From e2dcc912a1635315e61afac479a6ec996f276846 Mon Sep 17 00:00:00 2001 From: Steven Franke Date: Thu, 24 Dec 2015 19:44:30 +0000 Subject: [PATCH] Fix misprints in equation 4 and 5 git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6313 ab8295b8-cf94-4d9e-aec4-7959e3be5d79 --- lib/ftrsd/ftrsd_paper/ftrsd.lyx | 3246 +++++++++++++++---------------- 1 file changed, 1623 insertions(+), 1623 deletions(-) diff --git a/lib/ftrsd/ftrsd_paper/ftrsd.lyx b/lib/ftrsd/ftrsd_paper/ftrsd.lyx index 2f5a41365..5f3523913 100644 --- a/lib/ftrsd/ftrsd_paper/ftrsd.lyx +++ b/lib/ftrsd/ftrsd_paper/ftrsd.lyx @@ -1,1623 +1,1623 @@ -#LyX 2.1 created this file. For more info see http://www.lyx.org/ -\lyxformat 474 -\begin_document -\begin_header -\textclass paper -\use_default_options true -\maintain_unincluded_children false -\language english -\language_package default -\inputencoding auto -\fontencoding global -\font_roman default -\font_sans default -\font_typewriter default -\font_math auto -\font_default_family default -\use_non_tex_fonts false -\font_sc false -\font_osf false -\font_sf_scale 100 -\font_tt_scale 100 -\graphics default -\default_output_format default -\output_sync 0 -\bibtex_command default -\index_command default -\float_placement H -\paperfontsize 12 -\spacing onehalf -\use_hyperref false -\papersize default -\use_geometry true -\use_package amsmath 1 -\use_package amssymb 1 -\use_package cancel 1 -\use_package esint 1 -\use_package mathdots 1 -\use_package mathtools 1 -\use_package mhchem 1 -\use_package stackrel 1 -\use_package stmaryrd 1 -\use_package undertilde 1 -\cite_engine basic -\cite_engine_type default -\biblio_style plain -\use_bibtopic false -\use_indices false -\paperorientation portrait -\suppress_date false -\justification true -\use_refstyle 1 -\index Index -\shortcut idx -\color #008000 -\end_index -\leftmargin 1in -\topmargin 1in -\rightmargin 1in -\bottommargin 1in -\secnumdepth 3 -\tocdepth 3 -\paragraph_separation indent -\paragraph_indentation default -\quotes_language english -\papercolumns 1 -\papersides 1 -\paperpagestyle default -\tracking_changes false -\output_changes false -\html_math_output 0 -\html_css_as_file 0 -\html_be_strict false -\end_header - -\begin_body - -\begin_layout Title -A stochastic successive erasures soft-decision decoder for the JT65 (63,12) - Reed-Solomon code -\end_layout - -\begin_layout Author -Steven J. - Franke, K9AN and Joseph H. - Taylor, K1JT -\end_layout - -\begin_layout Abstract -The JT65 protocol has revolutionized amateur-radio weak-signal communication - by enabling amateur radio operators with small antennas and relatively - low-power transmitters to communicate over propagation paths not usable - with traditional technologies. - A major reason for the success and popularity of JT65 is its use of a strong - error-correction code: a short block-length, low-rate Reed-Solomon code - based on a 64-symbol alphabet. - Since 2004, most programs implementing JT65 have used the patented Koetter-Vard -y (KV) algebraic soft-decision decoder, licensed to K1JT and implemented - in a closed-source program for use in amateur radio applications. - We describe here a new open-source alternative called the Franke-Taylor - (FT, or K9AN-K1JT) algorithm. - It is conceptually simple, built around the well-known Berlekamp-Massey - errors-and-erasures algorithm, and in this application it performs even - better than the KV decoder. -\end_layout - -\begin_layout Section -Introduction -\end_layout - -\begin_layout Standard -JT65 message frames consist of a short compressed message encoded for transmissi -on with a Reed-Solomon code. - Reed-Solomon codes are block codes characterized by -\begin_inset Formula $n$ -\end_inset - -, the length of their codewords, -\begin_inset Formula $k$ -\end_inset - -, the number of message symbols conveyed by the codeword, and the number - of possible values for each symbol in the codewords. - The codeword length and the number of message symbols are specified with - the notation -\begin_inset Formula $(n,k)$ -\end_inset - -. - JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each - symbol. - Each of the 12 message symbols represents -\begin_inset Formula $\log_{2}64=6$ -\end_inset - - message bits. - The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist - of 72 information bits. - The JT65 code is systematic, which means that the 12 message symbols are - embedded in the codeword without modification and another 51 parity symbols - derived from the message symbols are added to form a codeword of 63 symbols. - -\end_layout - -\begin_layout Standard -The concept of Hamming distance is used as a measure of -\begin_inset Quotes eld -\end_inset - -distance -\begin_inset Quotes erd -\end_inset - - between different codewords, or between a received word and a codeword. - Hamming distance is the number of code symbols that differ in two words - being compared. - Reed-Solomon codes have minimum Hamming distance -\begin_inset Formula $d$ -\end_inset - -, where -\begin_inset Formula -\begin{equation} -d=n-k+1.\label{eq:minimum_distance} -\end{equation} - -\end_inset - -The minimum Hamming distance of the JT65 code is -\begin_inset Formula $d=52$ -\end_inset - -, which means that any particular codeword differs from all other codewords - in at least 52 symbol positions. - -\end_layout - -\begin_layout Standard -Given a received word containing some incorrect symbols (errors), the received - word can be decoded into the correct codeword using a deterministic, algebraic - algorithm provided that no more than -\begin_inset Formula $t$ -\end_inset - - symbols were received incorrectly, where -\begin_inset Formula -\begin{equation} -t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t} -\end{equation} - -\end_inset - -For the JT65 code -\begin_inset Formula $t=25$ -\end_inset - -, so it is always possible to decode a received word having 25 or fewer - symbol errors. - Any one of several well-known algebraic algorithms, such as the widely - used Berlekamp-Massey (BM) algorithm, can carry out the decoding. - Two steps are necessarily involved in this process. - We must (1) determine which symbols were received incorrectly, and (2) - find the correct value of the incorrect symbols. - If we somehow know that certain symbols are incorrect, that information - can be used to reduce the work involved in step 1 and allow step 2 to correct - more than -\begin_inset Formula $t$ -\end_inset - - errors. - In the unlikely event that the location of every error is known and if - no correct symbols are accidentally labeled as errors, the BM algorithm - can correct up to -\begin_inset Formula $d-1=n-k$ -\end_inset - - errors. - -\end_layout - -\begin_layout Standard -The FT algorithm creates lists of symbols suspected of being incorrect and - sends them to the BM decoder. - Symbols flagged in this way are called -\begin_inset Quotes eld -\end_inset - -erasures, -\begin_inset Quotes erd -\end_inset - - while other incorrect symbols will be called -\begin_inset Quotes eld -\end_inset - -errors. -\begin_inset Quotes erd -\end_inset - - With perfect erasure information up to 51 incorrect symbols can be corrected - for the JT65 code. - Imperfect erasure information means that some erased symbols may be correct, - and some other symbols in error. - If -\begin_inset Formula $s$ -\end_inset - - symbols are erased and the remaining -\begin_inset Formula $n-s$ -\end_inset - - symbols contain -\begin_inset Formula $e$ -\end_inset - - errors, the BM algorithm can find the correct codeword as long as -\begin_inset Formula -\begin{equation} -s+2e\le d-1.\label{eq:erasures_and_errors} -\end{equation} - -\end_inset - -If -\begin_inset Formula $s=0$ -\end_inset - -, the decoder is said to be an -\begin_inset Quotes eld -\end_inset - -errors-only -\begin_inset Quotes erd -\end_inset - - decoder. - If -\begin_inset Formula $0X$ -\end_inset - -. - Correspondingly, the FT algorithm works best when the probability of erasing - a symbol is somewhat larger than the probability that the symbol is incorrect. - We found empirically that good decoding performance is obtained when the - symbol erasure probability is about 1.3 times the symbol error probability. -\end_layout - -\begin_layout Standard -The FT algorithm tries successively to decode the received word using independen -t -\begin_inset Quotes eld -\end_inset - -educated guesses -\begin_inset Quotes erd -\end_inset - - to select symbols for erasure. - For each iteration a stochastic erasure vector is generated based on the - symbol erasure probabilities. - The erasure vector is sent to the BM decoder along with the full set of - 63 hard-decision symbol values. - When the BM decoder finds a candidate codeword it is assigned a quality - metric -\begin_inset Formula $d_{s}$ -\end_inset - -, the soft distance between the received word and the codeword: -\begin_inset Formula -\begin{equation} -d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,j}).\label{eq:soft_distance} -\end{equation} - -\end_inset - -Here -\begin_inset Formula $\alpha_{j}=0$ -\end_inset - - if received symbol -\begin_inset Formula $j$ -\end_inset - - is the same as the corresponding symbol in the codeword, -\begin_inset Formula $\alpha_{j}=1$ -\end_inset - - if the received symbol and codeword symbol are different, and -\begin_inset Formula $p_{1,j}$ -\end_inset - - is the fractional power associated with received symbol -\begin_inset Formula $j$ -\end_inset - -. - Think of the soft distance as made up of two terms: the first is the Hamming - distance between the received word and the codeword, and the second ensures - that if two candidate codewords have the same Hamming distance from the - received word, a smaller soft distance will be assigned to the one where - differences occur in symbols of lower estimated reliability. - -\end_layout - -\begin_layout Standard -In practice we find that -\begin_inset Formula $d_{s}$ -\end_inset - - can reliably indentify the correct codeword if the signal-to-noise ratio - for individual symbols is greater than about 4 in power units, or -\begin_inset Formula $E_{s}/N_{0}\apprge6$ -\end_inset - - dB. - We also find that weaker signals frequently can be decoded by using soft-symbol - information beyond that contained in -\begin_inset Formula $p_{1}$ -\end_inset - -and -\begin_inset Formula $p_{2}$ -\end_inset - -. - To this end we define an additional metric -\begin_inset Formula $u$ -\end_inset - -, the average signal-plus-noise power in all symbols, according to a candidate - codeword's symbol values: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\[ -u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\,j). -\] - -\end_inset - -Here the -\begin_inset Formula $c_{j}$ -\end_inset - -'s are the symbol values for the candidate codeword being tested. - -\end_layout - -\begin_layout Standard -The correct JT65 codeword produces a value for -\begin_inset Formula $u$ -\end_inset - - equal to average of -\begin_inset Formula $n=63$ -\end_inset - - bins containing both signal and noise power. - Incorrect codewords have at most -\begin_inset Formula $k=12$ -\end_inset - - such bins and at least -\begin_inset Formula $n-k=51$ -\end_inset - - bins containing noise only. - Thus, if the spectral array -\begin_inset Formula $S(i,\,j)$ -\end_inset - - has been normalized so that its median value (essentially the average noise - level) is unity, the correct codeword is expected to yield the metric value -\end_layout - -\begin_layout Standard -\begin_inset Formula -\[ -u=(1\pm n^{-\frac{1}{2}})(1+y)\approx(1.0\pm0.13)(1+y), -\] - -\end_inset - -where -\begin_inset Formula $y$ -\end_inset - - is the signal-to-noise ratio (in linear power units) and the quoted one-standar -d-deviation uncertainty range assumes Gaussian statistics. - Incorrect codewords will yield metric values no larger than -\end_layout - -\begin_layout Standard -\begin_inset Formula -\[ -u=\frac{n-k\pm\sqrt{n-k}}{n}+\frac{k\pm\sqrt{k}}{n}(1+y). -\] - -\end_inset - -For JT65 this expression evaluates to -\end_layout - -\begin_layout Standard -\begin_inset Formula -\[ -u\approx1\pm0.13+(0.19\pm0.06)\,y. -\] - -\end_inset - -As a specific example, consider signal strength -\begin_inset Formula $y=4$ -\end_inset - -, corresponding to -\begin_inset Formula $E_{s}/N_{0}=6$ -\end_inset - - dB. - For JT65, the corresponding SNR in 2500 Hz bandwidth is -\begin_inset Formula $-23.7$ -\end_inset - - dB. - The correct codeword is then expected to yield -\begin_inset Formula $u\approx5.0\pm$ -\end_inset - -0.6, while incorrect codewords will give -\begin_inset Formula $u\approx2.0\pm0.3$ -\end_inset - - or less. - We find that a threshold set at -\begin_inset Formula $u_{0}=4.4$ -\end_inset - - (about 8 standard deviations above the expected maximum for incorrect codewords -) reliably serves to distinguish correct codewords from all other candidates, - while ensuring a very small probability of false decodes. -\end_layout - -\begin_layout Standard -Technically the FT algorithm is a list decoder. - Among the list of candidate codewords found by the stochastic search algorithm, - only the one with the largest -\begin_inset Formula $u$ -\end_inset - - is retained. - As with all such algorithms, a stopping criterion is necessary. - FT accepts a codeword unconditionally if -\begin_inset Formula $u>u_{0}$ -\end_inset - -. - A timeout is used to limit the algorithm's execution time if no acceptable - codeword is found in a reasonable number of trials, -\begin_inset Formula $T$ -\end_inset - -. - Today's personal computers are fast enough that -\begin_inset Formula $T$ -\end_inset - - can be set as large as -\begin_inset Formula $10^{5},$ -\end_inset - - or even higher. -\end_layout - -\begin_layout Paragraph -Algorithm pseudo-code: -\end_layout - -\begin_layout Enumerate -For each received symbol, define the erasure probability as 1.3 times the - -\emph on -a priori -\emph default - symbol-error probability determined from soft-symbol information -\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$ -\end_inset - -. - -\end_layout - -\begin_layout Enumerate -Make independent stochastic decisions about whether to erase each symbol - by using the symbol's erasure probability, allowing a maximum of 51 erasures. -\end_layout - -\begin_layout Enumerate -Attempt errors-and-erasures decoding by using the BM algorithm and the set - of erasures determined in step 2. - If the BM decoder produces a candidate codeword, go to step 5. -\end_layout - -\begin_layout Enumerate -If BM decoding was not successful, go to step 2. -\end_layout - -\begin_layout Enumerate -Calculate the hard-decision Hamming distance between the candidate codeword - and the received symbols, the corresponding soft distance -\begin_inset Formula $d_{s}$ -\end_inset - -, and the quality metric -\begin_inset Formula $u$ -\end_inset - -. - If -\begin_inset Formula $u$ -\end_inset - - is the largest one encountered so far, set -\begin_inset Formula $u_{max}=u$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -If -\begin_inset Formula $u_{max}>u_{0}$ -\end_inset - -, go to step 8. - -\end_layout - -\begin_layout Enumerate -If the number of trials is less than the timeout limit -\begin_inset Formula $T,$ -\end_inset - - go to 2. - Otherwise, declare decoding failure and exit. -\end_layout - -\begin_layout Enumerate -An acceptable codeword with -\begin_inset Formula $u_{max}>u_{0}$ -\end_inset - - has been found. - Declare a successful decode and return this codeword . -\end_layout - -\begin_layout Section -Theory and Simulations -\end_layout - -\begin_layout Standard -The fraction of time that -\begin_inset Formula $X$ -\end_inset - -, the number of symbols received incorrectly, is expected to be less than - some number -\begin_inset Formula $D$ -\end_inset - - depends on signal-to-noise ratio. - For the case of additive white Gaussian noise (AWGN) and noncoherent 64-FSK - demodulation this probability is easy to calculate. - Representative examples for -\begin_inset Formula $D=25,$ -\end_inset - - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\uuline off -\uwave off -\noun off -\color none - -\begin_inset Formula $D=40$ -\end_inset - - -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\uuline default -\uwave default -\noun default -\color inherit -, and -\begin_inset Formula $D=43$ -\end_inset - - are plotted in Figure -\begin_inset CommandInset ref -LatexCommand ref -reference "fig:bodide" - -\end_inset - - for a range of SNRs as filled squares with connecting lines. - The rightmost such curve shows that on the AWGN channel the hard-decision - BM decoder should succeed about 90% of the time at -\begin_inset Formula $E_{s}/N_{0}=7.5$ -\end_inset - - dB, 99% of the time at 8 dB, and 99.98% at 8.5 dB. - For comparison, the righmost curve with open squares shows that simulated - results agree with theory to within less than 0.2 dB. - -\end_layout - -\begin_layout Standard -\begin_inset Float figure -wide false -sideways false -status open - -\begin_layout Plain Layout -\align center -\begin_inset Graphics - filename fig_bodide.pdf - -\end_inset - - -\begin_inset Caption Standard - -\begin_layout Plain Layout -\begin_inset CommandInset label -LatexCommand label -name "fig:bodide" - -\end_inset - -Word error rates as a function of -\begin_inset Formula $E_{s}/N_{0},$ -\end_inset - - the signal-to-noise ratio in bandwidth equal to the symbol rate. - Filled squares illustrate theoretical values for -\begin_inset Formula $D=25,$ -\end_inset - - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\uuline off -\uwave off -\noun off -\color none - -\begin_inset Formula $D=40$ -\end_inset - - -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\uuline default -\uwave default -\noun default -\color inherit -, and -\begin_inset Formula $D=43$ -\end_inset - -. - Open squares illustrate measured results for the BM and FT ( -\begin_inset Formula $T=10^{5}$ -\end_inset - -) decoders in program -\emph on -WSJT-X -\emph default -. -\end_layout - -\end_inset - - -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Received JT65 words with more than 25 incorrect symbols can be decoded if - sufficient information on individual symbol reliabilities is available. - Using values of -\begin_inset Formula $T$ -\end_inset - - that are practical with today's personal computers and the soft-symbol - information described above, we find that the FT algorithm nearly always - produces correct decodes up to -\begin_inset Formula $X=40$ -\end_inset - -, and some additional decodes are found in the range 41 to 43. - As an example, Figure -\begin_inset CommandInset ref -LatexCommand ref -reference "fig:N_vs_X" - -\end_inset - - plots the number of stochastic erasure trials required to find the correct - codeword versus the number of hard-decision errors for a run with 1000 - simulated transmissions at -\begin_inset Formula $SNR=-24$ -\end_inset - - dB, just slightly above the decoding threshold. - Note that both mean and variance of the required number of trials increase - steeply with the number of errors in the received word. - Execution time of the FT algorithm is roughly proportional to the number - of required trials. - -\end_layout - -\begin_layout Standard -\begin_inset Float figure -wide false -sideways false -status open - -\begin_layout Plain Layout -\align center -\begin_inset Graphics - filename fig_ntrials_vs_nhard.pdf - lyxscale 120 - scale 120 - -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Caption Standard - -\begin_layout Plain Layout -\begin_inset CommandInset label -LatexCommand label -name "fig:N_vs_X" - -\end_inset - -Number of trials needed to decode a received word versus Hamming distance - between the received word and the decoded codeword, for 1000 simulated - frames on an AWGN channel with no fading. - The SNR in 2500 Hz bandwidth is -24 dB ( -\begin_inset Formula $E_{s}/N_{o}=5.7$ -\end_inset - - dB). - -\end_layout - -\end_inset - - -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Section -Comparison with Berlekamp-Massey and Koetter-Vardy -\end_layout - -\begin_layout Standard -Comparisons of decoding performance are usually presented in the professional - literature as plots of word error rate versus -\begin_inset Formula $E_{b}/N_{0}$ -\end_inset - -, the signal-to-noise ratio per information bit. - Results of simulations using the Berlekamp-Massey, Koetter-Vardy, and Franke-Ta -ylor decoding algorithms on the (63,12) code are presented in this way in - Figure -\begin_inset CommandInset ref -LatexCommand ref -reference "fig:WER" - -\end_inset - -. - For these tests we generated 1000 signals at each signal-to-noise ratio, - assuming the additive white gaussian noise (AWGN) channel, and processed - the data using each algorithm. - As expected, the soft-decision algorithms FT and KV are about 2 dB better - than the hard-decision BM algorithm. - FT has a slight edge (about 0.2 dB) over KV with the default settings for - each algorithm, as implemented in our JT65 decoders. -\end_layout - -\begin_layout Standard -\begin_inset Float figure -wide false -sideways false -status open - -\begin_layout Plain Layout -\align center -\begin_inset Graphics - filename fig_wer.pdf - lyxscale 120 - scale 120 - -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Caption Standard - -\begin_layout Plain Layout -\begin_inset CommandInset label -LatexCommand label -name "fig:WER" - -\end_inset - -Word error rate (WER) as a function of -\begin_inset Formula $E_{b}/N_{0}$ -\end_inset - - for non-fading signals in AWGN. - -\end_layout - -\end_inset - - -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Because of the importance of error-free transmission in commercial applications, - plots like that in Figure -\begin_inset CommandInset ref -LatexCommand ref -reference "fig:WER" - -\end_inset - - often extend downward to much smaller error rates, say -\begin_inset Formula $10^{-6}$ -\end_inset - - or less, . - The circumstances for minimal amateur-radio QSOs are very different, however. - Error rates of order 0.1, or ever higher, may be acceptable. - In this case the essential information is better presented in a plot showing - the percentage of transmissions copied correctly as a function of signal-to-noi -se ratio. - -\end_layout - -\begin_layout Standard -Figure -\begin_inset CommandInset ref -LatexCommand ref -reference "fig:Psuccess" - -\end_inset - - presents the results of simulations for signal-to-noise ratios ranging - from -\begin_inset Formula $-18$ -\end_inset - - to -\begin_inset Formula $-30$ -\end_inset - - dB, again using 1000 simulated signals for each plotted point. - We include three curves for each decoding algorithm: one for the AWGN channel - and no fading, and two more for simulated Doppler spreads of 0.2 and 1.0 - Hz. - For reference, we note that the JT65 symbol rate is about 2.69 Hz. - The simulated Doppler spreads are comparable to those encountered on HF - ionospheric paths and for EME at VHF and lower UHF bands. -\end_layout - -\begin_layout Standard -\begin_inset Float figure -wide false -sideways false -status open - -\begin_layout Plain Layout -\align center -\begin_inset Graphics - filename fig_psuccess.pdf - lyxscale 90 - scale 90 - -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Caption Standard - -\begin_layout Plain Layout -\begin_inset CommandInset label -LatexCommand label -name "fig:Psuccess" - -\end_inset - -Percentage of JT65 messages successfully decoded as a function of SNR in - 2500 Hz bandwidth. - Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision - Franke-Taylor (FT) decoding algorithms. - Curves labeled DS correspond to the hinted-decode ( -\begin_inset Quotes eld -\end_inset - -Deep Search -\begin_inset Quotes erd -\end_inset - -) matched-filter algorithm. -\end_layout - -\end_inset - - -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Section -Hinted Decoding -\end_layout - -\begin_layout Standard -... - Still to come ... -\end_layout - -\begin_layout Section -Summary -\end_layout - -\begin_layout Standard -... - Still to come ... -\end_layout - -\begin_layout Bibliography -\begin_inset CommandInset bibitem -LatexCommand bibitem -key "key-1" - -\end_inset - -"Stochastic Chase Decoding of Reed-Solomon Codes", Camille Leroux, Saied - Hemati, Shie Mannor, Warren J. - Gross, IEEE Communications Letters, Vol. - 14, No. - 9, September 2010. -\end_layout - -\begin_layout Bibliography -\begin_inset CommandInset bibitem -LatexCommand bibitem -key "key-2" - -\end_inset - -"Soft-Decision Decoding of Reed-Solomon Codes Using Successive Error-and-Erasure - Decoding," Soo-Woong Lee and B. - V. - K. - Vijaya Kumar, IEEE -\begin_inset Quotes eld -\end_inset - -GLOBECOM -\begin_inset Quotes erd -\end_inset - - 2008 proceedings. -\end_layout - -\begin_layout Bibliography -\begin_inset CommandInset bibitem -LatexCommand bibitem -key "key-3" - -\end_inset - - -\begin_inset Quotes erd -\end_inset - -Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes, -\begin_inset Quotes erd -\end_inset - - Chang-Ming Lee and Yu T. - Su, IEEE Signal Processing Letters, Vol. - 16, No. - 8, August 2009. -\end_layout - -\begin_layout Bibliography -\begin_inset CommandInset bibitem -LatexCommand bibitem -key "key-4" - -\end_inset - -“Algebraic soft-decision decoding of Reed-Solomon codes,” R. - Köetter and A. - Vardy, IEEE Trans. - Inform. - Theory, Vol. - 49, Nov. - 2003. -\end_layout - -\begin_layout Bibliography -\begin_inset CommandInset bibitem -LatexCommand bibitem -key "key-5" - -\end_inset - -Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/ -\end_layout - -\end_body -\end_document +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\begin_document +\begin_header +\textclass paper +\use_default_options true +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman default +\font_sans default +\font_typewriter default +\font_math auto +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 +\font_tt_scale 100 +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\float_placement H +\paperfontsize 12 +\spacing onehalf +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1in +\topmargin 1in +\rightmargin 1in +\bottommargin 1in +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +A stochastic successive erasures soft-decision decoder for the JT65 (63,12) + Reed-Solomon code +\end_layout + +\begin_layout Author +Steven J. + Franke, K9AN and Joseph H. + Taylor, K1JT +\end_layout + +\begin_layout Abstract +The JT65 protocol has revolutionized amateur-radio weak-signal communication + by enabling amateur radio operators with small antennas and relatively + low-power transmitters to communicate over propagation paths not usable + with traditional technologies. + A major reason for the success and popularity of JT65 is its use of a strong + error-correction code: a short block-length, low-rate Reed-Solomon code + based on a 64-symbol alphabet. + Since 2004, most programs implementing JT65 have used the patented Koetter-Vard +y (KV) algebraic soft-decision decoder, licensed to K1JT and implemented + in a closed-source program for use in amateur radio applications. + We describe here a new open-source alternative called the Franke-Taylor + (FT, or K9AN-K1JT) algorithm. + It is conceptually simple, built around the well-known Berlekamp-Massey + errors-and-erasures algorithm, and in this application it performs even + better than the KV decoder. +\end_layout + +\begin_layout Section +Introduction +\end_layout + +\begin_layout Standard +JT65 message frames consist of a short compressed message encoded for transmissi +on with a Reed-Solomon code. + Reed-Solomon codes are block codes characterized by +\begin_inset Formula $n$ +\end_inset + +, the length of their codewords, +\begin_inset Formula $k$ +\end_inset + +, the number of message symbols conveyed by the codeword, and the number + of possible values for each symbol in the codewords. + The codeword length and the number of message symbols are specified with + the notation +\begin_inset Formula $(n,k)$ +\end_inset + +. + JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each + symbol. + Each of the 12 message symbols represents +\begin_inset Formula $\log_{2}64=6$ +\end_inset + + message bits. + The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist + of 72 information bits. + The JT65 code is systematic, which means that the 12 message symbols are + embedded in the codeword without modification and another 51 parity symbols + derived from the message symbols are added to form a codeword of 63 symbols. + +\end_layout + +\begin_layout Standard +The concept of Hamming distance is used as a measure of +\begin_inset Quotes eld +\end_inset + +distance +\begin_inset Quotes erd +\end_inset + + between different codewords, or between a received word and a codeword. + Hamming distance is the number of code symbols that differ in two words + being compared. + Reed-Solomon codes have minimum Hamming distance +\begin_inset Formula $d$ +\end_inset + +, where +\begin_inset Formula +\begin{equation} +d=n-k+1.\label{eq:minimum_distance} +\end{equation} + +\end_inset + +The minimum Hamming distance of the JT65 code is +\begin_inset Formula $d=52$ +\end_inset + +, which means that any particular codeword differs from all other codewords + in at least 52 symbol positions. + +\end_layout + +\begin_layout Standard +Given a received word containing some incorrect symbols (errors), the received + word can be decoded into the correct codeword using a deterministic, algebraic + algorithm provided that no more than +\begin_inset Formula $t$ +\end_inset + + symbols were received incorrectly, where +\begin_inset Formula +\begin{equation} +t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t} +\end{equation} + +\end_inset + +For the JT65 code +\begin_inset Formula $t=25$ +\end_inset + +, so it is always possible to decode a received word having 25 or fewer + symbol errors. + Any one of several well-known algebraic algorithms, such as the widely + used Berlekamp-Massey (BM) algorithm, can carry out the decoding. + Two steps are necessarily involved in this process. + We must (1) determine which symbols were received incorrectly, and (2) + find the correct value of the incorrect symbols. + If we somehow know that certain symbols are incorrect, that information + can be used to reduce the work involved in step 1 and allow step 2 to correct + more than +\begin_inset Formula $t$ +\end_inset + + errors. + In the unlikely event that the location of every error is known and if + no correct symbols are accidentally labeled as errors, the BM algorithm + can correct up to +\begin_inset Formula $d-1=n-k$ +\end_inset + + errors. + +\end_layout + +\begin_layout Standard +The FT algorithm creates lists of symbols suspected of being incorrect and + sends them to the BM decoder. + Symbols flagged in this way are called +\begin_inset Quotes eld +\end_inset + +erasures, +\begin_inset Quotes erd +\end_inset + + while other incorrect symbols will be called +\begin_inset Quotes eld +\end_inset + +errors. +\begin_inset Quotes erd +\end_inset + + With perfect erasure information up to 51 incorrect symbols can be corrected + for the JT65 code. + Imperfect erasure information means that some erased symbols may be correct, + and some other symbols in error. + If +\begin_inset Formula $s$ +\end_inset + + symbols are erased and the remaining +\begin_inset Formula $n-s$ +\end_inset + + symbols contain +\begin_inset Formula $e$ +\end_inset + + errors, the BM algorithm can find the correct codeword as long as +\begin_inset Formula +\begin{equation} +s+2e\le d-1.\label{eq:erasures_and_errors} +\end{equation} + +\end_inset + +If +\begin_inset Formula $s=0$ +\end_inset + +, the decoder is said to be an +\begin_inset Quotes eld +\end_inset + +errors-only +\begin_inset Quotes erd +\end_inset + + decoder. + If +\begin_inset Formula $0X$ +\end_inset + +. + Correspondingly, the FT algorithm works best when the probability of erasing + a symbol is somewhat larger than the probability that the symbol is incorrect. + We found empirically that good decoding performance is obtained when the + symbol erasure probability is about 1.3 times the symbol error probability. +\end_layout + +\begin_layout Standard +The FT algorithm tries successively to decode the received word using independen +t +\begin_inset Quotes eld +\end_inset + +educated guesses +\begin_inset Quotes erd +\end_inset + + to select symbols for erasure. + For each iteration a stochastic erasure vector is generated based on the + symbol erasure probabilities. + The erasure vector is sent to the BM decoder along with the full set of + 63 hard-decision symbol values. + When the BM decoder finds a candidate codeword it is assigned a quality + metric +\begin_inset Formula $d_{s}$ +\end_inset + +, the soft distance between the received word and the codeword: +\begin_inset Formula +\begin{equation} +d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,j}).\label{eq:soft_distance} +\end{equation} + +\end_inset + +Here +\begin_inset Formula $\alpha_{j}=0$ +\end_inset + + if received symbol +\begin_inset Formula $j$ +\end_inset + + is the same as the corresponding symbol in the codeword, +\begin_inset Formula $\alpha_{j}=1$ +\end_inset + + if the received symbol and codeword symbol are different, and +\begin_inset Formula $p_{1,j}$ +\end_inset + + is the fractional power associated with received symbol +\begin_inset Formula $j$ +\end_inset + +. + Think of the soft distance as made up of two terms: the first is the Hamming + distance between the received word and the codeword, and the second ensures + that if two candidate codewords have the same Hamming distance from the + received word, a smaller soft distance will be assigned to the one where + differences occur in symbols of lower estimated reliability. + +\end_layout + +\begin_layout Standard +In practice we find that +\begin_inset Formula $d_{s}$ +\end_inset + + can reliably indentify the correct codeword if the signal-to-noise ratio + for individual symbols is greater than about 4 in power units, or +\begin_inset Formula $E_{s}/N_{0}\apprge6$ +\end_inset + + dB. + We also find that weaker signals frequently can be decoded by using soft-symbol + information beyond that contained in +\begin_inset Formula $p_{1}$ +\end_inset + +and +\begin_inset Formula $p_{2}$ +\end_inset + +. + To this end we define an additional metric +\begin_inset Formula $u$ +\end_inset + +, the average signal-plus-noise power in all symbols, according to a candidate + codeword's symbol values: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\, j). +\] + +\end_inset + +Here the +\begin_inset Formula $c_{j}$ +\end_inset + +'s are the symbol values for the candidate codeword being tested. + +\end_layout + +\begin_layout Standard +The correct JT65 codeword produces a value for +\begin_inset Formula $u$ +\end_inset + + equal to average of +\begin_inset Formula $n=63$ +\end_inset + + bins containing both signal and noise power. + Incorrect codewords have at most +\begin_inset Formula $k=12$ +\end_inset + + such bins and at least +\begin_inset Formula $n-k=51$ +\end_inset + + bins containing noise only. + Thus, if the spectral array +\begin_inset Formula $S(i,\, j)$ +\end_inset + + has been normalized so that its median value (essentially the average noise + level) is unity, the correct codeword is expected to yield the metric value +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +u=(1\pm n^{-\frac{1}{2}})(1+y)\approx(1.0\pm0.13)(1+y), +\] + +\end_inset + +where +\begin_inset Formula $y$ +\end_inset + + is the signal-to-noise ratio (in linear power units) and the quoted one-standar +d-deviation uncertainty range assumes Gaussian statistics. + Incorrect codewords will yield metric values no larger than +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +u=\frac{n-k\pm\sqrt{n-k}}{n}+\frac{k\pm\sqrt{k}}{n}(1+y). +\] + +\end_inset + +For JT65 this expression evaluates to +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +u\approx1\pm0.13+(0.19\pm0.06)\, y. +\] + +\end_inset + +As a specific example, consider signal strength +\begin_inset Formula $y=4$ +\end_inset + +, corresponding to +\begin_inset Formula $E_{s}/N_{0}=6$ +\end_inset + + dB. + For JT65, the corresponding SNR in 2500 Hz bandwidth is +\begin_inset Formula $-23.7$ +\end_inset + + dB. + The correct codeword is then expected to yield +\begin_inset Formula $u\approx5.0\pm$ +\end_inset + +0.6, while incorrect codewords will give +\begin_inset Formula $u\approx2.0\pm0.3$ +\end_inset + + or less. + We find that a threshold set at +\begin_inset Formula $u_{0}=4.4$ +\end_inset + + (about 8 standard deviations above the expected maximum for incorrect codewords +) reliably serves to distinguish correct codewords from all other candidates, + while ensuring a very small probability of false decodes. +\end_layout + +\begin_layout Standard +Technically the FT algorithm is a list decoder. + Among the list of candidate codewords found by the stochastic search algorithm, + only the one with the largest +\begin_inset Formula $u$ +\end_inset + + is retained. + As with all such algorithms, a stopping criterion is necessary. + FT accepts a codeword unconditionally if +\begin_inset Formula $u>u_{0}$ +\end_inset + +. + A timeout is used to limit the algorithm's execution time if no acceptable + codeword is found in a reasonable number of trials, +\begin_inset Formula $T$ +\end_inset + +. + Today's personal computers are fast enough that +\begin_inset Formula $T$ +\end_inset + + can be set as large as +\begin_inset Formula $10^{5},$ +\end_inset + + or even higher. +\end_layout + +\begin_layout Paragraph +Algorithm pseudo-code: +\end_layout + +\begin_layout Enumerate +For each received symbol, define the erasure probability as 1.3 times the + +\emph on +a priori +\emph default + symbol-error probability determined from soft-symbol information +\begin_inset Formula $\{p_{1}\textrm{-rank},\, p_{2}/p_{1}\}$ +\end_inset + +. + +\end_layout + +\begin_layout Enumerate +Make independent stochastic decisions about whether to erase each symbol + by using the symbol's erasure probability, allowing a maximum of 51 erasures. +\end_layout + +\begin_layout Enumerate +Attempt errors-and-erasures decoding by using the BM algorithm and the set + of erasures determined in step 2. + If the BM decoder produces a candidate codeword, go to step 5. +\end_layout + +\begin_layout Enumerate +If BM decoding was not successful, go to step 2. +\end_layout + +\begin_layout Enumerate +Calculate the hard-decision Hamming distance between the candidate codeword + and the received symbols, the corresponding soft distance +\begin_inset Formula $d_{s}$ +\end_inset + +, and the quality metric +\begin_inset Formula $u$ +\end_inset + +. + If +\begin_inset Formula $u$ +\end_inset + + is the largest one encountered so far, set +\begin_inset Formula $u_{max}=u$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +If +\begin_inset Formula $u_{max}>u_{0}$ +\end_inset + +, go to step 8. + +\end_layout + +\begin_layout Enumerate +If the number of trials is less than the timeout limit +\begin_inset Formula $T,$ +\end_inset + + go to 2. + Otherwise, declare decoding failure and exit. +\end_layout + +\begin_layout Enumerate +An acceptable codeword with +\begin_inset Formula $u_{max}>u_{0}$ +\end_inset + + has been found. + Declare a successful decode and return this codeword . +\end_layout + +\begin_layout Section +Theory and Simulations +\end_layout + +\begin_layout Standard +The fraction of time that +\begin_inset Formula $X$ +\end_inset + +, the number of symbols received incorrectly, is expected to be less than + some number +\begin_inset Formula $D$ +\end_inset + + depends on signal-to-noise ratio. + For the case of additive white Gaussian noise (AWGN) and noncoherent 64-FSK + demodulation this probability is easy to calculate. + Representative examples for +\begin_inset Formula $D=25,$ +\end_inset + + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $D=40$ +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\uuline default +\uwave default +\noun default +\color inherit +, and +\begin_inset Formula $D=43$ +\end_inset + + are plotted in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:bodide" + +\end_inset + + for a range of SNRs as filled squares with connecting lines. + The rightmost such curve shows that on the AWGN channel the hard-decision + BM decoder should succeed about 90% of the time at +\begin_inset Formula $E_{s}/N_{0}=7.5$ +\end_inset + + dB, 99% of the time at 8 dB, and 99.98% at 8.5 dB. + For comparison, the righmost curve with open squares shows that simulated + results agree with theory to within less than 0.2 dB. + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename fig_bodide.pdf + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:bodide" + +\end_inset + +Word error rates as a function of +\begin_inset Formula $E_{s}/N_{0},$ +\end_inset + + the signal-to-noise ratio in bandwidth equal to the symbol rate. + Filled squares illustrate theoretical values for +\begin_inset Formula $D=25,$ +\end_inset + + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $D=40$ +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\uuline default +\uwave default +\noun default +\color inherit +, and +\begin_inset Formula $D=43$ +\end_inset + +. + Open squares illustrate measured results for the BM and FT ( +\begin_inset Formula $T=10^{5}$ +\end_inset + +) decoders in program +\emph on +WSJT-X +\emph default +. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Received JT65 words with more than 25 incorrect symbols can be decoded if + sufficient information on individual symbol reliabilities is available. + Using values of +\begin_inset Formula $T$ +\end_inset + + that are practical with today's personal computers and the soft-symbol + information described above, we find that the FT algorithm nearly always + produces correct decodes up to +\begin_inset Formula $X=40$ +\end_inset + +, and some additional decodes are found in the range 41 to 43. + As an example, Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:N_vs_X" + +\end_inset + + plots the number of stochastic erasure trials required to find the correct + codeword versus the number of hard-decision errors for a run with 1000 + simulated transmissions at +\begin_inset Formula $SNR=-24$ +\end_inset + + dB, just slightly above the decoding threshold. + Note that both mean and variance of the required number of trials increase + steeply with the number of errors in the received word. + Execution time of the FT algorithm is roughly proportional to the number + of required trials. + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename fig_ntrials_vs_nhard.pdf + lyxscale 120 + scale 120 + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:N_vs_X" + +\end_inset + +Number of trials needed to decode a received word versus Hamming distance + between the received word and the decoded codeword, for 1000 simulated + frames on an AWGN channel with no fading. + The SNR in 2500 Hz bandwidth is -24 dB ( +\begin_inset Formula $E_{s}/N_{o}=5.7$ +\end_inset + + dB). + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Comparison with Berlekamp-Massey and Koetter-Vardy +\end_layout + +\begin_layout Standard +Comparisons of decoding performance are usually presented in the professional + literature as plots of word error rate versus +\begin_inset Formula $E_{b}/N_{0}$ +\end_inset + +, the signal-to-noise ratio per information bit. + Results of simulations using the Berlekamp-Massey, Koetter-Vardy, and Franke-Ta +ylor decoding algorithms on the (63,12) code are presented in this way in + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:WER" + +\end_inset + +. + For these tests we generated 1000 signals at each signal-to-noise ratio, + assuming the additive white gaussian noise (AWGN) channel, and processed + the data using each algorithm. + As expected, the soft-decision algorithms FT and KV are about 2 dB better + than the hard-decision BM algorithm. + FT has a slight edge (about 0.2 dB) over KV with the default settings for + each algorithm, as implemented in our JT65 decoders. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename fig_wer.pdf + lyxscale 120 + scale 120 + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:WER" + +\end_inset + +Word error rate (WER) as a function of +\begin_inset Formula $E_{b}/N_{0}$ +\end_inset + + for non-fading signals in AWGN. + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Because of the importance of error-free transmission in commercial applications, + plots like that in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:WER" + +\end_inset + + often extend downward to much smaller error rates, say +\begin_inset Formula $10^{-6}$ +\end_inset + + or less, . + The circumstances for minimal amateur-radio QSOs are very different, however. + Error rates of order 0.1, or ever higher, may be acceptable. + In this case the essential information is better presented in a plot showing + the percentage of transmissions copied correctly as a function of signal-to-noi +se ratio. + +\end_layout + +\begin_layout Standard +Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:Psuccess" + +\end_inset + + presents the results of simulations for signal-to-noise ratios ranging + from +\begin_inset Formula $-18$ +\end_inset + + to +\begin_inset Formula $-30$ +\end_inset + + dB, again using 1000 simulated signals for each plotted point. + We include three curves for each decoding algorithm: one for the AWGN channel + and no fading, and two more for simulated Doppler spreads of 0.2 and 1.0 + Hz. + For reference, we note that the JT65 symbol rate is about 2.69 Hz. + The simulated Doppler spreads are comparable to those encountered on HF + ionospheric paths and for EME at VHF and lower UHF bands. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename fig_psuccess.pdf + lyxscale 90 + scale 90 + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:Psuccess" + +\end_inset + +Percentage of JT65 messages successfully decoded as a function of SNR in + 2500 Hz bandwidth. + Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision + Franke-Taylor (FT) decoding algorithms. + Curves labeled DS correspond to the hinted-decode ( +\begin_inset Quotes eld +\end_inset + +Deep Search +\begin_inset Quotes erd +\end_inset + +) matched-filter algorithm. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Hinted Decoding +\end_layout + +\begin_layout Standard +... + Still to come ... +\end_layout + +\begin_layout Section +Summary +\end_layout + +\begin_layout Standard +... + Still to come ... +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "key-1" + +\end_inset + +"Stochastic Chase Decoding of Reed-Solomon Codes", Camille Leroux, Saied + Hemati, Shie Mannor, Warren J. + Gross, IEEE Communications Letters, Vol. + 14, No. + 9, September 2010. +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "key-2" + +\end_inset + +"Soft-Decision Decoding of Reed-Solomon Codes Using Successive Error-and-Erasure + Decoding," Soo-Woong Lee and B. + V. + K. + Vijaya Kumar, IEEE +\begin_inset Quotes eld +\end_inset + +GLOBECOM +\begin_inset Quotes erd +\end_inset + + 2008 proceedings. +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "key-3" + +\end_inset + + +\begin_inset Quotes erd +\end_inset + +Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes, +\begin_inset Quotes erd +\end_inset + + Chang-Ming Lee and Yu T. + Su, IEEE Signal Processing Letters, Vol. + 16, No. + 8, August 2009. +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "key-4" + +\end_inset + +“Algebraic soft-decision decoding of Reed-Solomon codes,” R. + Köetter and A. + Vardy, IEEE Trans. + Inform. + Theory, Vol. + 49, Nov. + 2003. +\end_layout + +\begin_layout Bibliography +\begin_inset CommandInset bibitem +LatexCommand bibitem +key "key-5" + +\end_inset + +Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/ +\end_layout + +\end_body +\end_document