 99
Figure 2. Error rate and µ/σ2 evolution during the three first GOPs of Stefan video sequence, after MPEG-2
videocompressionat4Mbit/s.
those obtained from the previous, already processed frames. For the first group of combined
frames,theEGCtechnique isapplied.
Figure 2 presents the evolution of (µˆ /σˆ2) and of the bit error rate, for the three first
i i
Group of Pictures (GOP) of a TV sequence, after MPEG-2 video compression at 4 Mbit/s.
Asexpected, thehighestvaluesof(µˆ /σˆ2)areobtained forthelowesterrorrates(i.e.,forthe
i i
frameslessdistorted duetocompression).
Assuming again a gaussian channel, the bit error probability for MGC, P , is given by
bf
[10]:
(cid:13) 
(cid:14) (cid:4) (cid:7)
P = Q(cid:14)(cid:15)(cid:1)J µi 2. (20)
bf
σ
i=1 i
Togetsomeinsightintothebehaviorofthistypeofsignalcombination, letusassumethat
the correlators statistics (mean and variance) are constant, µ = µ and σ = σ, except for a
i i
fraction ε (0 ≤ ε ≤ 1) of the frames, for which µ = γµ (0 ≤ γ ≤ 1). With parameter
i
γ, we intend to model the reduction (fade) of the watermark mean level, which may occur
due to compression. In this case, the gain G in signal to noise ratio between MGC and EGC
combination techniques isgivenby[10]:
(cid:8)
1+(γ2−1)ε
G= . (21)
1+(γ −1)ε
Figure3presentstheevolutionofGasafunctionofγ andε.Fromthisfigure,weconclude
that the best performance of MGC (comparatively to EGC) will be achieved when a high
fraction of frames are severely degraded (ε → 1,γ → 0), as in high compression ratio
scenarios.
100 T.Brandãoetal.
Figure3. Qualitative(a)andquantitative(b)evolutionofG,asafunctionofγ andε.
Figure4. (a)Table-tennis;(b)Stefan;(c)Mobileandcalendar;(a),(b)and(c)areCCIR-601videosequences,
with300frameseach.
6. Results
Figure4presentsthevideosequencesusedinsimulations.Thebi-orthogonalsequencesneces-
saryforM-arymodulationweregenerated usingHadamard–Walshfunctions. Theperceptual
factor, α(m,n) in Equation (1), was obtained by filtering the image with a laplacian high-
pass filterandtaking absolute values. Thecoefficients ofthelaplacian filterwerescaled by a
factorβ,whichaccounts forthewatermarkinsertion strength. Aspre-detection filter,filterF
inFigure1(b),a3×3cross-shaped high-pass filterhasbeenused.
In the absence of compression, theoretical and experimental curves for the bit error rate
(BER)wereobtained as afunction of the pulse size, defined as DHV/N , which accounts for
b
theamountofpixelsusedtoembedoneinformationbit.Forthetheoreticalplots,thestatistical
parametersµandσ wereestimatedapplying Equations(2)and(3).
TheplotsdepictedinFigure5showacomparisonfortheperformanceachievedwith:mul-
tilevelsignaling withoutchannel coding andM = 2,256; multilevelsignaling withRS(14,8)
codes and M = 256; binary signaling with BCH(127,64) codes; convolutional coding with
cr = 1/2 and L = 7 (Conv(2,7) in the plots). The soft decision of the Viterbi decoder uses
256quantization levels.
Theoretical BER curves versus pulse size are represented in Figure 5(a). Each point on
these curves wasobtained using 100 different insertion keys. Inalltests, theembedded mark
has a length of N = 256 bits and is randomly generated. The parameter that regulates the
b
watermark insertion strength (β) was setto 0.4, a value that guarantees the invisibility of the
DiversityEnhancementofCodedSpreadSpectrum VideoWatermarking 101
Figure 5. BER for binary and non-binary codes, without video compression and J = 1: (a) Theoretical; (b)
Empirical.
mark. For the pulse size range displayed in the figure, the performance achieved by convo-
lutional codes is remarkable, but the curve corresponding to RS(14,8) codes with M = 256
crosses itat a pulse size value of ∼=140 (BER ∼= 3×10−6). Binary block codes exhibit poor
performance comparedtothesecases.
ExperimentalcurvesareshowninFigure5(b).Thenumberoftestsfortheuncodedbinary
casewas250,whilefortheremainingcaseswere1000,duetoanexpectedlowerBER.Over-
all,thereisagoodmatchbetweentheoretical andexperimentalcurves.Thebestperformance
is achieved by RS encoding, which doesn’t exhibit errors for pulse size values greater than
110. For the binary cases, convolutional encoding also performs well, and BCH encoding,
again, exhibits poor performance. Some instability occurs for the lowest BER rates (below
10−4)duetothelimitednumberoftests.
In Tables 1–3, results are presented for the percentage of frames (averaged over the three
tested sequences and after MPEG-2 video compression at 2, 4 and 6 Mbit/s, respectively)
in which the watermark was successfully retrieved, as a function of the number of signaling
levels, error correction code, diversity technique (ML, EGC or MGC) and diversity window
size (J). Each sequence was watermarked two times, using different embedding keys and
randomlygenerated watermarkswith64bitslength.Theinsertionstrength(β)wassetto0.2,
avaluethatguarantees thatthewatermarkisfarbelowvisibility.
Ascan be observed from Tables 1, 2, and 3, an increase in the number ofsignaling levels
brings an improvement in performance. The use of error correction is generally profitable,
particularly in the ase of soft decision convolutional coding or RS codes combined with
multilevel signaling. Considering a group of consecutive frames significantly increases the
detection rate, a result that is most evident for high compression ratios. Also, the system
performance increases withJ.Aspredicted fromFigure3,thebestperformance oftheMGC
combiningtechniqueisachievedforstrongcompression(MPEG-2@2Mbits/s).Inthiscase,
100% success on theextraction rateimplies theuse oftimediversity methods inconjunction
withmultilevelsignaling and/orerrorcorrection codes.
102 T.Brandãoetal.
Table1. PercentageratesofwatermarkextractionsuccessunderMPEG-2@2Mbits.
Levels/code J=1 J=3 J=5 J=15 J=25
ML EGC MGC ML EGC MGC ML EGC MGC ML EGC MGC
M=2/uncoded 1.11 0.00 3.33 7.33 2.22 16.67 24.44 50.00 76.67 80.00 83.33 91.67 91.67
M=16/uncoded 5.33 0.33 11.33 21.67 3.89 41.11 49.44 58.33 95.00 96.67 86.11 100.0 100.0
M=256/uncoded 11.44 0.67 35.67 47.00 20.56 66.11 66.67 71.67 95.00 95.00 86.11 100.0 100.0
M=2/BCH(127,64) 3.56 1.67 9.00 22.67 5.56 40.56 51.11 78.33 91.67 95.00 97.22 100.0 100.0
M=16/RS(14,8) 7.22 0.33 15.33 26.67 12.22 50.56 58.89 70.00 95.00 95.00 94.44 100.0 100.0
M=256/RS(14,8) 9.22 0.33 25.00 39.67 15.00 58.89 61.67 66.67 95.00 95.00 80.56 100.0 100.0
M=2/conv.hard 4.78 1.67 12.00 26.67 12.22 43.33 55.00 83.33 93.33 93.33 100.0 100.0 100.0
M=2/conv.soft 12.56 – 39.67 55.00 – 67.22 76.11 – 96.67 98.33 – 100.0 100.0
Table2. PercentageratesofwatermarkextractionsuccessunderMPEG-2@4Mbits.
Levels/code J=1 J=3 J=5 J=15 J=25
ML EGC MGC ML EGC MGC ML EGC MGC ML EGC MGC
M=2/uncoded 21.22 28.67 76.67 80.67 67.22 91.67 92.78 100.0 100.0 100.0 100.0 100.0 100.0
M=16/uncoded 27.56 32.67 93.67 96.0 85.0 99.44 99.44 100.0 100.0 100.0 100.0 100.0 100.0
M=256/uncoded 36.44 59.67 99.0 99.67 93.89 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/BCH(127,64) 26.89 78.67 92.33 93.67 93.89 98.89 99.44 100.0 100.0 100.0 100.0 100.0 100.0
M=16/RS(14,8) 28.78 61.0 95.0 97.0 89.44 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=256/RS(14,8) 31.22 61.67 99.0 99.33 91.11 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/conv.hard 27.67 85.0 94.67 95.67 96.11 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/conv.soft 38.89 – 99.67 99.67 – 100.0 100.0 – 100.0 100.0 – 100.0 100.0
7. Conclusions
In this paper it has been confirmed, theoretically and experimentally, that spread-spectrum
based video watermrking can benefit from multilevel signaling and/or error correction cod-
ing. The best performance is attained with binary convolutional codes and non-binary block
codes, a result that is also valid under MPEG-2 video compression. It has also been shown
that,forvideosequencesunderMPEG-2compression,betterperformancecanbeachievedby
retrieving the markoverawindow ofconsecutive frames, using timediversity techniques. In
general, the maximal gain combining method leads to the highest extraction success rates, a
Table3. PercentageratesofwatermarkextractionsuccessunderMPEG-2@6Mbits.
Levels/code J=1 J=3 J=5 J=15 J=25
ML EGC MGC ML EGC MGC ML EGC MGC ML EGC MGC
M=2/uncoded 37.33 92.0 99.33 98.0 99.44 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=16/uncoded 66.33 93.67 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=256/uncoded 90.0 99.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/BCH(127,64) 66.78 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=16/RS(14,8) 79.89 99.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=256/RS(14,8) 89.11 99.33 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/conv.hard 72.78 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
M=2/conv.soft 94.89 – 100.0 100.0 – 100.0 100.0 – 100.0 100.0 – 100.0 100.0