Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes

0.13 MB·English

by  Salah A. Aly

Quantum Convolutional Codes Derived from
Reed-Solomon and Reed-Muller Codes
Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli
Department of Computer Science, Texas A&M University
College Station, TX 77843-3112,USA
Email: {salah, klappi, pradeep}@cs.tamu.edu
7 Abstract—Convolutional stabilizer codes promise to make The weight wt(v(D)) of a polynomial v(D) in F [D] is
q
0 quantum communication more reliable with attractive online defined as the number of nonzero coefficients of v(D), and
0 encodingand decodingalgorithms. Thispaper introducesa new the weight of an element u(D) ∈ F [D]n is defined as
2 approach to convolutional stabilizer codes based on direct limit wt(u(D)) = n wt(u (D)). The freqe distance d of C
constructions. Two families of quantum convolutional codes are i=1 i f
n derived from generalized Reed-Solomon codes and from Reed- is defined as dPf =wt(C)=min{wt(u)|u∈C,u6=0}. We
a
Mullercodes.ASingletonboundforpureconvolutionalstabilizer say that an (n,k,δ) convolutional code with memory µ and
J q
codes is given. free distance d is an (n,k,δ;µ,d ) convolutional code.
2 f f q
Let N denote the set of nonnegative integers. Let Γ =
2 I. INTRODUCTION {v: N → Fq| all but finitely many coefficients of v areq0}.
2 A key obstacle to the communication of quantum infor- We define a vector space isomorphism σ: F [D]n →Γ that
q q
v mation is decoherence, the spontaneous interaction of the maps an element u(D) in F [D]n to the coefficient sequence
q
7 environment with the information-carrying quantum system. of the polynomial n−1Diu (Dn), that is, an element in
3 i=0 i
0 The protection of quantum information with quantum error- Fq[D]n is mappedPto its interleaved coefficient sequence.
1 correcting codes to reduce or perhaps nearly eliminate the Frequently,wewillrefertotheimageσ(C)ofaconvolutional
0 impactofdecoherencehasledtoahighlydevelopedtheoryof code (1) again as C, as it will be clear from the context
7 quantumerror-correctingblockcodes.Somewhatsurprisingly, whether we discuss the sequence or polynomial form of the
0 quantum convolutionalcodes have received less attention. code.LetG(D)=G +G D+···+G Dµ,whereG ∈Fk×n
/ 0 1 µ i q
h Ollivier and Tillich developed the stabilizer framework for for0≤i≤µ.WecanassociatetothegeneratormatrixG(D)
p quantum convolutional codes, and addressed encoding and its semi-infinite coefficient matrix
- decoding aspects of such codes [13], [14]. Almedia and
t G G ··· G
n Palazzo constructeda concatenatedconvolutionalcode of rate 0 1 µ
a G= G0 G1 ··· Gµ . (2)
1/4 with memory m = 3 [1]. Forney and Guha constructed
u quantumconvolutionalcodeswithrate1/3[5].Also,inajoint  ... ... ...
q  
: work with Grassl, they derived rate (n−2)/n convolutional If G(D) is the generator matrix of a convolutional code C,
v
stabilizer codes [4]. Grassl and Ro¨tteler constructed quantum
i then one easily checks that σ(C)=ΓqG.
X convolutional codes from product codes [8], and they gave a
In the literature, convolutional codes are often defined in
r general algorithm to obtain non-catastrophic encoders [7]. the form {p(D)G′(D)|p(D) ∈ F (D)k}, where G′(D) is
a q
In this paper,we give a new approachto quantumconvolu- a matrix of full rank in Fk×n[D]. In this case, one can
q
tional codes based on a direct limit construction, generalize
obtain a generator matrix G(D) in our sense by multiplying
some of the previously known results, and construct two G′(D) fromthe left with a suitable invertiblematrixU(D) in
families of quantum convolutional codes based on classical Fk×k(D), see [9].
q
generalized Reed-Solomon and Reed-Muller codes. We define the Euclidean inner product of two sequences
u and v in Γ by hu|vi = u v , and the Euclidean
II. BACKGROUND q i∈N i i
dual of a convolutional codePC ⊆ Γq by C⊥ = {u ∈
In this section, we give some background concerning clas-
Γ |hu|vi = 0 for all v ∈ C}. A convolutional code C is
q
sical convolutionalcodes, following [9, Chapter 14] and [12]. calledself-orthogonalif andonlyifC ⊆C⊥. Itis easyto see
Let Fq denote a finite field with q elements. An (n,k,δ)q that a convolutional code C is self-orthogonal if and only if
convolutionalcodeC isa submoduleofFq[D]n generatedby GGT =0.
a right-invertible matrix G(D)=(gij)∈Fq[D]k×n, Consider the finite field Fq2. The Hermitian inner product
C ={u(D)G(D)|u(D)∈F [D]k}, (1) of two sequences u and v in Γq2 is defined as hu|vih =
q i∈Nuiviq. We have C⊥h = {u ∈ Γq2|hu|vih =
such that ki=1νi = max{degγ|γ is a k-minor of G(D)} 0Pfor all v ∈ C}. As before, C ⊆ C⊥h if and only if
=:δ,wherPeνi =max1≤j≤n{deggij}.We sayδ isthedegree GG† = 0, where the Hermitian transpose † is defined as
ofC.ThememoryµofG(D)isdefinedasµ=max ν . (a )† =(aq ).
1≤i≤k i ij ji
III. QUANTUMCONVOLUTIONAL CODES generated by shifted versions of elements in S0.
Definition 1: Suppose that an abelian subgroup S of P
The state space of a q-ary quantum digit is given by the 0 0
complex vector space Cq. Let {|xi|x∈F } denote a fixed is chosen such that S1, S2, and S3 are satisfied. Then the
orthonormal basis of Cq, called the computqational basis. For +1-eigenspace of S = l−i→m(Si,ιij) in ∞i=0Cq defines a
a,b∈Fq, we define the unitary operators convolutionalstabilizer code with parameNters [(n,k,m)]q.
In practice, one works with a stabilizer S for some large
X(a)|xi=|x+ai and Z(b)|xi=exp(2πitr(bx)/p)|xi, t
(but previously unknown) t, rather than with S itself. We
where the addition is in F , p is the characteristic of F , and noticethattheratek/nofthequantumconvolutionalstabilizer
q q
tr(x) = xp +xp2 +···+xq is the absolute trace from F code defined by S is approached by the rate of the stabilizer
q
to Fp. The set E ={X(a),Z(b)|a,b∈Fq} is a basis of the block code St for large t. Indeed, St defines a stabilizer code
algebra of q×q matrices, called the error basis. with parameters[[(t+1)n+m,(t+1)k+m]]q; therefore,the
Aquantumconvolutionalcodeencodesastreamofquantum rates of these stabilizer block codes approach
digits. One does not know in advance how many qudits i.e., (t+1)k+m k+m/(t+1) k
lim = lim = .
quantumdigits will be sent, so the idea is to impose structure
t→∞(t+1)n+m t→∞n+m/(t+1) n
on the code that simplifies online encodingand decoding.Let
We say that an error E in P is detectable by a convolu-
n, m be positive integers. We will process n+m qudits at a ∞
tional stabilizer code with stabilizer S if and only if a scalar
time, m qudits will overlap from one step to the next, and n
multiple of E is contained in S or if E does not commute
qudits will be output.
For each t in N, we define the Pauli group P =hM|M ∈ with some element in S. The weight wt of an element in P∞
t
is defined as its number of non-identity tensor components.
E⊗(t+1)n+miasthegroupgeneratedbythe(t+1)n+m-fold
A quantum convolutional stabilizer code is said to have free
tensorproductofthe errorbasisE. LetI =X(0)be theq×q
identitymatrix.Fori,j ∈Nandi≤j,wedefinetheinclusion distance df if and only if it can detect all errors of weight
less than d , but cannot detect some error of weight d .
homomorphismι : P →P byι (M)=M⊗I⊗n(j−i).We f f
ij i j ij
Denote by Z(P ) the center of P and by C (S) the
haveι (M)=M andι =ι ◦ι fori≤j ≤k.Therefore, ∞ ∞ P∞
ii ik jk ij
centralizer of S in P . Then the free distance is given by
there exists a group ∞
d =min{wt(e)|e∈C (S)\Z(P )S}.
f P∞ ∞
P∞ =lim(Pi,ιij), Let(β,βq)denoteanormalbasisofFq2/Fq.Defineamap
−→ τ: P∞ → Γq2 by τ(ωcX(a0)Z(b0)⊗X(a1)Z(b1)⊗···) =
calledthedirectlimitofthegroupsPi overthetotallyordered (βa0+βqb0,βa1+βqb1,...). Forsequencesv andw in Γq2,
set (N,≤). For each nonnegative integer i, there exists a we define a trace-alternating form
homomorphism ι : P → P given by ι (M ) = M ⊗I⊗∞
i i ∞ i i i v·wq−vq ·w
fPor M=i ∈∞Piι, (aPnd);ιipu=t dιjiff◦eιreijnthlyo,ldPs focroanlslisits≤ofj.alWl ienfihnaivtee hv|wia =trq/p(cid:18) β2q −β2 (cid:19).
∞ i=0 i i ∞
tensor pSroducts of matrices in hM|M ∈Ei such that all but
Lemma 2: Let A and B be elements of P . Then A and
finitely many tensor components are equal to I. The direct ∞
B commute if and only if hτ(A)|τ(B)i =0.
limitstructurethatweintroducehereprovidesthepropercon- a
Proof: This follows from [11] and the direct limit
ceptualframeworkforthedefinitionofconvolutionalstabilizer
structure.
codes; see [16] for background on direct limits.
We will define the stabilizer of the quantum convolutional Lemma 3: Let Q be an Fq2-linear [(n,k,m)]q quantum
convolutional code with stabilizer S, where S = lim(S ,ι )
codealsothroughadirectlimit.LetS beanabeliansubgroup i ij
0 −→
ofP0.Forpositiveintegerst,werecursivelydefineasubgroup and S0 an abelian subgroup of P0 such that S1, S2, and S3
St of Pt by St =hN⊗I⊗n,I⊗tn⊗M|N ∈St−1,M ∈S0i. hold. Then C =σ−1τ(S) is an Fq2-linear (n,(n−k)/2;µ≤
LetZt denotethecenterofthegroupPt. We willassumethat ⌈m/n⌉)q2 convolutionalcodegeneratedbyσ−1τ(S0).Further,
S1) I⊗tn ⊗M and N ⊗I⊗tn commute for all N,M ∈ S C ⊆C⊥h.
0
and all positive integers t. Proof: Recall that σ : Fq2[D]n → Γq2, maps u(D) in
S2) StZt/Zt is an (t+1)(n−k)-dimensional vector space Fq2[D]n to in=−01Diui(Dn).Itisinvertible,thusσ−1τ(e)=
over Fq. σ−1 ◦ τ(e)Pis well defined for any e in P∞. Since S is
S3) St∩Zt contains only the identity matrix. generated by shifted versions of S0, it follows that C =
Assumption S1 ensures that St is an abelian subgroup of Pt, σ−1τ(S) is generated as the Fq2 span of σ−1τ(S0) and its
S2 implies that St is generated by t+1 shifted versions of shifts, i.e., Dlσ−1τ(S0), where l ∈ N. Since Q is an Fq2-
n−k generatorsof S0 andall these (t+1)(n−k)generators linear [(n,k,m)]q quantum convolutionalcode, S0 defines an
are independent, and S3 ensures that the stabilizer (or +1 [[n+m,k+m]]q stabilizer code with (n−k)/2 Fq2-linear
eigenspace) of St is nontrivial as long as k <n. generators. Since the maps σ and τ are linear σ−1τ(S0) is
The abelian subgroups St of Pt define an abelian group also Fq2-linear. As σ−1τ(e) is in Fq2[D]n we can define an
(n−k)/2×n polynomial generator matrix that generates C.
S =lim(Si,ιij)=hιt(I⊗tn⊗M)|t≥0,M ∈S0i Thisgeneratormatrixneednotberightinvertible,butweknow
−→
thatthereexistsarightinvertiblepolynomialgeneratormatrix quantum code is the degree of the underlying classical code.
thatgeneratesthiscode.ThusC isan(n,(n−k)/2;µ)q2 code. Asσ−1τ(S)=C, arguingasinCorollary4we canshowthat
SinceS isabelian,Lemma2andtheFq2-linearityofS imply σ−1τ(CP∞(S))=C⊥h and df =wt(C⊥h \C).
that C ⊆ C⊥h. Finally, observe that maximum degree of an Corollary 6: Let C be an (n,(n−k)/2,δ;µ)q code such
elementin σ−1τ(S ) is ⌈m/n⌉ owingto σ. Togetherwith [9, that C ⊆ C⊥. Then there exists an [(n,k,nµ;δ,d )] code
0 f q
Lemma14.3.8]thisimpliesthatthememoryofσ−1τ(S)must with d =wt(C⊥\C). It is pure if wt(C⊥\C)=wt(C⊥).
f
be µ≤⌈m/n⌉. Proof: Since C ⊆ C⊥, its generator matrix G as in
We define the degree of an Fq2-linear [(n,k,m)]q quan- equation (2) satisfies GGT =0. We can obtain an Fq2-linear
tum convolutional code Q with stabilizer S as the degree (n,(n−k)/2,δ;µ)q2 code, C′ from G as C′ =Γq2G. Since
of the classical convolutional code σ−1τ(S). We denote an G ∈ F(n−k)/2×n we have GG† = GGT = 0. Thus C′ ⊆
i q
[(n,k,m)]q quantumconvolutionalcodewithfreedistancedf C′⊥h. Further, it can checked that wt(C′⊥h\C′)=wt(C⊥\
and total constraint length δ as [(n,k,m;δ,df)]q. It must be C). The claim follows from Theorem 5.
pointedoutthisnotationisatvariancewiththeclassicalcodes Theorem 7 (Singleton bound): The free distance of an
in not just the order but the meaning of the parameters. [(n,k,m;δ,df)]q Fq2-linearpureconvolutionalstabilizercode
Corollary 4: AnFq2-linear[(n,k,m;δ,df)]q convolutional is bounded by
stabilizer code implies the existence of an (n,(n−k)/2;δ)q2
n−k 2δ
convolutional code C such that df =wt(C⊥h \C). df ≤ +1 +δ+1
Proof: AsbeforeletC =σ−1τ(S),byLemma2wecan 2 (cid:18)(cid:22)n+k(cid:23) (cid:19)
ceorrnocrluisdme athpaptedσ−to1τan(CePle∞m(eSn)t)in⊆CC⊥⊥hh\.CT.hWushialneτunisdeintejcetcatbivlee codePCroosuf:chBtyhaCtowrotl(lCar⊥yh4,\thCe)re=exdisft,saannd(nth,e(np−urkit)y/2o,fδt)hq2e
onS itisnotthecasewithC (S).Howeverwecanseethat code implies that wt(C⊥h)=df. The dual code C⊥ or C⊥h
tihfactitshienreCe⊥xhis\tsCa,ntheernrosrurejeicPnt∞ivCityo(fSτ)(\oZn(CPP∞)(SS)s)uicmhptlhieast ihsasanth(en,s(anm+e dke)g/r2e,eδ)aqs2ccoodnevo[1lu0t,ioTnhaelocroedmew2.i6t6h]f.rTeehudsi,stCan⊥che
P∞ ∞ d .BythegeneralizedSingletonbound[17,Theorem2.4]for
τ(e) = σ(c). As τ and σ are isometric e is an undetectable f
classical convolutional codes, we have
errorwithwt(c).Hence,wecanconcludethatdf =wt(C⊥h\
C). Combining with Lemma 3 we have the claim stated. δ
d ≤ (n−(n+k)/2) +1 +δ+1,
An[(n,k,m;δ,df)]q codeis saidto bea purecodeifthere f (cid:18)(cid:22)(n+k)/2(cid:23) (cid:19)
are no errors of weight less than d in the stabilizer of the
f which implies the claim.
code.Corollary4impliesthatdf =wt(C⊥h\C)=wt(C⊥h).
Theorem 5: Let C be (n,(n − k)/2,δ;µ)q2 convolu- IV. CONVOLUTIONAL RS STABILIZERCODES
tional code such that C ⊆ C⊥h. Then there exists an In this section we will use Piret’s construction of Reed-
[(n,k,nµ;δ,d )] convolutional stabilizer code, where d =
f q f Solomon convolutional codes [15] to derive quantum convo-
wt(C⊥h \C). The code is pure if df =wt(C⊥h). lutional codes. Let α ∈ Fq2 be a primitive nth root of unity,
Proof: [Sketch] Let G(D) be the polynomial generator wheren|q2−1.Letw=(w ,...,w ),γ =(γ ,...,γ )
0 n−1 0 n−1
matrixofC,withthesemi-infinitegeneratormatrixGdefined be in Fn where w 6=0 and all γ 6=0 are distinct. Then the
as in equation (2). Let Ct = hσ(G(D)),...,σ(DtG(D))i = generaliqz2ed Reed-Siolomon (GRS)icode over Fn is the code
hC ,σ(DtG(D))i, where σ is applied to every row in q2
t−1 with the parity check matrix, (cf. [9, pages 175–178])
G(D).Theself-orthogonalityofC impliesthatC isalsoself-
t
orthogonal. In particular C0 defines an [n+nµ,(n−k)/2]q2 w0 w1 ··· wn−1
self-orthogonal code. From the theory of stabilizer codes  w0γ0 w1γ1 ··· wn−1γn−1 
we know that there exists an abelian subgroup S ≤ P w γ2 w γ2 ··· w γ2(n−1)
0 0 Hγ,w = 0 0 1 1 n−1 n−1 .
such that τ(S0) = C0, where Pt is the Pauli group over  ... ... ... ... 
(t+1)n+m qudits; in this case m = nµ. This implies that  
τ(I⊗nt⊗S0)=σ(DtG(D)). Define St =hSt−1,I⊗nt⊗S0i,  w0γ0t−1 w1γ12(t−1) ··· wn−1γn(t−−11)(n−1) 
then τ(St) = hτ(St−1,σ(DtG(D))i. Proceeding recursively, Thecode is denotedby GRSn−t(γ,v), as its generatormatrix
we see that τ(St) = hσ(G(D)),...,σ(DtG(D))i = Ct. isoftheformHγ,v forsomev ∈Fnq2.Itisan[n,n−t,t+1]q2
By Lemma 2, the self-orthogonality of Ct implies that St is MDS code [9, Theorem 5.3.1]. If we choose w = αi, then
i
abelian, thus S1 holds. Note that τ(StZt/Zt)=Ct, where Zt w 6= 0. If gcd(n,2) = 1, then α2 is also a primitive nth
is the center of Pt. Combining this with Fq2-linearity of Ct roiot of unity; thus γi = α2i are all distinct and we have an
implies that StZt/Zt is a (t+1)(n−k) dimensional vector [n,n − t,t + 1]q2 GRS code with parity check matrix H0,
space over Fq; hence S2 holds. For S3, assume that z 6={1} where
isinS ∩Z .Thenz canbeexpressedasa linearcombination
t t 1 α α2 ··· αn−1
of the generators of S . But τ(z) = 0 implying that the
t  1 α3 α6 ··· α3(n−1) 
generators of S are dependent. Thus S ∩Z = {1} and S3
also holds. Thuts S = lim(St,ιtj) defintes ant [(n,k,nµ;δ)]q H0 = ... ... ... ... ... .
−→  
convolutional stabilizer code. By definition the degree of the  1 α2t−1 α2(2t−1) ··· α(2t−1)(n−1) 
 
Similarly if w = α−i and γ = α−2i, then we have another Lemma 10: Let 2≤2t≤⌊n/(q+1)⌋, where gcd(n,2)=
i i
[n,n−t,t+1]q2 GRS code with parity check matrix 1,n|q2−1andq+1<n≤q2−1.Thentheconvolutionalcode
1 α−1 α−2 ··· α−(n−1) C =Γq2H has free distance df ≥n−2t+1>2t+1=d⊥f,
where d⊥ =wt(C⊥h) is the free distance of C⊥h.
 1 α−3 α−6 ··· α−3(n−1)  f
H1 = .. .. .. .. .. . Proof: Since d⊥f = wt(C⊥h) = wt(C⊥), we compute
 . . . . .  wt(C⊥). Let c = (...,0,c ,...,c,0,...) be a codeword in
  0 l
 1 α−(2t−1) α−2(2t−1) ··· α−(2t−1)(n−1)  C⊥ with c ∈ Fn , c 6= 0, and c 6= 0. It follows from the
  i q2 0 l
parity check equations cHT = 0 that c HT = 0 = c HT
The [n,n−2t,2t+1]q2 GRS code with wi =α−i(2t−1) and holds. Thus, wt(c ),wt(c ) ≥ t+1. If l0>10, then wt(cl) ≥0
γ = α2i has a parity check matrix H∗ that is equivalent to 0 l
i wt(c )+wt(c )≥2t+2.Ifl=0,thenc isinthedualofH∗,
H0 up to a permutation of rows. 0 l 0
(cid:2)HO1u(cid:3)r goal is to show that under certain restrictions on n whichisan[n,n−2t,2t+1]q2 code.Thuswt(c)=wt(c0)≥
2t+1 and d⊥ ≥2t+1. But if c is in the dual of H∗, then
the following semi-infinite coefficient matrix H determines f x
an Fq2-linear Hermitian self-orthogonalconvolutional code (...,0,cx,0,...) is a codeword of C. Thus d⊥f =2t+1.
Let (...,c ,c ,c ,...) be a nonzero codeword in C.
i−1 i i+1
H0 H1 0 ··· ··· ObservingthestructureofC,weseethatanynonzeroci must
H = 0 H0 H1 0 ··· . (3) beinthespanofH∗.ButH∗ generatesan[n,2t,n−2t+1]q2
.. .. .. .. code. Hence d ≥ n − 2t + 1. If 2t ≤ ⌊n/(q+1)⌋, then
 . . . ··· .  f
  t≤n/6; thus d ≥n−2t+1>2t+1=d⊥ holds.
f f
ToshowthatH isHermitianself-orthogonal,itissufficientto The preceding proof generalizes [15, Corollary 4] where the
showthatH0,H1 arebothself-orthogonalandH0 andH1 are free distance of C⊥ was computed for q =2m.
orthogonalto each other. A portion of this result is contained Theorem 11: Letq beapowerofaprime,nanodddivisor
in [6, Lemma 8], viz., n = q2−1. We will prove a slightly of q2−1, such that q+1 < n ≤ q2−1 and 2 ≤ µ = 2t ≤
stronger result. ⌊n/(q+1)⌋. Then there exists a pure quantum convolutional
Lemma 8: Let n|q2−1 such that q+1<n≤q2−1 and code with parameters [(n,n−µ,n;µ/2,µ+1)] . This code
q
2≤µ=2t≤⌊n/(q+1)⌋, then is optimal, since it attains the Singleton bound with equality.
H =(αij) and H =(α−ij) Proof: The convolutional code generated by the coeffi-
0 1≤i<µ,0≤j<n 1 1≤i<µ,0≤j<n
cient matrix H in equation (3) has parameters (n,µ/2,δ ≤
areself-orthogonalwithrespecttotheHermitianinnerproduct. µ/2;1,df)q2.Inspectingthe correspondingpolynomialgener-
Further, H0 is orthogonal to H1. ator matrix shows that δ ≤ µ/2, since νi = 1 for 1 ≤ i ≤
Proof: Denote by H0,j =(1,αj,α2j,··· ,αj(n−1)) and µ/2. By Corollary 9, this code is Hermitian self-orthogonal;
H1,j =(1,α−j,α−2j,··· ,α−j(n−1)), where 1≤j ≤µ−1. moreover,Lemma 10 shows that the distance of its dual code
The Hermitian inner product of H0,i and H0,j is given by is given by d⊥f = µ + 1 < df. By Theorem 5, we can
n−1 α(i+jq)n−1 conclude that there exists a pure convolutionalstabilizer code
hH |H i = αilαjql = , with parameters [(n,n−µ,n;δ ≤ µ/2,µ+1)] . It follows
0,i 0,j h αi+jq −1 q
Xl=0 from Theorem 7 that
which vanishes if i+jq 6≡ 0 mod n. If 1 ≤ i,j ≤ µ−1 = µ+1≤(µ/2)(⌊2δ/(2n−µ)⌋+1)+δ+1
⌊n/(q+1)⌋−1,then q+1≤i+jq≤(q+1)⌊n/(q+1)⌋− ≤(µ/2)(⌊µ/(2n−µ)⌋+1)+δ+1.
(q + 1) < n; hence, hH |H i = 0. Thus, H is self-
0,i 0,j h 0 Since⌊µ/(2n−µ)⌋=0,therighthandsideequalsµ/2+δ+
orthogonal.Similarly,H isalsoself-orthogonal.Furthermore,
1 1, which implies δ =µ/2 and the optimality of the quantum
n−1 α(i−jq)n−1 code.
hH |H i = αilα−jql = .
0,i 1,j h αi−jq −1
Xl=0 V. CONVOLUTIONAL RM STABILIZERCODES
This inner product vanishes if αi−jq 6= 1 or, equivalently, if Inthissection,wederiveconvolutionalstabilizercodesfrom
i−jq6≡0 mod n.Since1≤i,j ≤⌊n/(q+1)⌋−1≤q−2, quasi-cyclic subcodes of binary Reed-Muller block codes [2],
we have 1 ≤ i ≤ ⌊n/(q+1)⌋−1 ≤ q−2 while q ≤ jq ≤ takingadvantageoftheframeworkdevelopedbyEsmaeiliand
q⌊n/(q+1)⌋−q < n. Thus i 6≡ jq mod n and this inner Gulliver for classical convolutional codes [3].
product also vanishes, which proves the claim. Let u = (u1,u2,...,un) and v = (v1,v2,...,vn) be
Since H is contained in H , we obtain the following: vectors in Fn; we define their boolean product as uv =
i i 2
Corollary 9: Let2≤µ=2t≤⌊n/(q+1)⌋, wheren|q2− (u1v1,u2v2,...,unvn).Theproductofisuchn-tuplesissaid
1 and q+1 < n ≤ q2−1. Then H and H are Hermitian to have degree i.
0 1
self-orthogonal.Further, H0 is orthogonalto H1 with respect Let b0 = (1,1,...,1) ∈ F22m. For m > 0 and 1 ≤ i ≤
to the Hermitian inner product. m, define bi ∈ F22m as concatenation of 2m−i blocks of the
Before we can construct quantum convolutional codes, we form 01 ∈ F2i, where 0 and 1 are the constant zero and
2
need to compute the free distances of C and C⊥h, where C one vectors in F2i−1, respectively. Let 0 ≤ r < m and B =
2
is the code generated by H. {b ,b ,...,b }⊆F2m.ThentherthorderReed-Mullercode
1 2 m 2
R(r,m) is the linear span of b and all products of elements and free distance 2m−r. By Corollary 6 there exists a linear
0
in B of degree r or less. The code R(r,m) has dimension [(2m−l,2m−l−2k(r),≤2m−l(2l−1))] convolutionalstabi-
2
k(r) = r m and minimum distance 2m−r; the dual of lizer code. For 0 ≤ r ≤ ⌊(m−l−1)/2⌋, the dual distance
i=0 i
R(r,m)Pis giv(cid:0)en(cid:1)by R(r,m)⊥ = R(m−1−r,m), and the 2r+1 <2m−r, hence the code is pure.
dual distance of R(r,m) is 2r+1, see [9] for details. After circulating the first version of this manuscript, Grassl
Let Bi denote the set of all products of elements in B of and Ro¨tteler kindly pointed out that the convolutional codes
m
degree i. For 0 ≤ i ≤ r < m, a generator matrix Gr of in [3] that are used here have degree0, hence, are a sequence
m
R(r,m) is given by (see [3] for details) of juxtaposed block codes disguised as convolutional codes.
Consequently, the codes constructed in the previous theorem
Br
m have parameters [(2m−l,2m−l−2k(r),0;0,2r+1)] .
 Br−1  2
m
Grm = ... . VI. CONCLUSION
  We developedanapproachtoconvolutionalstabilizercodes
 Bi+1 
 m  that is based on a direct limit construction, formalizing the
 Gi 
 m  arguments given in [14]. We proved a Singleton bound for
Let w = (110···0) ∈ F2µ. Let lw denote the vector pure convolutional stabilizer codes, and derived an optimal
µ 2 µ
obtained by concatenating l copies of wµ. For 0≤i≤l−1, family of quantum convolutional codes attaining this bound
let Mi,l = (2l−i−1wi+1)⊗Bmr−−il, which is a matrix of size from generalized Reed-Solomon codes. We illustrated how to
m−l ×2m, andletM = Gr−l 0 ··· 0 . One can use quasi-cyclic subcodes of Reed-Muller codes to construct
r−i l,l m−l
d(cid:0)erive(cid:1)aconvolutionalcodea(cid:2)stherowspanofthese(cid:3)mi-infinite a family of convolutionalstabilizer codes;this method can be
matrix G given in (2), where µ=2l−1 and the matricesGi, applied to other quasi-cyclic codes as well.
0≤i<2l, are defined by
Acknowledgments. This research was supported by NSF
M CAREER awardCCF 0347310and NSF grantCCF 0622201.
0,l
2 M1,l 3 REFERENCES
ˆ G0 G1 ··· G2l−1 ˜=66 ... 77. [1] A. C. A. de Almeida and R. Palazzo Jr. A concatenated [(4, 1, 3)]
6 M 7
6 l−1,l 7 quantum convolutional code. In Proc. IEEEInform. Theory Workshop
4 Ml,l 5 (SanAntonio,TX),Oct.2004.
[2] M.EsmaeiliandT.A.Gulliver. Quasi-cyclic structure ofReed-Muller
We note that G = Gr and that the rows of G , 1 ≤ i ≤
0 m−l i codes and their smallest regular trellis diagram. IEEE Trans. Inform.
2l − 1, are a subset of the rows in G . The convolutional Theory,43(3):1040–1052, 1997.
0
code generated by G is a (2m−l, r m−l ) code with [3] M.EsmaeiliandT.A.Gulliver. Alinkbetweenquasi-cyclic codesand
i=0 i 2 convolutionalcodes. IEEETrans.Inform.Theory,44(1):431–435,1998.
free distance 2m−r, see [3]. Even thPoughG(cid:0) is (cid:1)correspondsto [4] G.D.ForneyJr.,M.Grassl,andS.Guha. Convolutional andtail-biting
a catastrophic encoder, we can conclude the following: quasntumerror-correctingcodes.SubmittedIEEETrans.Inform.Theory,
Lemma 12: Let C = Γ G. Then d⊥, the free distance of Nov.2005.
2 f [5] G. D. Forney Jr. and S. Guha. Simple rate-1/3 convolutional and
the convolutionalcode C⊥ is 2r+1. tail-biting quantum error-correcting codes. In Proc. 2005 IEEE Intl.
Proof: Let c be a codeword in the dual of R(r,m−l) Symposium on Information Theory, Adelaide, Australia, pages 1028–
0
i.e.,c GT =0.AsG aresubmatricesofG wehavec GT = 1032,2005.
0 0 i 0 0 i [6] M. Grassl, T. Beth, and M. Ro¨tteler. On optimal quantum codes.
0. It follows that c=(...,0,c0,0,...) satisfies cGT =0 and Internat. J.Quantum Information, 2(1):757–775, 2004.
isinC⊥.Thusd⊥ ≤minwt(c )=wt(R(r,m−l)⊥)=2r+1. [7] M. Grassl and M. Ro¨tteler. Non-catastrophic encoders and encoder
f 0
inverses for quantum convolutional codes. In Proc. 2006 IEEE Intl.
Let c = (...,0,c ,...,c ,0,...) be a codeword of min-
0 l SymposiumonInformationTheory,Seattle,WA,USA,pages1109–1113,
imum weight in C⊥. Since cGT = 0, we can infer that 2006.
c GT = 0 = c GT. Since c is in the dual space of G , [8] M.GrasslandM.Ro¨tteler.Quantumblockandconvolutionalcodesfrom
it0ha2sl−a1minimumlwe0ight of 2r+l 1. Therefore, min{wt(c )+0 self-orthogonal productcodes. InProc.2005IEEEIntl.Symposium on
0 InformationTheory,Adelaide, Australia,pages1018–1022,Sept.2005.
2r+1}≤d⊥ ≤2r+1; hence d⊥ =2r+1. [9] W.C.HuffmanandV.Pless. FundamentalsofError-CorrectingCodes.
f f
Lemma 13: Let 1≤l ≤m and 0≤r ≤⌊(m−l−1)/2⌋, University Press,Cambridge, 2003.
[10] R.JohannessonandK.Sh.Zigangirov. FundamentalsofConvolutional
thentheconvolutionalcodegeneratedbyGisself-orthogonal.
coding. Digital andMobileCommunication, Wiley-IEEEPress,1999.
Proof: It is sufficient to show that GiGTj = 0 for 0 ≤ [11] A. Ketkar, A. Klappenecker, S. Kumar, and P.K. Sarvepalli. Nonbi-
i,j ≤2l−1. Since the rowsofG are a subsetof the rowsof nary stabilizer codes over finite fields. IEEE Trans. Inform. Theory,
i
52(11):4892 –4914,November,2006.
G itsufficestoshowthatG isself-orthogonal.ForG tobe
0 0 0 [12] K.Lally. Algebraiclowerboundsonthefreedistance ofconvolutional
self-orthogonalwerequirethatr ≤(m−l)−r−1whichholds. codes. IEEETrans.Inform.Theory,52(5):2101–2110, 2006.
Hence, G generates a self-orthogonalconvolutional code. [13] H. Ollivier and J.-P. Tillich. Description of a quantum convolutional
code. Phys.Rev.Lett.,91(17):1779021–4, 2003.
Theorem 14: Let 1≤l ≤m and 0≤r ≤⌊(m−l−1)/2⌋,
[14] H.Ollivier andJ.-P.Tillich. Quantumconvolutional codes:Fundamen-
then there exist pure linear quantumconvolutionalcodes with tals. ArXiv:quant-ph/0401134,2004.
the parameters [(2m−l,2m−l−2k(r),≤2m−l(2l−1))] and [15] Ph. Piret. A convolutional equivalent toReed-Solomon codes. Philips
2
free distance 2r+1, where k(r)= r m−l . J.Res.,43:441–458,1988.
i=0 i [16] L.Ribes andP.Zalesskii. Profinitegroups. Springer, 2000.
Proof: By Lemma13,G defiPnesa l(cid:0)inear(cid:1)self-orthogonal [17] J.RosenthalandR.Smarandache. ConstructionsofMDS-convolutional
convolutional code with parameters (2m−l,k(r),≤ 2l −1) codes. IEEETrans.Inform.Theory,47(5):2045–2049, 2001.
2