Channels with Synchronization/Substitution Errors and Computation of Error Control Codes

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by  Stavros Konstantinidis

Channels with Synchronization/Substitution Errors
and Computation of Error Control Codes
Stavros Konstantinidis Nelma Moreira and Roge´rio Reis
Department of Mathematics and Computing Science CMUP & DCC
Saint Mary’s University Faculdade de Cieˆncias da Universidade do Porto
Halifax, Nova Scotia, Canada Rua do Campo Alegre, 4169007 Porto Portugal
Email: [email protected] Email: {nam,rvr}@dcc.fc.up.pt
6 (research supported by NSERC) (research supported by FCT project UID/MAT/00144/2013)
1
0
2 Abstract—We introduce the concept of an f-maximal error- inputtoalgorithms.InSectionIII,wepresenttworandomized

l detecting block code, for some parameter f between 0 and 1, in algorithms: the first one decides (up to a certain degree of
u order to formalize the situation where a block code is close to confidence) whether a given block code C is maximal er-
J maximal with respect to being error-detecting. Our motivation detectingforagivenchanneler.Thesecondalgorithmisgiven
9 for this is that constructing a maximal error-detecting code achanneler,aner-detectingblockcodeC ⊆Aℓ (whichcould
2 is a computationally hard problem. We present a randomized
be empty), and integer N > 0, and attempts to add to C N
algorithm that takes as input two positive integers N,ℓ, a
new words of length ℓ resulting into a new er-detecting code.
] probability value f, and a specification of the errors permitted
T If less than N words get added then either the new code is
in some application, and generates an error-detecting, or error-
I correcting, block code having up to N codewords of length ℓ. If 95%-maximalor the chance thata randomlychosenword can
s. thealgorithmfindslessthanN codewords,thenthosecodewords be added is less than 5%. Our motivation for considering a
c constitute a code that is f-maximal with high probability. The randomized algorithm is that embedding a given er-detecting
[ error specification (also called channel) is modelled as a trans- block code C into a maximal er-detecting block code is a

2 ducer, which allows one to model any rational combination of computationally hard problem—this is shown in Section IV.
v substitution and synchronization errors. We also present some In Section V, we discuss briefly some capabilities of the
2 elements of our implementation of various error-detecting prop- new module codes.py in the open source software package
erties and their associated methods. Then, we show several tests
1 FAdo [1], [4] and we discuss some tests of the randomized
of the implemented randomized algorithm on various channels.
3 algorithms on various channels. In Section VI, we discuss
Amethodologicalcontributionisthepresentationofhowvarious
6 a few more points on channel modelling and conclude with
desirable error combinations can be expressed formally and
0 processed algorithmically. directions for future research.
.
1
We note that, while there are various algorithms for com-
0
I. INTRODUCTION puting error-control codes, to our knowledge these work for
6
1 We consider block codes C, that is, sets of words of the specific channels and implementations are generally not open
: same length ℓ, for some integer ℓ>0. The elements of C are source.
v
i calledcodewordsorC-words.WeuseAtodenotethealphabet
X used for making words and II. CHANNELS AND ERROR CONTROL CODES
r
a Aℓ =the set of all words of length ℓ. We need a mathematical model for channels that is use-
ful for answering algorithmic questions pertaining to error
Our typical alphabet will be the binary one {0,1}. We shall control codes. While many models of channels and codes
use the variables u,v,w,x,y,z to denote words over A (not for substitution-type errors use a rich set of mathematical
necessarily in C). The empty word is denoted by ε. structures, this is not the case for channels involving syn-
We also consider error specifications er, which we call chronizationerrors[13]. We believethe appropriatemodelfor
combinatorial channels, or simply channels. A channel er our purposes is that of a transducer. We note that transducers
specifies, for each allowed input word x, the set er(x) of all have been defined as early as in [18], and are a powerful
computational tool for processing sets of words—see [2] and
possible outputwords. We assume that error-freecommunica-
tion is always possible, so x ∈ er(x). On the other hand, if pg 41–110 of [17].
y ∈ er(x) and y 6= x then the channel introduces errors into Definition 1. A transducer is a 5-tuple1 t = (S,A,I,T,F)
x. Informally, a block code C is er-detecting if the channel such that A is the alphabet, S is the finite set of states, I ⊆S
cannot turn a given C-word into a different C-word. It is er- is the set of initial states, F ⊆S is the set of final states, and
correcting if the channel cannot turn two different C-words T is the finite set of transitions. Each transition is a 4-tuple
into the same word. (s ,x /y ,t ), where s ,t ∈ S and x ,y are words over A.
i i i i i i i i
In Section II, we make the aboveconceptsmathematically
1The general definition of transducer allows two alphabets: the input and
precise, and show how known examples of combinatorial
the output alphabet. Here, however, we assume that both alphabets are the
channels can be defined formally so that they can be used as same.
The word xi is the input label and the word yi is the output 0/0,1/1 0/0,1/1 0/0,1/1
label of the transition. For two words x,y we write y ∈t(x)
to mean that y is a possible output of t when x is used as
0/1 0/1
input. More precisely, there is a sequence sub2= s t1 t2
1/0 1/0
(s0,x1/y1,s1), (s1,x2/y2,s2),..., (sn−1,xn/yn,sn)
a/a a/a a/a
of transitions such that s0 ∈ I, sn ∈ F, x = x1···xn and
y = y1···yn. The relation R(t) realized by t is the set of
word pairs (x,y) such that y ∈t(x). A relation ρ⊆A∗×A∗ a/ε a/ε
iscalledrationalifitisrealizedbyatransducer.Ifeveryinput id2= s t1 t2
ε/a ε/a
and every output label of t is in A∪{ε}, then we say that
t is in standard form. The domain of the transducer t is the
a/a a/a
set of words x such that t(x) 6= ∅. The transducer is called
input-preserving if x ∈ t(x), for all words x in the domain
of t. The inverse of t, denoted by t−1, is the transducer that a/ε ε/a
is simply obtained by making a copy of t and changing each del1= s r t
transition (s,x/y,t) to (s,y/x,t). Then
x∈t−1(y) if and only if y ∈t(x). a/a a/a
We note that every transducer can be converted (in linear ε/a a/ε
ins1= s r t
time) to one in standard form realizing the same relation.
In our objective to model channels er as transducers, we
require that a transducerer is a channelif it allows error-free Figure 1. Examples of (combinatorial) channels: sub2,id2,del1,ins1.
communication, that is, er is input-preserving. Notation: A short arrow with no label points to an initial state (e.g., state
s), and a double line indicates a final state (e.g., state t). An arrow with
Definition 2. An error specification er is an input-preserving label a/a represents multiple transitions, each with label a/a, for a ∈ A;
transducer. The (combinatorial) channel specified by er is and similarly for an arrow with label a/ε—recall, ε = empty word. Two or
R(er), thatis, the relation realizedby er. For the purposesof morelabelsononearrowfromsomestateptosomestateqrepresentmultiple
transitionsbetweenpandqhavingtheselabels.Channel sub2:usesthebinary
thispaper,however,wesimplyidentifytheconceptofchannel
alphabet {0,1}. On input x, sub2 outputs x, or any word that results by
with that of error specification. performingoneortwosubstitutions inx.Thelattercaseiswhensub2 takes
the transition (s,0/1,t1) or (s,1/0,t1), corresponding to one error, and
A piece of notation that is useful in this work is the then possibly (t1,0/1,t2)or(t1,1/0,t2), corresponding toasecond error.
A block code C is sub2-detecting iff the min. Hamming distance of C is
following, where W is any set of words,
>2.Channelid2:alphabetAnotspecified.Oninputx,id2outputsaword
that results by inserting and/or deleting at most 2 symbols in x. A block
er(W) = [ er(w) (1) code C is id2-detecting iff the min. Levenshtein distance of C is > 2 [8].
w∈W Channels del1,ins1:considered in[15], here alphabet Anot specified. On
input x, del1 outputs either x, or any word that results by deleting exactly
Thus, er(W) is the set of all possible outputs of er when the onesymbolinxandtheninserting asymbolattheendofx.
input is any word from W. For example, if er = id1 = the
channel that allows up to 1 symbol to be deleted or inserted
in the input word, then id1({00,11}) = An er-detecting block code C is called maximal er-detecting
if C ∪{w} is not er-detecting for any word of length ℓ that
{00,0,000,100,010,001,11,1,011,101,110,111}.
is not in C. The concept of a maximal er-correcting code is
Fig. 4 considers examples of channels that have been defined similar.
in past research when designing error control codes. Here
these channels are shown as transducers, which can be used From a logical point of view (see Lemma 4 below) error-
as inputs to algorithmsfor computingerrorcontrolcodes. For detection subsumes the concept of error-correction. This con-
the channel sub2, we have 00101∈sub2(00000) because on nection is stated already in [7] but without making use of
input00000,the channelsub2 can read the first two input0’s it there. Here we add the fact that maximal error-detection
at state s and output 0, 0; then, still at state s, read the 3rd 0 subsumes maximal error-correction. Due to this observation,
and output 1 and go to state t1; etc. in this paper we focus only on error-detecting codes.
The concepts of error-detectionand -correction mentioned Note:Theoperation‘◦’betweentwotransducerstandsis
in the introduction are phrased below more rigorously. calledcompositionandreturnsanewtransducers◦tsuchthat
z ∈(s◦t)(x) if and only if y ∈t(x)andz ∈s(y),for some y.
Definition 3. Let C be a block code and let er be a channel.
We say that C is er-detecting if Lemma 4. Let C ⊆Aℓ be a block code ander be a channel.
Then C is er-correcting if and only if it is (er−1 ◦ er)-
v ∈C, w ∈C and w ∈er(v) imply v =w. detecting. Moreover, C is maximal er-correcting if and only
We say that C is er-correcting if if it is maximal (er−1◦er)-detecting.
v ∈C, w ∈C and x∈er(v)∩er(w) imply v =w. Proof:Thefirststatementisalreadyin[7].Forthesecond
2
statement, first assume that C is maximal er-correcting and as in the case of a channel, but each transition has only an
consider any word w ∈Aℓ\C. If C∪{w} were (er◦er−1)- input label, that is, it is of the form (s,x,t) with x being one
detectingthenC∪{w}wouldalsobeer-correctingand,hence, alphabetsymbolor the emptyword ε. The languageaccepted
C would be non-maximal; a contradiction. Thus, C must be by a is denoted by L(a) and consists of all words formed by
maximal (er◦er−1)-detecting. The converse can be shown concatenating the labels in any path from an initial to a final
analogously. state. The automatonis called deterministic, or DFA for short,
ifI consistsofasinglestate,therearenotransitionswithlabel
Theoperation‘∨’betweenanytwotransdcucerstandsis
ε, and there are no two distinct transitions with same labels
obtainedbysimplytakingtheunionoftheirfivecorresponding
going out of the same state. Special cases of automata are
components (states, alphabet, initial states, transitions, final
constraint systems in which normally all states are final (pg
states) after a renaming, if necessary, of the states such that
1635–1764 of [16]), and trellises. A trellis is an automaton
the two transdcucers have no states in common. Then
accepting a block code, and has one initial and one final state
(t∨s)(x) = t(x) ∪ s(x). (pg1989–2117of[16]).Inthecaseofatrellisawetalkabout
the code represented by a, and we denote it as C(a), which
Let er be a channel, let C ⊆ Aℓ be an er-detecting block is equal to L(a).
code, and let w∈Aℓ\C. In [3], the authors show that
For computationalcomplexity considerations, the size |m|
C∪{w} is er-detecting iff w∈/ (er∨er−1)(C). (2) of a finite state machine (automaton or transducer) m is the
Definition 5. Let C ⊆Aℓ be an er-detecting block code. We number of states plus the sum of the sizes of the transitions.
saythatawordwcanbeaddedintoC ifw ∈/ (er∨er−1)(C). The size of a transition is 1 plus the length of the label(s) on
the transition. We assume that the alphabet A is small so we
do not include its size in our estimates.
Statement (2) above implies that
C is maximal er-detecting iff Aℓ \ (er∨er−1)(C) = ∅. An important operation between an automaton a and a
(3) transducert,heredenotedby‘✄’,returnsanautomaton(a✄t)
that acceptsthe set of all possible outputsof t when the input
Definition 6. The maximality index of a block code C ⊆ Aℓ is any word from L(a), that is,
w.r.t. a channel er is the quantity
|Aℓ∩(er∨er−1)(C)| L(a✄t)=t(L(a)).
maxind(C,er)= .
|Aℓ| Remark 8. We recall here the construction of (a✄t) from
Let f be a real number in [0,1]. An er-detecting block code given a = (S1,A,I1,T1,F1) and t = (S2,A,I2,T2,F2),
C is called f-maximal er-detecting if maxind(C,er)≥f. where we assume that a contains no transition with label ε.
First, if necessary, we convert t to standard form. Second,
if t contains any transition whose input label is ε, then we
The maximality index of C is the proportion of the ‘used
up’ words of length ℓ over all words of length ℓ. One can add into T1 transitions (q,ε,q), for all states q ∈ S1. Let T1
denote now the updated set of transitions. Then, we construct
verify the following useful lemma.
the automaton
Lemma 7. Let er be a channel and let C ⊆ Aℓ be an er-
detecting block code. b = (S1×S2, A, I1×I2, T, F1×F2)
1) maxind(C,er) = 1 if and only if C is maximal er- such that ((p1,p2),y,(q1,q2)) ∈ T, exactly when there are
detecting. transitions (p1,x,q1)∈T1 and (p2,x/y,q2)∈T2. The above
constructioncanbedoneintimeO(|a||t|) andthesizeofbis
2) Assuming that words are chosen uniformly at random
from Aℓ, the maximality index is the probability that a O(|a||t|).Therequiredautomaton(a✄t)isthetrimversionof
b,whichcanbecomputedintimeO(|b|).(Thetrimversionof
randomly chosen word w of length ℓ cannot be added
into C preserving its being er-detecting, that is, an automaton m is the automaton resulting when we remove
any states of m thatdo notoccurin some path froman initial
maxind(C,er) = Pr w cannot be added into C . to a final state of m.)
h i
Proof: The first statement follows from Definition 6 and nonMax (er,a,f,ε)
condition(3).Thesecondstatementfollowswhenwenotethat b := (a✄(er∨er−1));
the event that a randomly chosen word w from Aℓ cannot be n := 1 + 1/ 4ε(1−f)2 ;
added into C is the same as the event that w ∈ Aℓ ∩(er∨ j (cid:16) (cid:17)k
er−1)(C). ℓ := the length of the words in C(a);
tr := 1;
while (tr ≤ n):
III. GENERATING ERROR CONTROL CODES
w := pickFrom(A,ℓ);
We turn now our attention to algorithms processing chan- if (w not in L(b)) return w;
nels and sets of words. A set of words is called a language, tr := tr+1;
with a block code being a particular example of language. return None;
A powerfulmethod of representing languagesis via finite au-
Figure2. Algorithm nonMax—seeTheorem9.
tomata[17].A(finite)automatonaisa5-tuple(S,A,I,T,F)
3
Next we present our randomized algorithms—we use [14] as Remark10. Wementiontheimportantobservationthatonecan
referenceforbasicconcepts.Weassumethatwehaveavailable modify the algorithm nonMax by removing the construction
touseinouralgorithmsanidealmethodpickFrom(A,ℓ)that of b and replacing the ‘if’ line in the loop with
chooses uniformly at random a word in Aℓ. A randomized
if (C(a)∪{w} is er-detecting) return w;
algorithm R(···) with specific values for its parameters can
beviewedasarandomvariablewhosevalueiswhatevervalue While with this change the output would still be correct,
is returned by executing R on the specific values. the time complexity of the algorithm would increase to
Theorem9. ConsiderthealgorithmnonMax inFig.2,which O(cid:16)|a|2|er|.(cid:0)ε(1 − f)2(cid:1)(cid:17). This is because testing whether
takes as input a channel er, a trellis a accepting an er- L(v) is er-detecting, for any given automaton v and channel
detecting code, and two numbers f,ε∈[0,1]. er, can be done in time O(|v|2|er|), and in practice |v| is
much larger than ℓ.
1) The algorithm either returns a word w ∈Aℓ\C(a) such
that the code C(a)∪{w} is er-detecting, or it returns In Fig. 3, we present the main algorithm for adding new
None. words into a given deterministic trellis a.
2) If C(a) is not f-maximal er-detecting, then
makeCode (er,a,N)
Pr nonMax returns None <ε. W := empty list; c:= a
h i
cnt := 0; more := True;
while (cnt < N and more)
3) The time complexity of nonMax is w := nonMax(er,c,0.95,0.05);
if (w is None) more := False;
O(cid:16)ℓ|a||er|.(ε(1−f)2)(cid:17). else {add w to c and to W; cnt := cnt+1;}
return c, W;
Proof: The first statement follows from statement (2) in Figure 3. Algorithm makeCode—see Theorem 12. The trellis a can be
the previous section, as any w returned by the algorithm is omittedsothatthealgorithm wouldstartwithanemptysetofcodewords.In
not in (er∨er−1)(C(a)). For the second statement, suppose this case, however, the algorithm would require as extra input the codeword
thatthe codeC(a) isnotf-maximaler-detecting.LetCnt be lengthℓandthedesiredalphabetA.Weusedthefixedvalues0.95and0.05,
astheyseemtoworkwellinpractical testing.
the random variable whose value is the value of tr − 1 at the
endofexecutionoftherandomizedalgorithmnonMax.Then,
Cnt counts the number of words that are in C(a) out of n Remark 11. Insomesense,algorithmmakeCode generalizes
randomlychosenwordsw. ThusCnt is binomial:the number to arbitrary channels the idea used in the proof of the well-
of successes (words w in L(b)) in n trials. So E(Cnt)=np, known Gilbert-Varshamov bound [12] for the largest possible
where p=Prhw∈L(b)i. By the definitionof n in nonMax, blockcodeM ⊆Aℓ thatis subk-correcting,forsome number
we get 1/(4n(1 − f)2) < ε. Now consider the Chebyshev kofsubstitutionerrors.Inthatproof,awordcanbeaddedinto
the code M if the word is outside of the union of the “balls”
inequality, Prh|X −E(X)|≥ai ≤ σ2/a2, where a > 0 is sub2k(u), for all u∈M. In that case, we have that sub−k1 =
arbitrary and σ2 is the variance of some random variable X. subk and (sub−k1◦subk)=sub2k(u). The present algorithm
For X =Cnt the variance is np(1−p), and we get adds new words w to the constructed trellis c such that each
new wordw isoutsideof the “union-ball”(er∨er−1)(C(c)).
Pr |Cnt/n−p|≥1−f <ε,
h i Theorem 12. Algorithm makeCode in Fig. 3 takes as input
a channel er, a deterministic trellis a of some length ℓ, and
wherewe useda=n(1−f)andthefactthatp(1−p)≤1/4. anintegerN >0suchthatthecodeC(a)iser-detecting,and
Using Lemma 7 and the assumption that C(a) is not f- returns a deterministic trellis c and a list W of words such
maximal, we have that maxind(C(a),er)<f, which implies that the following statements hold true:
Prhw ∈L(b)i<f; hence, p<f. Then 1) C(c)=C(a)∪W and C(c) is er-detecting,
2) If W has less than N words, then either
Prh nonMax returns Nonei=PrhCnt=ni= maxind(C(c),er) ≥ 0.95 or the probability that a
Pr Cnt/n=1 =Pr Cnt/n−p=1−p ≤ randomly chosen word from Aℓ can be added in C(c) is
h i h i <0.05.
Prh|Cnt/n−p|≥1−pi≤Prh|Cnt/n−p|≥1−fi <ε, 3) The algorithm runs in time O ℓN|er||a|+ℓ2N2|er| .
(cid:16) (cid:17)
as required.
Proof: Let c be the value of the trellis c at the end of
i
For the third statement, we use standard results from the i-th iterationofthe while loop.The firststatementfollows
automaton theory, [17], and Remark 8. In particular, com- fromTheorem9:anywordwreturnedbynonMax issuchthat
puting b can be done in time O(|a|·|er|) such that |b| = C(c )∪{w}iser-detecting.Forthesecondstatement,assume
i
O(|a| · |er|). Testing whether w ∈ L(b) can be done in that, at the end of execution, W has <N words and C(c) is
time O(|w||b|)=O(ℓ|b|). Thus, the algorithm works in time not 95%-maximal. By the previous theorem, this means that
O(ℓ|a||er|/(ε(1−f)2)). the random process nonMax(er,c,0.95,0.05) returns None
4
withprobability<0.05,asrequired.Forthethirdstatement,as such that a accepts Aℓ if and only if C(d) is a maximal er-
2
the loop in the algorithm nonMax performs a fixed number detectingblockcodeoflengthℓ.Therestoftheproofconsists
of iterations (=2000), we have that the cost of nonMax is of 5 parts: construction of deterministic trellis d accepting
O(ℓ|c ||er|). The cost of adding a new word w of length ℓ words of length ℓ, construction of er, facts about d and er,
i
to ci−1 is O(ℓ) and increases its size by O(ℓ), so each ci is proving that C(d) is er-detecting, proving that a accepts Aℓ2
of size O(|a|+iℓ). Thus, the cost of the i-th iteration of the if and only if C(d) is maximal er-detecting.
while loop in makeCode is O(ℓ|er|(|a|+iℓ)). As there are Construction of d: Let A be the alphabetA2∪˙T, where T
up to N iterations the total cost is is the set of transitions of a. The required deterministic trellis
N d is any deterministic trellis accepting Aℓ\Aℓ2, that is,
XO(cid:16)ℓ|er|·(|a|+iℓ)(cid:17) = O(cid:16)ℓN|er||a|+ℓ2N2|er|(cid:17). C(d) = Aℓ\Aℓ2.
i=1
Thiscanbe constructed,forinstance,bymakingdeterministic
trellises d1 and d2 accepting, respectively, Aℓ and Aℓ2, and
Remark 13. In the algorithm makeCode, attempting to add then intersecting d1 with the complement of d2. Note that
only one word into C(a) (case of N = 1), requires time any word in C(d) contains at least one symbol in T.
O(ℓ|er||a|+ℓ2|er|), which is of polynomial magnitude. This
case is equivalent to testing whether C(a) is maximal er- Construction of er: This is of the form (er1 ∨ er2) as
detecting, which is shown to be a hard decision problem in follows.Thetransducerer2hasonlyonestatesandtransitions
(s,α/α,s), for all α ∈ A, and realizes the identity relation
Theorem 15.
{(x,x) | x ∈ A∗}. Thus, we have that er2(x) = {x}, for all
wRehmeraerkthe14i.nitIinal tthreellivserasioisnomofitttehde, tahlegotriimthemcommapkleexCitoydies twhoartdTs′′xco∈nsAis.tsTohfeetxraancstldyutcheerterarn1s=itio(nSs,(Ap,,s(p,T,a′′,,qF)/)ai,sq)sufcohr
O(ℓ2N2|er|). We also note that the algorithm would work which (p,a,q) is a transition of a.
with the same time complexity if the given trellis a is not
deterministic.In this case, however,the resulting trellis would Factsaboutdander:Thefollowingfactsarehelpfulinthe
not be (in general) deterministic either. restoftheproof.Someofthesefactsrefertothedeterministic
trellis d1 = (S,T,s,T1,F) resulting by omitting the output
partsofthetransitionlabelsofer1,thatis,(p,(p,a,q),q)∈T1
IV. WHY NOT USE ADETERMINISTIC ALGORITHM exactly when (p,(p,a,q)/a,q) ∈ T′′. Then, C(d1) ⊆ (A\
Our motivation for considering randomized algorithms is A2)ℓ ⊆C(d).
that the embedding problem is computationally hard: given F0: L(d1✄er1)=L(a).
a deterministic trellis d and a channel er, compute (using a F1: The domain of er1 is C(d1), a subset of (A\A2)ℓ.
ddeetteercmtiningisctoicdealcgoonrittahimni)ngatCre(ldli)s.tBhaytcroepmrpesuetanttisoanamllayxhimaradl,ewre- FF23:: Iefr1v(∈C(edr)1)(u=)Lth(ean).v ∈Aℓ2 and v 6=u.
mean that a decision version of the embedding problem is F4: er−1(C(d))=∅.
1
coNP-hard. This is shown next.
For fact F0, note that the product construction described
Theorem 15. The following decision problem is coNP-hard. in Remark 8 produces in (d1 ✄er1) exactly the transitions
((p,p),a,(q,q)), where(p,a,q)isatransitionina,bymatch-
IAnnsstawnecre::dwehteetrhmeirniCst(idc)triesllmisadximanadl ecrh-adnenteeclteinrg.. (inpg,(apn,ya,trqa)n/sait,iqo)no(pf,e(rp1,.aF,qa)c,tqF)1offodll1oownslybywitthhethceontrsatrnuscittiioonn
of er1 and the definition of d1: in any accepting computation
Proof: Let us call the decision problem in question of er1, the input labels appear in an accepting computation
MAXED, and let FULLBLOCK be the problem of deciding of d1 that uses the same sequence of states. F3 is shown as
whether a given trellis over the alphabet A2 ={0,1} with no follows: As the domain of er1 is C(d1) and C(d1) ⊆ C(d),
ε-labeled transitions accepts Aℓ2, for some ℓ. The statement is we have that er1(C(d))=er1(C(d1)), which is L(a) by F0.
a logical consequence of the following claims. Fact F4 followsby notingthat the domainof er−1 is a subset
1
of Aℓ but C(d) contains no words in Aℓ.
Claim 1: FULLBLOCK is coNP-complete. 2 2
Claim 2: FULLBLOCK is polynomially reducible to C(d) is er-detecting: Let u,v ∈ C(d) such that v ∈
MAXED. er(u) = er1(u)∪{u}. We need to show that v = u, that is,
The first claim follows from the proof of the following to show that v ∈/ er1(u). Indeed, if v ∈ er1(u) then v ∈ Aℓ2,
which contradicts v ∈C(d)=Aℓ\Aℓ.
factonpage329of[10]:Decidingwhethertwogivenstar-free 2
regularexpressionsoverA2areinequivalentisanNP-complete a accepts Aℓ2 if and only if C(d) is maximaler-detecting:
problem. Indeed, in that proof the first regular expression can By statement (3) we have that C(d) is maximal er-detecting,
be arbitrary, but the second regular expression represents the if and only if (er∨er−1)(C(d))=Aℓ. We have:
alanstgaur-afgreeeArℓe,gfuolrarsoemxperpesossiiotinvetoinatnegaecryℓc.liMcoaruetoovmear,tocnonwvietrhtinnog (er∨er−1)(C(d)) = C(d)∪er1(C(d))∪er−11(C(d))
ε-labeled transitions is a polynomial time problem. = (Aℓ\Aℓ2)∪L(a)∪∅
For the second claim, consider any trellis a = = (Aℓ\Aℓ2)∪˙ L(a).
(S,A2,s,T,F)withnoε-labeledtransitionsin T.We needto Thus,C(d)ismaximaler-detecting,ifandonlyifL(a)=Aℓ2,
construct in polynomial time an instance (d,er) of MAXED as required.
5
V. IMPLEMENTATIONAND USE Definition 3 and using standard logical arguments, it follows
that
Allmainalgorithmictoolshavebeenimplementedoverthe
years in the Python package FAdo [1], [4], [6]. Many aspects 1) L is er1-detecting and er2-detecting, if and only if L is
of the new module FAdo.codes are presented in [6]. Here we (er1∨er2)-detecting;
presentmethodsofthatmodulepertainingtogeneratingcodes.
2) L is er−1-detecting, if and only if it is er-detecting, if
Assume that the string d1 contains a description of the and only if it is (er−1∨er)-detecting.
transducerdel1 in FAdo format.In particular,d1 beginswith
the type of FAdo object being described, the final states, and The inverseof del1 is ins1 andis shownin Fig. 4, where
the initial states (after the character *). Then, d1 containsthe recall it results by simply exchanging the order of the two
listoftransitions,witheachoneoftheform“sxyt\n”,where words in all the labels in del1. By statement 2 of the above
‘\n’ is the new-line character. This shown in the following remark, the del1-detecting codes are the same as the ins1-
Python script. detecting ones, and the same as the (del1∨ins1)-detecting
ones—this is shown in [15] as well. The method of using
import FAdo.codes as codes transducers to model channels is quite general and one can
d1 = ’@Transducer 0 2 * 0\n’ give many more examples of past channels as transducers, as
’0 0 0 0\n0 1 1 0\n0 0 @epsilon 1\n’ well as channels not studied before. Some further examples
’0 1 @epsilon 1\n1 0 0 1\n1 1 1 1\n’ are shown in the next figures, Fig. 4-6.
’1 @epsilon 0 2\n1 @epsilon 1 2\n’
One can go beyond the classical error control properties
pd1 = codes.buildErrorDetectPropS(d1)
and define certain synchronization properties via transducers.
a = pd1.makeCode(100, 8, 2)
Let OF be the set of all overlap-freewords, that is, all words
print pd1.notSatisfiesW(a)
w such that a proper and nonempty prefix of w cannot be
print pd1.nonMaximalW(a, m)
a suffix of w. A block code C ⊆ OF is a solid code if
s2 = ...string for transducer sub_2
any proper and nonempty prefix of a C-word cannot be a
ps2 = codes.buildErrorDetectPropS(s2)
suffix of a C-word. For example, {0100, 1001} is not a
pd1s2 = pd1 & ps2
block solid code, as 01 is a prefix and a suffix of some
b = pd1s2.makeCode(100, 8, 2)
codewords and 01 is nonempty and a proper prefix (shorter
than the codewords).Solid codescan also be non-blockcodes
The above script uses the string d1 to create the object pd1 by extending appropriatelythe above definition [19] (they are
representing the del1-detection property over the alphabet also called codes without overlaps in [9]). The transducer ov
{0,1}.Then,itconstructsanautomatonarepresentingadel1- in Fig. 6 is such that any block code C ⊆OF is a solid code,
detectingblockcodeoflength8withupto100wordsoverthe if and only if C is an ‘ov-detecting’block code. We note that
2-symbolalphabet{0,1}.ThemethodnotSatisfiesW(a) solid codes have instantaneous synchronization capability (in
tests whether the code C(a) is del1-detecting and returns particular all solid codes are comma-free codes) as well as
a witness of non-error-detection (= pair of codewords u,v synchronization in the presence of noise [5].
with v ∈ del1(u)), or (None, None)—of course, in the
above example it would return (None, None). The method
nonMaximalW(a, m) tests whether the code C(a) is max- 0b
imaldel1-detectingandreturnseithera wordv ∈L(m)\C(a)
such that C(a)∪{v} is del1-detecting, or None if C(a) is 0/1 a/a
already maximal. The object m is any automaton—here it is
the trellis representingAℓ. This methodis used only forsmall 1/0
a/a 1/0
codes,asingeneralthemaximalityproblemisalgorithmically 0a 1
hard (recall Theorem 15), which motivated us to consider the 0/1
randomized version nonMax in this paper. For any channel
er and trellis a, the method notSatisfiesW(a) can be bsid2= 0
ε/a, a/ε
made to work in time O(|er||a|2), which is of polynomial
ε/a, a/ε
complexity. The operation ‘&’ combines error-detection prop- 0/1 1/0
erties. Thus, the second call to makeCode constructs a code
that is del1-detecting and sub2-detecting (=sub1-correcting). 2 a/a
1a 1b
0/1
VI. MORE ON CHANNEL MODELLING,TESTING
1/0
In this section, we consider further examples of channels
and show how operationson channels can result in new ones.
We also show the results of testing our codes generation Figure 4. The channel specified by bsid2 allows up to two errors in the
inputword.Eachoftheseerrorscanbeadeletion,aninsertion,orabitshift:
algorithm for several different channels.
a10becomes01,ora01becomes10.Thealphabet is{0,1}.
Remark 16. We note that the definition of error-detecting
(or error-correcting) block code C is trivially extended to
any language L, that is, one replaces in Definition 3 ‘block For ε = 0.05 and f = 0.95, the value of n in nonMax
code C’ with ‘language L’. Let er,er1,er2 be channels. By is 2000. We performed several executions of the algorithm
6
N=,ℓ=,end= id2 del1 sub2 bsid2 ov
100, 8, 18,20,23 37,42,51 15,16,18 17,19,21 01,07,08
100, 7, 10,12,13 20,23,28 09,10,13 11,11,13 03,04,05
100, 8, 1 11,13,14 39,50,64 09,10,11 09,12,13 01,05,06
100, 8, 01 06,07,08 64,64,64 04,06,08 06,07,09 01,04,05
500, 12, 177,182,188 500,500,500 148,157,162 169,173,178 51,59,63
500, 13, 318,327,334 272,273,278 302,303,309 43,111,120
a/ε maximum, that is, having the largest possible number of
tostatet1 f0 codewords, for given er and ℓ. It seems maximum codes are
a/a a/a,a/ε rare, but there are many random maximal ones having lower
rates. The del1-detectingcodeof [15] has higherrate thanall
the random ones generated here.
segd4= s1 s2 s3
a/a a/a
For the case of block solid codes (last column of the
s0 a/a table), we note that the function pickFrom in the algorithm
a/ε a/ε
nonMax has to be modified as the randomly chosen word w
a/ε
t1 t2 t3 should be in OF.
a/a a/a
tostates1 a/ε a/a VII. CONCLUSIONS
f1
a/a We have presented a unified method for generating error
control codes, for any rational combination of errors. The
method cannot of course replace innovative code design, but
Figure 5. Transducer for the segmented deletion channel of [11] with
should be helpful in computing various examples of codes.
parameter b= 4. In each of the length b consecutive segments of the input
word, at most one deletion error occurs. The length of the input word is a The implementation codes.py is available to anyone for
multiple of b. By Lemma 4, segd4-correction is equivalent to (segd−41 ◦ download and use [4]. In the implementation for generating
segd4)-detection. codes,weallowonetospecifythatgeneratedwordsonlycome
froma certaindesirablesubsetM of Aℓ, which is represented
by a deterministic trellis. This requires changing the function
a/ε a/a ε/a
pickFrom in nonMax so that it chooses randomly words
from M. There are a few directions for future research. One
a/a ε/a is to work on the efficiency of the implementations, possibly
ov= 0 1 2 allowing parallel processing, so as to allow generation of
block codes having longer block length. Another direction is
to somehow find a way to specify that the set of generated
Figure6. Thisinput-preservingtransducerdeletesaprefixoftheinputword codewords is a ‘systematic’ code so as to allow efficient
(apossibly emptyprefix)andtheninserts apossiblyemptysuffixattheend encodingofinformation.Athirddirectionistodoasystematic
oftheinputword. study on how one can map a stochastic channel sc, like the
binary symmetric channel or one with memory, to a channel
er (representing a combinatorial channel), so as the available
makeCode on various channels using n = 2000, no initial algorithms on er have a useful meaning on sc as well.
trellis, and alphabet A = {0,1}. In the above table, the
first column gives the values of N and ℓ, and if present and
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