The dual of convolutional codes over $\mathbb{Z}_{p^r}$

0.09 MB·

by  Mohammed El Oued

Chapter 1
Z
The dual of convolutional codes over
pr
MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
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Abstract An importantclass of codes widely used in applicationsis the class of

convolutionalcodes.Mostoftheliteratureofconvolutionalcodesisdevotedtocon-
]
A volutional codes over finite fields. The extension of the concept of convolutional
R codesfromfinitefieldstofiniteringshaveattractedmuchattentioninrecentyears
. duetofactthattheyarethemostappropriatecodesforphasemodulation.However
h
convolutionalcodes over finite rings are more involved and not fully understood.
t
a Many results and features that are well-knownfor convolutionalcodes over finite
m
fields have not been fully investigated in the context of finite rings. In this paper
[ we focus in one of these unexploredareas, namely, we investigate the dual codes

ofconvolutionalcodesoverfinite rings.Inparticularwestudythe p-dimensionof
1
v the dual code of a convolutionalcode over a finite ring. This contribution can be
0 considered a generalization and an extension, to the rings case, of the work done
2 byForneyandMcElieceonthedimensionofthedualcodeofaconvolutionalcode
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overafinitefield.
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v MohammedElOued
i FSMMath Department, University of Monastir, Monastir 5050, Tunisia e-mail:
X
[email protected]
r
a DiegoNapp
CIDMA-CenterforResearchandDevelopmentinMathematicsandApplications,Departmentof
Mathematics,UniversityofAveiro,Aveiro,Portugale-mail:[email protected]
RaquelPinto
CIDMA-CenterforResearchandDevelopmentinMathematicsandApplications,Departmentof
Mathematics,UniversityofAveiro,Aveiro,Portugale-mail:[email protected]
MarisaToste
CIDMA - Center for Research and Development in Mathematics and Applications, Instituto
PolitcnicodeCoimbra-ESTGOH,Coimbra,Portugale-mail:[email protected]
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2 MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
1.1 Introduction
Codesplayanimportantroleinourdays.Theyareimplementedinmostofallcom-
munications systems in order to detect and correct errors that can be introduced
duringthetransmissionofinformation.Convolutionalcodesoverfiniteringswere
firstintroducedby [10] andhavebecomingmorerelevantforcommunicationsys-
temsthatcombinecodingandmodulation.
We will consider convolutionalcodesconstituted by left compactsequencesin
Zpr, where p is a prime andr an integer greaterthan 1, i.e., the codewordsof the
codewillbeoftheform
w:Z→Zn
pr
t 7→ w
t
where w =0 for t <k for some k ∈Z. These sequences can be represented by
t
¥
Laurent series, w(D) = (cid:229) wtDt. Let us denote by Zpr((D)) the ring of Laurent
t=k
series over Zpr. Moreover,we will represent the ring of polynomialsover Zpr by
Zpr[D]andtheringofrationalmatricesoverZpr byZpr(D).Moreprecisely,Zpr(D)
istheset
p(D)
{ :p(D),q(D)∈Zpr[D]andthecoefficientofthesmallestpowerofDinq(D)isaunit}
q(D)
modulotheequivalencerelation
p(D) p (D)
∼ 1 ifandonlyif p(D)q (D)= f (D)q(D).
1 1
q(D) q (D)
1
Convolutional codes over finite rings behave very differently from convolutional
codesoverfinitefieldsduetotheexistenceofzerodivisors.Onemaindifferenceis
thataconvolutionalcodeoverafinitefieldFisalwaysafreemoduleoverF((D))
whichdoesnothappenintheringcase.Inordertodealwiththisproblemwewill
consideranewtypeofbasis,forZpr[D]-submodulesofZnpr[D],whichwillallowus
todefineakindofbasisforeveryconvolutionalcode,called p-basis,andarelated
typeofdimension,called p-dimension.Thisnotionshavebeenextensivelyusedin
thelastdecade[4,6,7,8,9,12,2,3],extendingtheideasof p-adicexpansion, p-
dimension, p-basis, etc, used in the contextof Zpr-submodulesof Znpr, [1, 13, 14,
15].
InthispaperwewillstudythedualofaconvolutionalcodeoverZpr[D].Inpar-
ticular, we will show that the dual of a convolutionalcode is also a convolutional
codeandwewillrelatethep-dimensionsofaconvolutionalcodeanditsdual.Hence
thisworkgeneralizestheresultsderivedbyForneyandMcEliece[5,11]ofconvo-
lutionalcodesoverfinitefields.
1 ThedualofconvolutionalcodesoverZpr 3
1.2 The moduleZn [D]
pr
AnyelementinZn canbewrittenuniquelyasalinearcombinationof1,p,p2,...
pr
...,pr−1,withcoefficientsinAp={0,1,...,p−1}⊂Zpr (calledthep-adicexpan-
sionoftheelement)[1].NotethatallelementsofA \{0}areunits.Thisproperty
p
providesakindoflinearindependenceontheelementsofA .In[15],theauthors
p
considered this propertyto define a special type of linear combinationof vectors,
called p-linearcombination,whichallowedtodefinethenotionof p-generatorse-
quence, p-basisand p-dimensionforeverysubmoduleofZn . Thesenotionswere
pr
extendedforvectorsin[9]andwerecalltheminthissection.
k
Definition1. [9] Let v (D),...,v (D) be in Zn [D]. The vector (cid:229) a (D)v (D),
1 k pr j j
j=1
with a (D) ∈ A [D], is said to be a p-linear combination of v (D),...,v (D)
j p 1 k
and the setof all p-linearcombinationof v (D),...,v (D) is called the p-span of
1 k
{v (D),...,v (D)},denotedby p-span(v (D),...,v (D)).
1 k 1 k
Note that the p-span of a set of vectors is not always a module. We need to
introduceanextraconditiontobefulfilledbythevectors.
Definition2. [9]Anorderedsetofvectors(v (D),...,v (D))inZn [D]issaidtobe
1 k pr
a p-generatorsequenceif pv(D)isa p-linearcombinationofv (D),...,v (D),
i i+1 k
i=1,...,k−1,and pv (D)=0.
k
Lemma1.[9] If (v (D),...,v (D)) is a p-generator sequence in Zn [D] it holds
1 k pr
thatp-span(v (D),...,v (D))=span(v (D),...,v (D)),andconsequentlywehave
1 k 1 k
that p-span(v1(D),...,vk(D))isaZpr-submoduleofZnpr[D].
NotethatifM=span(v1(D),...,vk(D))isasubmoduleofZpr[D],then
(v (D),pv (D)...,pr−1v (D),v (D),pv (D),...,
1 1 1 2 2
(1.1)
...,pr−1v (D),...,v (D),pv (D)...,pr−1v (D)).
2 l k k
isa p-generatorsequenceofM..
Definition3. [9] The vectors v (D),...,v (D) in Zn [D] are said to be p-linearly
1 k pr
independentiftheonly p-linearcombinationofv (D),...,v (D)thatisequalto0
1 k
isthetrivialone.
Definition4. [9]Anorderedsetofvectors(v (D),...,v (D))whichisa p-linearly
1 k
independent p-generatorsequence of a submodule M of Zn [D] is said to be a p-
pr
basisofM.
It is proved in [8] that two p-bases of a Zpr-submodule M of Znpr[D] have the
same number of elements. This number of elements is called p-dimension of M
andisdenotedby p-dim(M).
ThesamenotionsandresultsaresatisfiedforthemoduleZn in[15].Infact,as
pr
mentionedbefore,thesenotionswerefirstintroducedinthispaperforsuchmodules
andlaterextendedforthemoduleZn [D]in[9].
pr
4 MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
1.3 ConvolutionalCodes overZ
pr
Definition5. A convolutional code C of length n is a Zpr((D))-submodule of
Zn ((D))forwhichthereexistsapolynomialmatrixG(D)∈Zk˜×n[D]suchthat
pr pr
C = ImZpr((D))G(D)
= u(D)G(D)∈Zn ((D)): u(D)∈Zk((D)) .
pr p
n e o
The matrix G(D) is called a generator matrix of C. If G(D) is full row rank
thenitiscalledanencoderofC.
Moreover,if
C =ImApr((D))G(D)
= u(D)G(D)∈Zn ((D)): u(D)∈Ak((D)) .
pr p
n o
with G(D)∈Zk×n[D] a polynomialmatrix whose rowsform a p-basis of C, then
pr
G(D)isa p-encoderofC andwesaythatC has p-dimensionk.
IfthereexistsaconstantmatrixGsuchthat
C = u(D)G∈Zn ((D)): u(D)∈Zk((D)) ,
pr p
n e o
thenC iscalledablockcode.
Obviously,blockcodesareaparticularcaseofconvolutionalcodes.Everyblock
codeC admitsageneratormatrixinstandardform[14]
I A0 A0 A0 ··· A0 A0
k0 1,0 2,0 3,0 r−1,0 r,0
 0 pI pA1 pA1 ··· pA1 pA1 
k1 2,1 3,1 r−1,1 r,1
0 0 p2I p2A2 ··· p2A2 p2A2
 k2 3,2 r−1,2 r,2 
G= 0 0 0 pr−1I ··· 0 pr−1A3 . (1.2)
 k3 r,3 
 ... ... ... ... ... ... ... 
 0 0 0 0 ··· pr−1Ikr−1 pr−1Arr,−r−11
The integersk ,k ,...,k arecalled the parametersof G. Allencodersof C in
0 1 r−1
standardformhavethesameparametersk ,k ,...,k .
0 1 r−1
NotethatifG(D)isageneratormatrixofaconvolutionalcodeC andX(D)isanin-
vertiblerationalmatrixsuchthatX(D)G(D)ispolynomial,thenImZpr((D))G(D)=
ImZpr((D))X(D)G(D),whichmeansthatX(D)G(D)isalsoageneratormatrixofC.
Thus, the next straightforwardresult follows. We include its prooffor the sake of
completeness.
1 ThedualofconvolutionalcodesoverZpr 5
Lemma2.LetC beasubmoduleofZnpr((D))givenbyC =ImZpr((D))N(D),where
N(D)∈Zk˜×n(D).ThenC isaconvolutionalcode,andifN(D)isfullrowrank,C
pr
isafreecodeofdimensionk.
Proof. Write N(D)=hqpiijj((DD))i, where pij(D),qij(D)∈Zpr[D], and the coefficient
of the smallest power of D in q (D) is a unit. Consider the diagonal matrix
ij
Y(D) ∈ Zk˜×k˜[D] whose element of the row i is the least common multiple of
pr
qi1(D),qi2(D),...,qik˜(D). Thus Y(D) is invertible and N(D)=Y(D)−1X(D) for
somepolynomialmatrixX(D)∈Zk˜p×rn[D].ThenImZpr((D))N(D)=ImZpr((D))X(D),
whichmeansthatX(D)isageneratormatrixofC.Thelaststatementofthelemma
followsfromthefactthatN(D)isfullrowrankifandonlyifX(D)isfullrowrank.
⊓⊔
Nextwe will considera decompositionof a convolutionalcode into simplercom-
ponents.Forthatweneedthefollowinglemma.
Lemma3.Let M be a submoduleof Zn ((D)). Then, there exists a uniquefamily
pr
M ,...,M offreesubmodulesofZn ((D))suchthat
0 r−1 pr
M=M ⊕pM ⊕...⊕pr−1M . (1.3)
0 1 r−1
Proof. Let M be the projection of M over Z ((D)) and denote its dimension by
p
k0(M). Let M0 be the free code over Zpr((D)) of rank k0 satisfying M=M0 and
M ⊂M.AsZn ((D))isasemisimplemodule,M admitsacomplementcodeM′ in
0 pr 0 0
M.Necessarily,thereexistsacodeM′ suchthatM′ =pM′.WehaveM=M ⊕pM′.
1 0 1 0 1
Thenbyinductionwehavetheresult. ⊓⊔
NotethatifC isablockcode,thisdecompositionasC =C ⊕pC ⊕...⊕pr−1C
0 1 r−1
whereC ,...,C arefreeblockcodes,isdirectlyderivedfromageneratormatrix
0 r−1
in standard form. In fact, if G, of the form (1.2), is a generator matrix of C then
piCi=ImZpr((D))piGi,whereGi=[0···0Iki Ai2,i···Air,i],i=0,...,r−1.
NextwewillshowthatanyconvolutionalcodeC alsoadmitssuchdecomposition.
LetG(D) be a generatormatrix of C. If G(D) is full rowrankthen C is free and
C =C .
0
Let us assume now that G(D) is not full row rank. Then the projection of G(D)
intoZ [D],G(D)∈Zk×n[D],isalsonotfullrowrankandthereexistsanonsingu-
p p
G (D)
lar matrix F (D)∈Zk×k[D] such that F (D)G(D)= 0 modulo p, where
0 p 0 (cid:20) 0 (cid:21)
e
G (D) is full row rank with rank k . Considering F (D) ∈ Zk×k[D], it follows
0 0 0 pr
G (D)
tehatF (D)G(D)= 0 ,whereG (D)∈Zk0×n issuchthatG (D)=G (D).
0 (cid:20)pG1(D)(cid:21) 0 pr 0 0
G (D) e
Moreover,sinceF (D)bisinvertible, 0 isalsoageneratormatrixofC.
0 (cid:20)pG1(D)(cid:21)
b
6 MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
LetusnowconsiderF (D)∈Z(k−k0)×(k−k0)[D]suchthatF (D)G (D)= G1(D)
1 p 1 1 (cid:20) 0 (cid:21)
e
modulo p, whereG (D) is fullrow rankwith rankk . Then,cobnsideringF (D)∈
1 1 1
Z(k−k0)×(k−k0)[D],ietfollowsthatF (D)G (D)= G′1(D) ,whereG′(D)∈Zk˜1×n
pr 1 1 (cid:20)pG2(D)(cid:21) 1 pr
issuchthatG′(D)=G(D),andtherefobre
1 b
e
G (D)
0
I 0
(cid:20) 0k0 F1(D)(cid:21)F0(D)G(D)=pp2GG′1((DD)).
2
 
b
G (D)
If 0 is not full row rank, then there exists a permutation matrix P and a
(cid:20)G′(D)(cid:21)
1
rationalmatrixL(D)∈Zk˜1×k0(D)suchthat
pr
G (D)
I 0 G (D) 0
P k0 0 = pG′′(D) ,
(cid:20)L1(D)Ik1(cid:21)(cid:20)pG′1(D)(cid:21) p2G1′(D)
 2 
where G′′(D) ∈ Zk1×n(D) and G′(D) ∈ Z(k˜1−k1)×n(D) are rational matrices and
1 pr 2 pr
G (D) I 0
0 isafullrowrankrationalmatrix.NotethatsinceP k0 isnon-
(cid:20)G′1′(D)(cid:21) (cid:20)L1(D)Ik1(cid:21)
singularitfollowsthat
G (D)
G (D) 0
ImZpr((D))(cid:20)pG0′(D)(cid:21)=ImZpr((D)) pG′1′(D) .
1 p2G′(D)
 2 
LetG (D)Zk1×n[D]andG′′(D)∈Z(k˜1−k1)×n[D]bepolynomialmatrices(seeLemma
1 pr 2 pr
2)suchthat
G (D) G (D)
0 0
ImZpr((D)) pG′1′(D) =ImZpr((D)) pG1(D) .
p2G′(D) p2G′′(D)
 2   2 
Then
G (D)
0
 pG1(D) 
p2G′′(D)
 2 
p2G (D)
2
 
b G (D)
isstillageneratormatrixofC suchthat 0 isfullrowrank.
(cid:20)G1(D)(cid:21)
ProceedinginthesamewayweobtainageneratormatrixofC oftheform
1 ThedualofconvolutionalcodesoverZpr 7
G (D)
0
 pG1(D) 
. ,
.
 . 
 
pr−1Gr−1(D)
 
andsuchthat
G (D)
0
 G1(D) 
.
.
 . 
 
Gr−1(D)
 
isfullrowrank.ThusC :=ImG(D)isafreeconvolutionalcode,i=0,1,...,r−1,
i i
and C =C ⊕pC ⊕···⊕pr−1C . If we denote by k the rank of C then the
0 1 r−1 i i
family {k ,...,k } is a characteristic of the code. Moreover,it’s clear that C is
0 r−1
freeifandonlyifk =0fori=1...r−1.
i
Thefollowinglemmaswillbeveryusefulforderivingtheresultsoftheremaining
sections.
Lemma4.LetC beafreeconvolutionalcodeoflengthnoverZpr((D))andrank
k.Then, p-dim(piC)=(r−i)k.
Proof. Let G(D)∈Zk×n[D] be an encoder of C. The result follows from the fact
pr
piG(D)
pi+1G(D)
that . isan p-encoderofC,sinceG(D)isfullrowrank. ⊓⊔
.
 . 
 
pr−1G(D)
 
Lemma5.LetC1 andC2betwoconvolutionalcodesoverZpr((D)).Thenwehave
p-dim(C +C )=p-dimC +p-dimC −p-dim(C ∩C ).
1 2 1 2 1 2
Ifthesumisdirect,wehave
p-dim(C ⊕C )=p-dimC +p-dimC .
1 2 1 2
Proof. SupposethatC andC areindirectsum,i.e,C +C =C ⊕C .
1 2 1 2 1 2
IfB isap-basisofC andB isap-basisofC ,then(B ,B )isap-basisofC ⊕C
1 1 2 2 1 2 1 2
whichgivestheresult.
Forthegeneralcase,LetdenotebyAthecomplementofC ∩C inC ,i.e.,C =
1 2 1 1
A⊕C ∩C ,andletBsuchthatC =B⊕C ∩C .Thenwehave
1 2 2 1 2
C +C =A⊕C ∩C ⊕B
1 2 1 2
andtheresultisimmediate. ⊓⊔
NextcorollaryfollowsimmediatelyfromLemmas4and5.
8 MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
Corollary1.LetC beaconvolutionalcodeoflengthnsuchthat
C =C ⊕pC ⊕···⊕pr−1C
0 1 r−1
withC afreeconvolutionalcodewithrankk,i=0,1,...,r−1.Then
i i
r−1
p-dim(C)= (cid:229) (r−i)k.
i
i=0
1.4 Dual Codes
Let C be a convolutionalcode of length n over Zpr((D)). The orthogonal of C,
denotedbyC⊥,isdefinedas
C⊥={y∈Zn :[y,x]=0forallx∈C},
pr
where[y,x]denotestheinnerproductoverZn .
pr
Inthissectionwewillshowthatthedualofaconvolutionalcodeisstillaconvolu-
tionalcode.Thenexttheoremprovesthisstatementforfreeconvolutionalcodes.
Theorem1.Let C be a free convolutionalcode with length n over Zpr((D)) and
rankk˜.ThenC⊥isalsoafreeconvolutionalcodeoflengthnandrankn−k˜.
Proof. Let G(D)∈Zk˜×n be an encoder of C. Since G(D) is full row rank there
pr
exists a polynomialmatrix L(D)∈Z(n−k˜)×n[D] such that G(D) is nonsingular.
pr (cid:20)L(D)(cid:21)
Let [X(D)Y(D)], with X(D)∈Zn×k˜(D) andY(D)∈Zn×(n−k˜)(D), be the inverse
pr pr
G(D)
of(cid:20)L(D)(cid:21).ThenC⊥=ImZpr((D))Y(D)t, whichmeansbyLemma2thatC⊥ isa
convolutionalcode.Moreover,sinceY(D)isfullcolumnrank,thereexistsafullrow
rank polynomial matrix G⊥(D)∈Z(pnr−k˜)×n[D] such that C⊥ =ImZpr((D))G⊥(D).
ThusC⊥ isafreeconvolutionalcodeorrankn−k˜. ⊓⊔
IfC isafreecodeofrankk˜,then p-dim(C)=k˜r.Thisgivesusthenextcorollary.
Corollary2.LetC beafreeconvolutionalcodeoflengthnoverZpr.Thenwehave
p-dim(C)+p-dim(C⊥)=nr.
In the sequel of this work we propose to established this result for any code over
Zpr((D)).
1 ThedualofconvolutionalcodesoverZpr 9
Thefollowingauxiliarylemmaswillbefundamentalintheproofofnexttheorem.
They were proved in [2] for block codes over Zpr and are trivially extended for
convolutionalcodesZpr((D)).Wewritetheirproofsforcompleteness.
Lemma6.[2] Let C be a free convolutional code over Zpr((D)). For any given
integeri∈{0,...r−1}wehave
C ∩piZn ((D))=piC.
pr
Proof. The inclusion piC ⊂ C ∩piZn ((D)) is trivial. Conversely, let y(D) ∈
pr
piZn ((D))∩C. Let {x (D),...,x (D)} be a basis of C and its projection over
pr 1 k
Z ((D)){x (D),...,x (D)}beabasisofC.Then,thereexistsa (D),...,a (D)∈
p 1 k 1 k
k
Zpr((D))suchthaty(D)= (cid:229) aj(D)xj(D).Asy(D)∈piZnpr((D)),wehavey(D)=
j=1
k
(cid:229)
a (D)x (D)=0, and therefore a (D)=0, j=1...k. Then, for all j=1...k,
j j j
j=1
aj(D)canbewrittenintheformpbj(D)wherebj(D)∈Zpr((D)).Byrepeatingthe
procedureitimes,weobtaina (D)=pia (D), ∀j=1...k,whichgives
j j
k
y(D)=pi(cid:229) a (D)x (D)∈piC.
j j
j=1
⊓⊔
Lemma7.[2]SupposethatC isafreecode.Lety(D)∈Zpr((D))n andletibean
integerin{0,...,r−1},suchthatpiy(D)∈C.Theny(D)∈C +pr−iZn ((D)).
pr
Proof. Bytheprecedinglemma,thereexistsx(D)∈C suchthat piy(D)= pix(D).
This implies that y(D) = x(D). Thus there exists y1(D)∈ C, y2(D) ∈ Zpr((D))
satisfying y(D)=y (D)+py (D). We have piy(D)= piy (D)+pi+1y (D), then
1 2 1 2
piy(D)−piy (D)=pi+1y (D)∈C.Theny (D)=y (D)+py (D)wherey (D)∈
1 2 2 3 4 3
C and y (D)∈Zn ((D)). Then y(D)=y (D)+py (D)+p2y (D). By repeating
4 pr 1 3 4
∈C
this procedure r−i times, we obtain y(D|)=x1({Dz)+pr}−ix2(D) with x1(D)∈C.
⊓⊔
Lemma8.[2] Let C be a free convolutionalcode over Zpr((D)). For all integer
i∈{0,...r−1}wehave
(piC)⊥=C⊥+pr−iZn ((D)).
pr
Proof. It’sclearthatC⊥+pr−iZn ((D))⊂(piC)⊥.Conversely,lety(D)∈(piC)⊥.
pr
Thenforallx(D)∈C wehave[y(D),pix(D)]=[piy(D),x(D)]=0,thus piy(D)∈
C⊥.AsC⊥isafreecode,weconcludebyLemma7thaty(D)∈C⊥+pr−iZn ((D)).
pr
⊓⊔
10 MohammedElOuedandDiegoNappandRaquelPintoandMarisaToste
Theorem2.LetC =C ⊕pC ⊕...⊕pr−1C beaconvolutionalcodeoflength
0 1 r−1
noverZpr((D)),suchthatCiisfree,i=0,1,...,r−1,withC0⊕C1⊕...⊕Cr−1=
C +C +...+C a free convolutionalcode.Then,there exists a family of free
0 1 r−1
convolutionalcodesoflengthnoverZpr((D)),Bi,i=0,...,r−1,suchthatC⊥=
B ⊕pB ⊕...⊕pr−1B ,and
0 1 r−1
1. B =(C ⊕...⊕C )⊥.
0 0 r−1
2. Fori∈{1,...,r−1},rank(B)=rank(C ).
i r−i
Proof. Supposethatrank(C)=k fori=0,...,r−1.Wefirstbeginbylookingfor
i i
thedualofC ⊕pC .
0 1
(C ⊕pC )⊥ = C⊥∩(pC )⊥=C⊥∩(C⊥+pr−1Zn )
0 1 0 1 0 1 pr
= C⊥∩C⊥+pr−1C⊥
0 1 0
= (C ⊕C )⊥+pr−1C⊥.
0 1 0
ByTheorem1,wecanconcludethatthereexistsafreecodeB suchthat
r−1
(C ⊕pC )⊥=(C ⊕C )⊥⊕pr−1B .
0 1 0 1 r−1
Supposerank(B )=l ,thenwehave:
r−1 r−1
p-dim[(C ⊕pC )⊥]= p-dim(C ⊕C )⊥+p-dim(pr−1B )
0 1 0 1 r−1
= nr−(k +k )r+l .
0 1 r−1
Ontheotherhand, p-dim[(C ⊕pC )⊥]=nr−(k r+(r−1)k ).Weconcludethat
0 1 0 1
rank(B )=k .WerepeatthesameprocedurewithC ⊕pC ⊕p2C .
r−1 1 0 1 2
(C ⊕pC ⊕p2C )⊥ = (C ⊕pC )⊥∩(p2C )⊥=[(C ⊕C )⊥⊕pr−1B ]∩(C⊥+pr−2Zn )
0 1 2 0 1 2 0 1 r−1 2 pr
= (C ⊕C ⊕C )⊥⊕pr−1(B ∩C⊥)+pr−2(C ⊕C )⊥+pr−1B
0 1 2 r−1 2 0 1 r−1
= (C ⊕C ⊕C )⊥⊕pr−1B +pr−2(C ⊕C )⊥.
0 1 2 r−1 0 1
ByTheorem1,thereexistsafreeconvolutionalcodeB suchthat
r−2
(C ⊕pC ⊕p2C )⊥=(C ⊕C ⊕C )⊥⊕pr−1B ⊕pr−2B .
0 1 2 0 1 2 r−1 r−2
Supposethatrank(B )=l ,thenwehave
r−2 r−2
p-dim(C ⊕pC ⊕p2C )⊥ = p-dim[(C ⊕C ⊕C )⊥]+p-dim(pr−1B )+p-dim(pr−2B )
0 1 2 0 1 2 r−1 r−2
= nr−(k +k +k )r+k +2l .
0 1 2 1 r−2
Ontheotherhand
p-dim(C ⊕pC ⊕p2C )⊥ = nr−[k r+k (r−1)+k (r−2)]
0 1 2 0 1 2
= (n−k −k )r+k +2k .
0 1 1 2