Table Of ContentNoname manuscript No.
(will be inserted by the editor)
On MDS convolutional Codes over
Z
pr
Diego Napp · Raquel Pinto · Marisa
Toste
Received: date/Accepted: date
6
1
0 Resumo Maximum Distance Separable (MDS) convolutional codes are cha-
2
racterized through the property that the free distance meets the generalized
n Singletonbound.The existence offree MDS convolutionalcodesoverZ was
pr
a
recently discovered in [26] via the Hensel lift of a cyclic code. In this paper
J
8 we further investigate this important class of convolutional codes over Zpr
from a new perspective. We introduce the notions of p-standard form and r-
1
optimal parameters to derive a novel upper bound of Singleton type on the
] freedistance.Moreover,wepresentaconstructivemethodforbuildinggeneral
T
(non necessarily free) MDS convolutional codes over Zpr for any given set of
I
. parameters.
s
c
Keywords Convolutionalcodesoverfinite rings·freedistance·MDScodes·
[
Singleton bound · p-basis
1
v
7
1 Introduction
0
5
4 The extension of the concept of convolutionalcodes from finite fields to finite
0 rings was first developed in [18] and have attracted much attention in recent
.
1 years. This interest is mainly due to the discover that the most appropriate
0 codesforphasemodulationarethelinearcodesovertheresidueclassringZ ,
M
6 M a positive integer. It was immediately apparent that convolutional codes
1
overringsbehaveverydifferentfromconvolutionalcodesoverfinite fields.For
:
v instance,incontrastwiththefieldcase,(linear)convolutionalcodesoverfinite
Xi rings R are not necessarily free modules over R.
ar RaquelPinto·DiegoNapp·
DepartmentofMathematics, UniversityofAveiro,Portugal
E-mail:[email protected]
MarisaToste
SuperiorSchoolofTechnologiesandManagementofOlveiradoHospital,PolytechnicInsti-
tuteofCoimbra,Coimbra,Portugal
2 DiegoNappetal.
Fundamental results of the structural properties of convolutional codes
over finite rings can be found in, for instance, [7,11,23,24]. In particular, the
propertiesofnoncatastrophic,rightinvertible,basicandsystematicringconvo-
lutionalencoderswerethoroughlydiscussed.Theproblemofderivingminimal
encoders (left prime and row-reduced) was posed in [6,30]. This problem was
solvedin[17,16]usingtheconceptofminimalp-encoder,whichisanextension
of the concept of p-basis introduced in [31] to the polynomial context.
In[2,14]thesearchforanddesignofunit-memoryconvolutionalcodesover
Z that gives rise to binary trellis codes with high free distances was investi-
4
gated and several concrete constructions were reported. In [12] two 16-state
trellis codes of rate 2, again over Z , were found by computer search. Howe-
4 4
ver, in contrast to the block code case [10,25] little is known about distance
properties and constructions of convolutional codes over large rings, see for
instance [30].
Recently, in [26] a bound on the free distance of convolutional codes over
Z was derived, generalizing the bound given in [28] for convolutional codes
pr
overfinite fields.CodesachievingsuchaboundwerecalledMaximalDistance
Separable(MDS).TheconcreteconstructionsofMDSconvolutionalcodesover
Z presented in [26] were restricted to free codes and cannot be extended to
pr
the general case. An explicit general construction of nonfree MDS codes over
finite rings was left as an open problem.
In this paper we adopt a simple but novel approach to further investigate
this important class of convolutional codes over Z . In particular, we derive
pr
newupper-boundsonthefreedistanceandprovideexplicitnovelconstructions
ofnonfreeMDSconvolutionalcodesoverZ foreverysetofgivenparameters.
pr
In the proof of these results, an essential role is played by the theory of p-
basis and in particularof a canonicalform of the p-encoders.In contrastwith
the papers [25,26] where the Hensel lift of a cyclic code was used, in this
paper a direct lifting is employed to build convolutional codes over Zpr from
knownconstructionsofconvolutionalcodesoverZ .NotethatbytheChinese
p
Remainder Theorem,results on codes overZ can be extended to codes over
pr
Z , see also [5,11,19].
M
The paper is organized as follows: In the next section we introduce some
preliminaries on p-basis of Z [D]-submodules of Zn [D]. After presenting
pr pr
block codes over Z we introduce the new notions of p-standard form and
pr
r-optimalparameters.UsingthesenotionsanovelSingleton-typeupper-bound
is derived. In section 3 we consider convolutionalcodes and provide the basic
definitions. An upper bound for their free distance is presented. Finally, we
propose a method to build MDS convolutional codes over Z for any given
pr
set of parameters in section 4.
OnMDSconvolutional CodesoverZpr 3
2 Preliminaries
2.1 P-basis and p-dimension
AnyelementinZn canbewrittenuniquelyasalinearcombinationof1,p,p2,...
pr
...,pr−1, with coefficients in A = {0,1,...,p−1} ⊂ Z (called the p-adic
p pr
expansionoftheelement)[3].NotethatallelementsofA \{0}areunits.This
p
set will play an important role throughout the paper since it will allow us to
introduce the notion of p-basisof Z -submodule of Zn , whichwill be crucial
pr pr
in the analysis and construction of optimal convolutional codes over Z .
pr
k
Letv (D),...,v (D)beinZn [D].Thevector a (D)v (D),witha (D)∈
1 k pr j j j
j=1
X
A [D], is said to be a p-linear combination of v (D),...,v (D) and the
p 1 k
set of all p-linear combination of v (D),...,v (D) is called the p-span of
1 k
{v (D),...,v (D)}, denoted by p-span (v (D),...,v (D)). An ordered set of
1 k 1 k
vectors (v (D),...,v (D)) in Zn [D] is said to be a p-generator sequence
1 k pr
if pv (D) is a p-linear combination of v (D),...,v (D), i = 1,...,k−1,
i i+1 k
and pv (D)=0.
k
If (v (D),...,v (D)) is a p-generator sequence it holds (see for instance
1 k
[17])thatp-span(v (D),...,v (D))=span(v (D),...,v (D)),andconsequen-
1 k 1 k
tly the p-span(v (D),...,v (D)) is a Z -submodule of Zn [D]. Note that if
1 k pr pr
M =span(v (D),...,v (D)),
1 k
(v (D),pv (D)...,pr−1v (D),v (D),pv (D),...,
1 1 1 2 2
(1)
...,pr−1v (D),...,v (D),pv (D)...,pr−1v (D)).
2 l k k
is a p-generator sequence of M.
The vectors v (D),...,v (D) in Zn [D] are said to be p-linearly inde-
1 k pr
pendentif the only p-linearcombinationofv (D),...,v (D) thatis equalto
1 k
0 is the trivial one.
An ordered set of vectors (v (D),...,v (D)) which is a p-generator se-
1 k
quence of M and p-linearly independent is said to be a p-basis of M. It is
provedin[16]thattwop-basesofaZpr-submoduleM ofZnpr[D]havethesame
number of elements. This number of elements is called p-dimension of M.
ν
A nonzero polynomial vector v(D) in Zn [D], written as v(D)= v Dt,
pr t
t=0
with vt ∈Znpr, and vν 6=0, is said to have degree ν, denoted by degv(PD)=ν,
and v is called the leading coefficient vector of v(D), denoted by vlc. For
ν
a given matrix G(D) ∈ Zk×n[D] we denote by Glc ∈ Zk×n the matrix whose
pr pr
rows are constituted by the leading coefficient of the rows of GD). A p-basis
(v (D),...,v (D)) is called a reduced p-basis if the vectors vlc,...,vlc are
1 k 1 k
p-linearly independent in Zn .
pr
By [17] every submodule M of Zn [D] has a reduced p-basis. Note that
pr
M does not always admit a (reduced) basis. Moreover, any reduced p-basis
4 DiegoNappetal.
(v (D),...,v (D)) of M exhibits the p-predictable degree property [17]:
1 k
k
deg a (D)v (D) = max (dega (D)+degv (D))
i i j j
i=1 ! j:aj(D)∈Ap[D]\{0}
X
The degrees of the vectors of two reduced p-bases of M are the same (up
to permutation) and their sum is called the p-degree of M.
2.2 Block Codes
A(linear) block code C oflengthn overZ is aZ -submoduleofZn and
pr pr pr
theelementsofC arecalledcodewords.AgeneratormatrixG∈Zk×n ofC isa
per
polynomialmatrixwhoserowsformaminimalsetofgeneratorsofC overZ .
pr
IfGhasfullrowrank,thenitiscalledanencoderofC andCeis afreemodule.
If C has p-dimension k, a p-encoder G ∈ Zk×n of C is a matrix whose rows
pr
forem a p-basis of C and therefore
C =Im G={v =uG∈Zn :u∈Ak}.
Ap pr p
Next we introduce the notion of p-standard form that will play an im-
portant role in the sequel. Given a p-basis (v ,...,v ) of C there are certain
1 k
elementary operations that can be applied to (v ,...,v ) so that we obtain
1 k
another p-basis of C. These are described in the following lemma which is not
difficult to prove.
Lemma 21 Let (v ,...,v ) be a p-basis of a submodule M of Zn . Then,
1 k pr
1. If v′ =v + k a v , with a ∈Z , then (v ,...,v ,v′,v ,...,v )
i i j=i+1 j j j pr 1 i−1 i i+1 k
is a p-basis of M;
P
2. If pv is a p-linear combination of v ,v ,...,v , for some j > i, then
i j j+1 k
(v ,...,v ,v ,...,v ,v ,v ,...,v ) is a p-basis of M.
1 i−1 i+1 j−1 i j k
Given a generator matrix of C in standard form as in [3,25], it is easy to
see that we can extend it as in (1) and apply the elementary row operations
(as defined in Lemma 21 and deleting the zero rows)to obtain a p-encoder G
OnMDSconvolutional CodesoverZpr 5
in the following form:
Ik0 A01,0 A02,0 A03,0 ··· A0r−1,0 A0r,0
−−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−−
pIk0 0 pA02,1 pA03,1 ··· pA0r−1,1 pA0r,1
0 pIk1 pA12,1 pA13,1 ··· pA1r−1,1 pA1r,1
−−p−2Ik−0−− −−−0−−− −−−0−−− −−p2−A−03,2−− −−−··−· −− −p−2A−0r−−1−,2− −−p2−A−0r,2−−
−−−00−−− −−p−20Ik−1−− −−p−20Ik−2−− −−pp22−AA−2313,,22−− −−−····−·· −− −pp−22AA−2r1r−−−11−,,22− −−pp22−AA−2r1r,,22−−
... ... ... ... ··· ... ...
−−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−−
pr−1Ik0 0 0 0 ··· 0 pr−1A0r,r−1
0 pr−1Ik1 0 0 ··· 0 pr−1A1r,r−1
00 00 pr−01Ik2 pr−01Ik3 ······ 00 pprr−−11AA2r3r,,rr−−11
... ... ... ... ... ... ...
0 0 0 0 ··· pr−1Ikr−1 pr−1Arr,−r1−1
where I denotes the identity matrix of size ℓ. One can verify that the scalars
ℓ
k ,i=0,1,...,r−1,areequalforallp-encodersofC inthisform,i.e.,theyare
i
uniquelydeterminedforagivencodeC ⊂Zn .Wecallk ,k ,...,k thepa-
pr 0 1 r−1
rametersofC.Clearly,ifChasp-dimensionequaltokthenk = r−1k (r−i).
i=0 i
IfGisinsuchaformwesaythatGisinthep-standardform.Thep-standard
P
form will be a useful tool to prove our results in the same way the standard
form was for previous results in the literature, see for instance [3,25].
The free distance d(C) of a linear block code C is given by
d(C)=min{wt(v),v ∈C,v 6=0}
where wt(v) is the Hamming weight of v over Z .
pr
Considering the last row of any p-encoder in the p-standard form as a co-
deword,thenextresultonthegeneralizedSingletonboundonthefreedistance
of codes over Z readily follows.
pr
Theorem 21 [25]GivenalinearblockcodeC ⊂Zn withparametersk ,...,k ,
pr 0 r−1
it must hold that
d(C)≤n−(k +···+k )+1.
0 r−1
Among block codes of length n and p-dimension k, we are interested in
the ones with largest possible distance. Hence, given an integer r ≥ 1 and a
non-negative integer k we call the ordered set (k ,k ,··· ,k ), k ∈ N, i =
0 1 r−1 i
0,··· ,r−1 an r-optimal set of parameters of k if
k +k +···+k = min (k′ +k′ +···+k′ ).
0 1 r−1 k=rk0′+(r−1)k1′+···+kr′−1 0 1 r−1
Note that when C is free, k must divide r and k = k, i.e., (k ,0,...,,0)
0 r 0
is the unique r-optimal set of parameters of k. However, the r-optimal set of
6 DiegoNappetal.
parameters of k is not necessarily unique for given k,r. For instance if k=25
and r = 6, (4,0,0,0,0,1)and (0,5,0,0,0,0)are two possible 6-optimal set of
parameters of 25. The computation of the r-optimal set of parameters is the
well-known change making problem [4].
Lemma 22 Let (k ,k ,··· ,k ) be an r-optimal set of parameters of k.
0 1 r−1
Then, k +k +···+k = k .
0 1 r−1 r
(cid:6) (cid:7)
Proof Write k =rb+a, where b,a∈N anda<r. Note that a canbe written
as a = r−i, for some 1 ≤ i ≤ r. If r|k then a = 0 and necessarily k = k
0 r
and k = 0, for 1 ≤ j ≤ r−1. If r ∤ k, we can select k = b, k = 1 and
j 0 r−a
k =0,forj ∈{1,...r−1}\{r−a}.Hencek +k +···+k =b+1= k .
j 0 1 r−1 r
It is easy to verify that these values minimize k +k +···+k subject to
0 1 r−1
k =rk +(r−1)k +···+k . (cid:6) (cid:3)(cid:7)
0 1 r−1
Using the previous lemma the Singleton bound of codes over Z in terms
pr
of the p-dimension reads as follows.
Corollary 21 Given a block code C ⊂Zn and p-dimension k,
pr
k
d(C)≤n− +1.
r
(cid:24) (cid:25)
Using completely different approach this result was also derived in [26,
Theorem 3.1] without using the notions of p-standard form nor the r-optimal
set of parameters. Note, however, that our approach and in particular these
two notions will turn out to be crucial to derive our results in the following
two sections.
3 Convolutional Codes
A convolutional code C of length n is a Z [D]-submodule of Zn [D]. A
pr pr
generator matrix G(D) ∈ Zk×n[D] of C is a polynomial matrix whose rows
per
form a minimal set of generators of C over Zpr[D]. If G(D) has full row rank,
then it is called aneencoder of C and C is a free code.
IfC hasp-dimensionk,ap-encoderG(D)∈Zk×ne[D]ofC isapolynomial
pr
matrix whose rows form a p-basis of C and therefore
C = Im G(D)= u(D)G(D)∈Zn [D]: u(D)∈Ak[D] .
Ap[D] pr p
(cid:8) (cid:9)
If the rows of G(D) (G(D)) form a reduced p-basis (basis) then we say
the G(D) (G(D)) is in reduced form. The p-degree of C, denoted by δ, is the
sum of the row degrees ofeany p-encoder in reduced form. In the sequel, we
willadoptthee notationusedbyMcEliece[20,p. 1082]anddenote by(n,k,δ)-
convolutional code a code C ⊂Zn [D] with p-dimension k and p-degree δ.
pr
OnMDSconvolutional CodesoverZpr 7
Remark 1 We emphasize that in this paper we do not assume that C is free.
Note that convolutional codes C ⊂ Zn [D] always admit a p-encoder however
pr
they may not admit a full row rank generator matrix, i.e., an encoder. The
differenceisthattheinputvectortakesvaluesinA [D]forp-encoderswhereas
p
for generator matrices takes values in Z [D]. This idea of using a p-adic
pr
expansion for the information input vector is already present in, for instance,
[3] and was further developed in [31] introducing the notion of p-generator
sequence of vectors in Z . In [16,17] this notion was extended to polynomial
pr
vectors.
Remark 2 Conform [1,15,21,27,30] we have decided to define our codes as
finite support convolutional codes. There exists however a considerable body
of literature in which code sequences are semi-infinite Laurentseries [7,11,18,
23,24]. We note that for the issues treated in this paper there is no difference
and all our results apply to both approaches.
The weight of v(D) is given by wt(v(D)) = wt(v ) and the free
i≥0 i
distance of a convolutional code C is defined as
P
d(C)=min{wt(v(D)): v(D)∈C, v(D)6=0}.
The j-th row distance dr of a p-encoder in reduced form G(D) [13] is
j
defined as the minimum of the weights of all finite codewords resulting from
an information sequence u(D)∈Ak[D] with deg(u(D))≤j, i.e.,
p
dr = min wt(u(D)G(D)).
j
deg(u(D))≤j
Clearly, if C = Im G(D),
Ap[D]
d(C)≤···≤dr ≤···≤dr ≤dr. (2)
j 1 0
Let C be a (n,k,δ)-convolutionalcode defined over Z . Let G(D)=G +
pr 0
G D+···+G Dν1 be a p-encoderin reducedformwith orderedrowdegrees
1 ν1
ν ≥ ν ··· ≥ ν , and let ν = min{ν ,ν ,...,ν } denote the value of the
1 2 k 1 2 k
smallest row degree and ℓ the number of rows with row degree equal to ν.
After applying row permutation and elementary row operations we can bring
the last ℓ rows of the matrix G into the p-standard form with parameters
ν
ℓ ,ℓ ,...,ℓ . This transformation has no effect on the row space of G(D)
0 1 r−1
and it also does not affect the row degrees ν . We have the following upper
i
bound on the free distance of the code.
Theorem 1 Let G(D) = G + G D + ··· + G Dν1 be a p-encoder of an
0 1 ν1
(n,k,δ)-convolutional code C in reduced form and row degrees ν ≥ ν ··· >
1 2
ν = ··· = ν and define ν = ν . Assume that the last ℓ rows of G are
k−ℓ−1 k k ν
in p-standard form with parameters ℓ ,ℓ ,...,ℓ . Then the free distance of
0 1 r−1
C must satisfy
d(C)≤n(ν+1)−(ℓ +ℓ +···+ℓ )+1. (3)
0 1 r−1
8 DiegoNappetal.
Proof We show that the upper bound in (3) is actually an upper bound of
dr and therefore the result readily follows from (2). Write G(D) = G +
0 0
G D +··· +G Dν1 and denote by G′ the last ℓ rows of G . As matrices
1 ν1 i i
G′ ,G′ ,···G′ are zero, G′(D)=G′ +G′D+···+G′Dν are the last ℓ
ν+1 ν+2 ν1 0 1 ν
rows of G(D). Using that G′ is in the p-standard form, i.e., G′ is equal to
ν ν
Iℓ0 A01,0 A02,0 A03,0 ··· A0r−1,0 A0r,0
−−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−−
pIℓ0 0 pA02,1 pA03,1 ··· pA0r−1,1 pA0r,1
0 pIℓ1 pA12,1 pA13,1 ··· pA1r−1,1 pA1r,1
−−p−2I−ℓ0−− −−−0−−− −−−0−−− −−p2−A−03,2−− −−−··−· −− −p−2A−0r−−1−,2− −−p2−A−0r,2−−
−−−00−−− −−p−20I−ℓ1−− −−p−20I−ℓ2−− −−pp22−AA−2313,,22−− −−−····−·· −− −pp−22AA−2r1r−−−11−,,22− −−pp22−AA−2r1r,,22−−
... ... ... ... ··· ... ... ,
−−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−− −−−−−−
pr−1Iℓ0 0 0 0 ··· 0 pr−1A0r,r−1
000 pr−001Iℓ1 pr−001Iℓ2 pr−001Iℓ3 ········· 000 ppprrr−−−111AAA2r1r3r,,,rrr−−−111
... ... ... ... ... ... ...
0 0 0 0 ··· pr−1Iℓr−1 pr−1Arr,−r1−1
it is easy to see that the input vector u = (0,0,··· ,0,1) ∈ Ak[D] gives a
p
codeword v(D) = uG(D) = u′G′(D) with u′ = (0,··· ,0,1) ∈ Aℓ[D]. The
p
polynomial vector v(D) has the last n−(ℓ +ℓ +···+ℓ )+1 coordinates
0 1 r−1
with weight at most ν +1 and the first ℓ +ℓ +···+ℓ −1 coordinates
0 1 r−1
with weight at most ν. Therefore,
dr ≤[n−(ℓ +ℓ +···+ℓ )+1](ν+1)+(ℓ +ℓ +···+ℓ −1)ν
0 0 1 r−1 0 1 r−1
=n(ν+1)−(ℓ +ℓ +···+ℓ )+1,
0 1 r−1
which concludes the proof. (cid:3)
Remark 3 Wenotethatℓ +ℓ +···+ℓ areinvariantsofC.Indeed,ifG(D)
0 1 r−1
isanotherp-encoderofC inreducedformitmustalsohaveℓrowsofdegreeν.
LetG′(D)=G′+G′D+···+G′Dν beconstitutedbytheserowsandG′(D)=
0 1 ν
G′ +G′D+···+G′Dν be as in proof of Theorem 1. Then one can use the
0 1 ν
predictable degree property to show that Im G′(D) = Im G′(D)
Ap[D] Ap[D]
and furthermore Im G′ = Im G′ which shows the claim that the ℓ +
Ap ν Ap ν 0
ℓ +···+ℓ are invariants of C, see [17] for more details.
1 r−1
Takingthemaximumofthebound(3)overall(n,k,δ)-convolutionalcodes
we obtain the main result of [26, Theorem 4.10].
Corollary 31 The free distance of an (n,k,δ) convolutional code C satisfies
δ k δ δ
d(C)≤n +1 − +1 − +1. (4)
k r k r
(cid:18)(cid:22) (cid:23) (cid:19) (cid:24) (cid:18)(cid:22) (cid:23) (cid:19) (cid:25)
OnMDSconvolutional CodesoverZpr 9
Proof Let G(D) be as in Theorem 1. The highest value of (3) is obtained by
consideringthemaximumvalueofν andtheminimumvalueof(ℓ +ℓ +···+
0 1
ℓ ). It is easy to see that the maximum value of ν is when ν = δ and
r−1 k
ν =ν =···=ν = δ +1. From this it follows that
1 2 k−ℓ k (cid:4) (cid:5)
(cid:4) (cid:5) δ δ
δ =(k−ℓ) +1 +ℓ
k k
(cid:18)(cid:22) (cid:23) (cid:19) (cid:22) (cid:23)
and, thus
δ
ℓ=k +1 −δ.
k
(cid:18)(cid:22) (cid:23) (cid:19)
Ontheotherhand,thevaluesof(ℓ ,ℓ ,...,ℓ )thatminimizeℓ +ℓ +···+
0 1 r−1 0 1
ℓ andsuchthat ℓ= r (r−i)ℓ arethe r-optimal setofparametersofℓ.
r−1 i=0 i
By Lemma 22, ℓ +ℓ +···+ℓ = ℓ . Finally,
0 1 P r−1 r
(cid:6) (cid:7)
δ k( δ +1)−δ
d(C) ≤n +1 − k +1
k r
(cid:18)(cid:22) (cid:23) (cid:19) & (cid:4) (cid:5) '
δ k δ δ
≤n +1 − +1 − +1.
k r k r
(cid:18)(cid:22) (cid:23) (cid:19) (cid:24) (cid:18)(cid:22) (cid:23) (cid:19) (cid:25)
(cid:3)
An(n,k,δ)-convolutionalcodeoverZ issaidtobeMaximumDistance
pr
Separable (MDS) if
δ k δ δ
d(C)=n +1 − +1 − +1.
k r k r
(cid:18)(cid:22) (cid:23) (cid:19) (cid:24) (cid:18)(cid:22) (cid:23) (cid:19) (cid:25)
Remark 4 It is important to remark that the Singleton-type upper bound
presented in (4) is derived as a corollary of the Theorem 1 by taking an r-
optimal set parameters of ℓ = k δ +1 −δ and therefore it follows that
k
MDSconvolutionalcodesoverZ musthavetheseoptimalsetofparameters.
pr
(cid:0)(cid:4) (cid:5) (cid:1)
4 General constructions of MDS convolutional codes over Zpr
In this section we present a general procedure for building (non necessarily
free) MDS convolutionalcodes overZpr. The idea is to start from well-known
constructions of MDS convolutional codes over Z and then lift them to Z
p pr
in such a way that the resulting convolutional code is MDS over Z . This
pr
method is direct and works for any given set of parameters (n,k,δ).
For the sake of simplicity of exposition, we first assume that k | δ and
consequently the row degrees of any p-encoder G(D) of C are ν =ν = ··· =
1
ν = δ and thus ℓ = k. The general case will be treated at the end of the
k k
10 DiegoNappetal.
section. Hence, the MDS (n,k,δ)-convolutional C that we aim to construct
must satisfy
δ k δ δ
d(C)=n +1 − +1 − +1.
k r k r
(cid:18)(cid:22) (cid:23) (cid:19) (cid:24) (cid:18)(cid:22) (cid:23) (cid:19) (cid:25)
Note that
δ k δ δ
n +1 − +1 − +1=n(ν+1)−(k +k +···+k )+1
k r k r 0 1 r−1
(cid:18)(cid:22) (cid:23) (cid:19) (cid:24) (cid:18)(cid:22) (cid:23) (cid:19) (cid:25)
where k ,k ,...,k is an r-optimal set of parameters of k.
0 1 r−1
Take (k =k +k +···+k ) and δ =νk, and let us consider any of the
0 1 r−1
well-known construction of MDS convolutional codes C (see [9,22,29]) with
length n, deimension k and degree δ overea fieled Z .
p
e
The distance of Ceequals (see [2e8])
e
δ
d(C)=(n−k) +1 +δ+1.
$k% !
e
e e e
e
Let
G (D)
k0
−−−−
Ge (D)
k1
G(D)= −−−− ∈Z [D]k×n (5)
p
e .. e
.
e −−−−
G (D)
kr−1
e
be an encoder of C in reduced form, where G (D) is a k ×n matrix, i =
ki i
0,1,...,r−1, .
By Lemma 22,ek = k and since δ =νk wee get that
r
(cid:6) (cid:7)
e e e
k
d(C)=n(ν+1)− +1. (6)
r
(cid:24) (cid:25)
e
