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A few more tweaks of the draft paper.

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@ -89,16 +89,16 @@ The JT65 mode 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 strong
error-correction coding: a short block-length, low-rate, Reed-Solomon code
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 JT65 decoders have used the patented Koetter-Vardy (KV)
algebraic soft-decision decoder.
The KV decoder is implemented in a closed-source program licensed to K1JT
for use in amateur radio applications.
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 FT algotithm.
It is conceptually simple, is built around the well-known Berlekamp-Massey
errors-and-erasures algorithm, and perform at least as well as the KV decoder.
It is conceptually simple, built around the well-known Berlekamp-Massey
errors-and-erasures algorithm, and performs at least as well as the KV
decoder.
\end_layout
\begin_layout Section
@ -106,32 +106,32 @@ Introduction
\end_layout
\begin_layout Standard
JT65 message frames consist of a short, compressed message encoded for transmiss
ion with a Reed-Solomon code.
Reed-Solomon codes are block codes; as such they are characterized by the
length of their codewords,
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 number of message symbols conveyed by the codeword,
, the length of their codewords,
\begin_inset Formula $k$
\end_inset
, and the number of possible values for each symbol in the codewords.
The codeword length and the number of message symbols are specified as
a tuple in the form
, 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 using
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 symbol represents
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 consist
The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist
of 72 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
@ -191,30 +191,31 @@ For the JT65 code,
\begin_inset Formula $t=25$
\end_inset
: it is always possible to efficiently decode a received word having no
more than 25 symbol errors.
, so it is always possible to efficiently decode a received word having
no more than 25 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 ncessarily involved in this process, namely
\end_layout
\begin_layout Enumerate
determine which symbols were received incorrectly
Determine which symbols were received incorrectly.
\end_layout
\begin_layout Enumerate
determine the correct value of the incorrect symbols
Find the correct value of the incorrect symbols.
\end_layout
\begin_layout Standard
If we somehow know that certain symbols are incorrect, this information
can be used to reduce the work in step 1 and allow step 2 to correct more
than
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
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 $d$
@ -275,15 +276,7 @@ errors-only
\begin_inset Quotes erd
\end_inset
decoder and can correct up to
\begin_inset Formula $t$
\end_inset
errors (
\begin_inset Formula $t$
\end_inset
=25 for JT65).
decoder.
If
\begin_inset Formula $0<s\le d-1$
\end_inset
@ -303,7 +296,15 @@ errors-and-erasures
decoder.
The possibility of doing errors-and-erasures decoding lies at the heart
of the FT algorithm.
On that foundation we build a capability for using
\begin_inset Quotes eld
\end_inset
soft
\begin_inset Quotes erd
\end_inset
information on symbol reliability.
\end_layout
\begin_layout Section
@ -320,8 +321,8 @@ Do I feel lucky?
The FT algorithm uses the estimated quality of received symbols to generate
lists of symbols considered likely to be in error, thereby enabling reliable
decoding of received words with more than 25 errors.
As a specific example, consider a received JT65 signal producing 23 correct
symbols and 40 errors.
As a specific example, consider a received JT65 word with 23 correct symbols
and 40 errors.
We do not know which symbols are in error.
Suppose that the decoder randomly selects
\begin_inset Formula $s=40$
@ -466,7 +467,7 @@ Suppose a codeword contains
of the erased symbols are actually incorrect is then
\begin_inset Formula
\[
P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}\simeq2.4\times10^{-7}.
\]
\end_inset
@ -478,7 +479,7 @@ Similarly, the probability that
of the erased symbols are incorrect is
\begin_inset Formula
\[
P(x=36)=8.610\times10^{-9}.
P(x=36)\simeq8.6\times10^{-9}.
\]
\end_inset
@ -623,7 +624,7 @@ reference "eq:hypergeometric_pdf"
.
The odds for successful decoding on the first try are now about 1 in 38.
A few hundred independently randomized tries would be enough to all-but-guarant
ee production of a valid codeword from the BM decoder.
ee production of a valid codeword by the BM decoder.
\end_layout
\begin_layout Section
@ -653,7 +654,7 @@ The FT algorithm uses two quality indices made available by a noncoherent
on which of 64 frequency bins contains the the largest signal-plus-noise
power.
The fraction of total power in the two bins containing the largest and
second-largest powers (denoted by,
second-largest powers (denoted by
\begin_inset Formula $p_{1}$
\end_inset
@ -735,14 +736,22 @@ a-priori
.
Correspondingly, the FT algorithm works best when the probability of erasing
a symbol is somewhat larger than the probability that the symbol is incorrect.
Empirically, we found good decoding performance when the symbol erasure
probability is about 1.3 times the symbol error probability.
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 educated guesses to select symbols for erasure.
For each iteration an stochastic erasure vector is generated based on the
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 received symbols.
@ -751,7 +760,8 @@ t educated guesses to select symbols for erasure.
\begin_inset Formula $d_{s}$
\end_inset
defined as the soft distance between the received word and the codeword:
defined as the soft distance between the received word and the codeword,
where
\begin_inset Formula
\begin{equation}
d_{s}=\sum_{i=1}^{n}\alpha_{i}\,(1+p_{1,i}).\label{eq:soft_distance}
@ -780,19 +790,18 @@ Here
\end_inset
.
Think of the soft distance as 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 distance will be assigned to the one where differences
occur in symbols of lower quality.
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
Technically the FT algorithm is a list decoder, potentially generating a
list of candidate codewords.
Among the list of candidate codewords found by this stochastic search algorithm
, only the one with the smallest soft-distance from the received word is
Among the list of candidate codewords found by the stochastic search algorithm,
only the one with the smallest soft distance from the received word is
retained.
As with all such algorithms, a stopping criterion is necessary.
FT accepts a codeword unconditionally if its soft distance is smaller than
@ -801,7 +810,7 @@ Technically the FT algorithm is a list decoder, potentially generating a
\end_inset
.
A timeout is employed to limit the algorithm's execution time if no codewords
A timeout is used to limit the algorithm's execution time if no codewords
within soft distance
\begin_inset Formula $d_{a}$
\end_inset
@ -814,12 +823,12 @@ Algorithm pseudo-code:
\end_layout
\begin_layout Enumerate
For each received symbol, define the erasure probability to be 1.3 times
the
For each received symbol, define the erasure probability as 1.3 times the
\emph on
a priori
\emph default
symbol-error probability determined by the soft-symbol information
symbol-error probability determined from soft-symbol information
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
\end_inset
@ -833,8 +842,8 @@ Make independent stochastic decisions about whether to erase each symbol
\end_layout
\begin_layout Enumerate
Attempt errors-and-erasures decoding with the BM algorithm and the set of
eraseures determined in step 2.
Attempt errors-and-erasures decoding by using the BM algorithm and the set
of eraseures determined in step 2.
If the BM decoder is successful go to step 5.
\end_layout
@ -870,12 +879,20 @@ If the number of trials is less than the maximum allowed number, go to 2.
\end_layout
\begin_layout Enumerate
A codeword with
\begin_inset Formula $d_{s}\le d_{a}$
A
\begin_inset Quotes eld
\end_inset
best
\begin_inset Quotes erd
\end_inset
codeword with
\begin_inset Formula $d_{s,min}\le d_{a}$
\end_inset
has been found.
Declare a successful decode and return the codeword .
Declare a successful decode and return this codeword .
\end_layout
\begin_layout Section