Subsystem Code Constructions

0.11 MB·English

by  Salah A. Aly

1
Subsystem Code Constructions
Salah A. Aly and Andreas Klappenecker
Department of Computer Science
Texas A&M University, College Station, TX 77843-3112, USA
Email: {salah, klappi}@cs.tamu.edu
Abstract—Subsystem codes are the most versatile class of of a pure ((n,K,R,d)) subsystem code implies the
q
8 quantum error-correcting codes known to date that combine existence of a pure ((n,pK,R/p,d)) code.
q
0 the best features of all known passive and active error-control
ii) We show that all pure MDS subsystem codes are derived
0 schemes. The subsystem code is a subspace of the quantum
from MDS stabilizer codes. We establish here for the
2 state space that is decomposed into a tensor product of two
vector spaces: the subsystem and the co-subsystem. A generic first time the existence of numerous families of MDS
n
methodtoderivesubsystemcodesfromexistingsubsystemcodes subsystem codes.
a
J is given that allows one to trade the dimensions of subsystem iii) Wederivetwopropagationrulesthatyieldnewsubsystem
and co-subsystem whilemaintainingor improvingthe minimum
9 codes by extending or shortening the length existing
distance. As a consequence, it is shown that all pure MDS
codes.
subsystem codes are derived from MDS stabilizer codes. The
h] existence of numerous families of MDS subsystem codes is iv) We derivetwo propagationrulesthatyielda new subsys-
p established. Propagation rules are derived that allow one to tem code by combining two subsystem codes.
- obtainlongerandshortersubsystemcodesfromgivensubsystem
t codes. Furthermore, propagation rules are derived that allow
n one to construct a new subsystem code by combining two given II. SUBSYSTEMCODE CONSTRUCTIONS
a
subsystem codes.
u First we recall the following fact that is key to most
q constructions of subsystem codes (see below for notations):
[
I. INTRODUCTION

3 Subsystem codesare a relativelynew constructionof quan- Theorem 1: Let C be a classical additive subcode of F2qn
v tum codes that combine the features of decoherence free suchthatC 6={0}andletDdenoteitssubcodeD =C∩C⊥s.
1
2 subspaces [1], noiseless subsystems [2], and quantum error- If x = |C| and y = |D|, then there exists a subsystem code
3 correctingcodes [3], [4]. Such codes promise to offer appeal- Q=A⊗B such that
4 ing features, such as simplified syndrome calculation and a i) dimA=qn/(xy)1/2,
2. widevarietyofeasily implementablefault-tolerantoperations, ii) dimB =(x/y)1/2.
1 see [5]–[8]. The minimum distance of subsystem A is given by
7 An ((n,K,R,d)) subsystem code is a KR-dimensional (a) d=swt((C+C⊥s)−C)=swt(D⊥s−C) if D⊥s 6=C;
q
0 subspace Q of Cqn that is decomposed into a tensor product (b) d=swt(D⊥s) if D⊥s =C.
:
v Q = A⊗B of a K-dimensional vector space A and an R- Thus,thesubsystemAcandetectallerrorsinE ofweightless
Xi dimensionalvectorspace B such thatallerrorsof weightless thand,andcancorrectallerrorsinE ofweight≤⌊(d−1)/2⌋.
than d can be detected by A. The vector spaces A and B are Proof: See [9, Theorem 5].
r
a respectively called the subsystem A and the co-subsystem B. A subsystem code that is derived with the help of the
Forsomebackgroundonsubsystemcodes,seeforinstance[6], previous theorem is called a Clifford subsystem code. We
[9], [10]. will assume throughout this paper that all subsystem codes
A special feature of subsystem codes is that any classical are Clifford subsystem codes. In particular, this means that
additive code C can be used to construct a subsystem code. the existence of an ((n,K,R,d))q subsystem code implies
One should contrast this with stabilizer codes, where the the existence of an additive code C ≤ F2n with subcode
q
classical codes are required to satisfy a self-orthogonality D = C ∩C⊥s such that |C| = qnR/K, |D| = qn/(KR),
condition. and d=swt(D⊥s −C).
We assume that the reader is familiar with the relation AsubsystemcodederivedfromanadditiveclassicalcodeC
between classical and quantum stabilizer codes, see [3], [11]. iscalledpuretod′ ifthereisnoelementofsymplecticweight
In [6], [9], the authors gave an introduction to subsystem lessthand′ in C.A subsystemcodeiscalledpureif itispure
codes, established upper and lower bounds on subsystem to the minimum distance d. We require that an ((n,1,R,d))
q
code parameters, and provided two methods for constructing subsystem code must be pure.
subsystemcodes.Themainresultsonthispaperareasfollows: We also use the bracket notation [[n,k,r,d]] to write the
q
i) If q is a power of a prime p, then we show that a parameters of an ((n,qk,qr,d)) subsystem code in simpler
q
subsystem code with parameters ((n,K/p,pR,≥ d)) form. Some authors say that an [[n,k,r,d]] subsystem code
q q
can be obtained from a subsystem code with parameters has r gaugequdits, but this terminologyis slightly confusing,
((n,K,R,d)) . Furthermore, we show that the existence as the co-subsystem typically does not correspond to a state
q
2
spaceofr quditsexceptperhapsintrivialcases.Wewillavoid some integers r ≥ 1, and s ≥ 0. There exists an F -basis of
p
this misleading terminology. An ((n,K,1,d)) subsystem C of the form
q
code is also an ((n,K,d)) stabilizer code and vice versa.
Notation.Letq be a powqerof a primeintegerp. We denote C =spanFp{z1,...,zs,xs+1,zs+1,...,xs+r,zs+r}
by F the finite field with q elements. We use the notation that can be extended to a symplectic basis
q
(x|y) = (x1,...,xn|y1,...,yn) to denote the concatenation {x1,z1,...,xnm,znm} of F2qn, that is, hxk | xℓis = 0,
of two vectors x and y in Fn. The symplectic weight of hz | z i = 0, hx | z i = δ for all 1 ≤ k,ℓ ≤ nm,
q k ℓ s k ℓ s k,ℓ
(x|y)∈F2n is defined as see [12, Theorem 8.10.1].
q
Define an additive code
swt(x|y)={(x ,y )6=(0,0)|1≤i≤n}.
i i
Cm =spanF {z1,...,zs,xs+1,zs+1,...,xs+r+1,zs+r+1}.
p
We define swt(X) = min{swt(x)|x ∈ X,x 6= 0} for any
It follows that
nonempty subset X 6={0} of F2n.
q
⊥
v =Th(ea′t|rba′c)ei-nsyFm2pnleisctidcefipnroedduacst of two vectors u=(a|b) and Cms =spanFp{z1,...,zs,xs+r+2,zs+r+2,...,xnm,znm}
q and
hu|vis =trq/p(a′·b−a·b′), D =Cm∩Cm⊥s =spanFp{z1,...,zs}.
By definition, the code C is a subset of C .
where x · y denotes the dot product and tr denotes the m
q/p The subsystem code defined by C has the parameters
trace fromF to the subfieldF . The trace-symplecticdualof m
q p (n,K ,R ,d ), where K = qn/(ps+2r+2ps)1/2 = K/p
a code C ⊆F2n is defined as m m m m
q and R =(ps+2r+2/ps)1/2 =pR. For the claims concerning
m
C⊥s ={v ∈F2n |hv|wi =0 for all w ∈C}. minimum distance and purity, we distinguish two cases:
q s (a) If Cm 6=D⊥s, then K >p and dm =swt(D⊥s−Cm)≥
WedefinetheEuclideaninnerproducthx|yi=Pni=1xiyi and swt(D⊥s−C)=d.Sincebyhypothesisswt(D⊥s−C)=
the Euclidean dual of C ⊆Fnq as d and swt(C) ≥ d′, and D ⊆ C ⊂ Cm ⊆ D⊥s by
construction, we have swt(C ) ≥ min{d,d′}; thus, the
C⊥ ={x∈Fnq |hx|yi=0 for all y ∈C}. subsystem code is pure to mimn{d,d′}.
CiWne⊆FanqFl2snoasdaeshfixn|yeithhe=HePrmni=it1iaxnqiyininaenrdprtohdeuHctefromritviaenctodrusalx,oyf (b) sfIofwlCtlo(mwDs⊥=fsr)oDm=⊥tsdh,met.ahsesnumKemdp=ur1ity=thKat/dp,=thsawtti(sD, K⊥s−=Cp);=it
q2 This proves the claim.
C⊥h ={x∈Fnq2 |hx|yih =0 for all y ∈C}. theFoprrevFiqo-ulisnethaerosruembsywstheimchcaossdeerstsththearet oenxeisctsanacvoanrsitartuioctnthoef
resulting subsystem code such that it is again F -linear.
q
III. TRADING DIMENSIONS OF SUBSYSTEMAND Theorem 3: Let q be a power of a prime p. If there exists
CO-SUBSYSTEM CODES an F -linear [[n,k,r,d]] subsystem code with k > 1 that is
q q
puretod′,thenthereexistsanF -linear[[n,k−1,r+1,≥d]]
In this section we show how one can trade the dimensions q q
subsystem code that is pure to min{d,d′}. If a pure F -linear
of subsystem and co-subsystem to obtain new codes from a q
[[n,1,r,d]] subsystem code exists, then there exists an F -
givensubsystemorstabilizercode.Theresultsareobtainedby q q
linear [[n,0,r+1,d]] subsystem code.
exploitingthesymplecticgeometryofthespace.Aremarkable q
Proof:Theproofisanalogoustotheproofoftheprevious
consequence is that nearly any stabilizer code yields a series
theorem,exceptthatF -basesareusedinsteadofF -bases.
of subsystem codes. q p
There exists a partial converse of Theorem 2, namely if
Our first result shows that one can decrease the dimension
the subsystem code is pure, then it is possible to increase the
of the subsystem andincrease atthe same time the dimension
dimensionofthesubsystemanddecreasethedimensionofthe
oftheco-subsystemwhilekeepingorincreasingtheminimum
co-subsystem while maintaining the same minimum distance.
distance of the subsystem code.
Theorem 4: Let q be a power of a prime p. If there exists
Theorem 2: Letqbeapowerofaprimep.Ifthereexistsan
((n,K,R,d)) subsystem codewith K >p thatis pureto d′, a pure ((n,K,R,d))q subsystem codewith R>1, then there
q exists a pure ((n,pK,R/p,d)) subsystem code.
thenthereexistsan((n,K/p,pR,≥d)) subsystemcodethat q
ispuretomin{d,d′}.Ifapure((n,p,Rq,d)) subsystemcode Proof: Suppose that the ((n,K,R,d))q Clifford subsys-
q tem code is associated with a classical additive code
exists, then there exists a ((n,1,pR,d)) subsystem code.
q
Proof: By definition, an ((n,K,R,d))q Clifford subsys- Cm =spanFp{z1,...,zs,xs+1,zs+1,...,xs+r+1,zs+r+1}.
temcodeisassociatedwitha classicaladditivecodeC ⊆F2n
and its subcode D = C ∩C⊥s such that x = |C|, y = |Dq|, Let D = Cm ∩Cm⊥s. We have x = |Cm| = ps+2r+2, y =
CK6==Dqn⊥/s(,xoyt)h1e/r2w,iRse=d=(xs/wyt)(1D/2⊥, sa)ndifdD=⊥ssw=t(CD.⊥s −C) if d|D=| =swpt(sD, h⊥esn)c.e K = qn/pr+s and R = pr+1. Furthermore,
The code
We have q =pm forsome positiveintegerm. Since K and
R are positive integers, we have x = ps+2r and y = ps for C =spanF {z1,...,zs,xs+1,zs+1,...,xs+r,zs+r}
p
3
has the subcode D = C ∩C⊥s. Since |C| = |Cm|/p2, the proof that the Singleton bound holds for general subsystem
parameters of the Clifford subsystem code associated with C codes remains elusive.
are((n,pK,R/p,d′))q.SinceC ⊂Cm,theminimumdistance In the next lemma, we give a few examples of MDS sub-
d′ satisfies systemcodesthatcanbeobtainedfromTheorem7. Theseare
′ ⊥ ⊥ ⊥ the first families of MDS subsystem codes (though sporadic
d =swt(D s −C)≤swt(D s −C )=swt(D s)=d.
m examples of MDS subsystem codes have been established
On the other hand, d′ = swt(D⊥s −C) ≥ swt(D⊥s) = d, before, see e.g. [6], [7]).
whence d=d′. Furthermore, the resulting code is pure since Lemma 8: i) AnFq-linearpure[[n,n−2d+2−r,r,d]]q
d=swt(D⊥s)=swt(D⊥s −C). MDS subsystem code exists for all n, d, and r such that
ReplacingF -basesbyF -basesintheproofoftheprevious 3≤n≤q, 1≤d≤n/2+1, and 0≤r ≤n−2d+1.
p q
theoremyieldsthefollowingvariationoftheprevioustheorem ii) AnFq-linearpure[[(ν+1)q,(ν+1)q−2ν−2−r,r,ν+2]]q
for F -linear subsystem codes. MDS subsystem code exists for all ν and r such that
q
Theorem 5: Letq beapowerofaprimep.Ifthereexistsa 0≤ν ≤q−2 and 0≤r ≤(ν+1)q−2ν−3.
pure Fq-linear [[n,k,r,d]]q subsystem code with r > 0, then iii) An Fq-linear pure [[q−1,q−1−2δ−r,r,δ+1]]q MDS
there exists a pure F -linear [[n,k+1,r−1,d]] subsystem subsystem code exists for all δ and r such that 0≤ δ <
q q
code. (q−1)/2 and 0≤r ≤q−2δ−1.
The purity hypothesis in Theorems 4 and 5 is essential, as iv) An Fq-linear pure [[q,q−2δ−2−r′,r′,δ+2]]q MDS
the next remark shows. subsystem code exists for all 0 ≤ δ < (q − 1)/2 and
Remark 1: TheBacon-Shorcodeisanimpure[[9,1,4,3]]2 0≤r′ <q−2δ−2.
subsystem code. However, there does not exist any [[9,5,3]]2 v) An Fq-linear pure [[q2 −1,q2 −2δ −1−r,r,δ +1]]q
stabilizer code. Thus, in general one cannot omit the purity MDS subsystem code exists for all δ and r in the range
assumption from Theorems 4 and 5. 0≤δ <q−1 and 0≤r <q2−2δ−1.
[[nA,kn,0[[,nd,]k],dsu]]bqsysstatebmilizceordec.oWdeecreacnoradlstohisbeimrpegoartradnetdspaesciaanl vi) sAunbsFyqs-tleimneacrodpeureex[i[sqt2s,fqo2r−al2lδδ−a2nd−rr′′,irn′,tδhe+r2a]n]gqeM0D≤S
case of theqprevious theorems in the next corollary. δ <q−1 and 0≤r′ <q2−2δ−2.
Corollary 6: If there exists an (F -linear) [[n,k,d]] sta- Proof: i) By [15, Theorem 14], there exist Fq-linear
q q
bilizer code that is pure to d′, then there exists for all r [[n,n−2d+2,d]]q stabilizer codes for all n and d such that
in the range 0 ≤ r < k an (F -linear) [[n,k − r,r,≥ d]] 3 ≤ n ≤ q and 1 ≤ d ≤ n/2+1. The claim follows from
q q
subsystem code that is pure to min{d,d′} . If a pure (F - Theorem 7.
q
linear) [[n,k,r,d]] subsystem code exists, then a pure (F - ii) By [16, Theorem 5], there exist a [[(ν +1)q,(ν +1)q−
q q
2ν−2,ν+2]] stabilizercode.Inthiscase,thecodeisderived
linear) [[n,k+r,d]] stabilizer code exists. q
q from an Fq2-linear code X of length n over Fq2 such that
X ⊆X⊥h.TheclaimfollowsfromLemma15andTheorem7.
IV. MDSSUBSYSTEMCODES iii),iv) There exist F -linear stabilizer codes with parameters
q
Recallthatan[[n,k,r,d]]q subsystemcodederivedfroman [[q−1,q−2δ−1,δ+1]]q and [[q,q−2δ−2,δ +2]]q for
Fq-linearclassicalcodeC ≤F2qn satisfiestheSingletonbound 0 ≤ δ < (q−1)/2, see [15, Theorem 9]. Theorem 7 yields
k+r ≤n−2d+2, see [13,Theorem3.6].A subsystemcode the claim.
attaining the Singleton bound with equality is called an MDS v),vi) There exist F -linear stabilizer codes with parameters
q
subsystem code. [[q2−1,q2−2δ−1,δ+1]] and [[q2,q2−2δ−2,δ+2]] .
q q
An important consequence of the previous theorems is the for 0 ≤ δ < q−1 by [15, Theorem 10]. The claim follows
following simple observation which yields an easy construc- from Theorem 7.
tion of subsystem codes that are optimal among the F -linear The existence of the codes in i) are merely established by a
q
Clifford subsystem codes. non-constructive Gilbert-Varshamov type counting argument.
Theorem 7: If there exists an F -linear [[n,k,d]] MDS However, the result is interesting, as it asserts that there exist
q q
stabilizer code, then there exists a pure Fq-linear [[n,k − forexample[[6,1,1,3]]q subsystemcodesforallprimepowers
r,r,d]]q MDSsubsystemcodeforallrintherange0≤r ≤k. q ≥7,[[7,1,2,3]]q subsystemcodesforallprimepowersq ≥
Proof: An MDS stabilizer code must be pure, see [11, 7, and other short subsystem codes that one should compare
Theorem 2] or [14, Corollary 60]. By Corollary 6, a pure Fq- with a [[5,1,3]]q stabilizer code. If the syndrome calculation
linear [[n,k,d]] stabilizer code implies the existence of an is simpler, then such subsystem codes could be of practical
q
F -linear [[n,k−r,r,d ≥ d]] subsystem code that is pure value.
q r q
todforanyrintherange0≤r≤k.Sincethestabilizercode Thesubsystemcodesgivenin ii)-vi)of thepreviouslemma
isMDS,wehavek =n−2d+2.BytheSingletonbound,the are constructively established. The subsystem codes in ii)
parametersoftheresultingF -linear[[n,n−2d+2−r,r,d ]] are derived from Reed-Muller codes, and in iii)-vi) from
q r q
subsystemcodesmustsatisfy(n−2d+2−r)+r≤n−2d +2, Reed-Solomon codes. There exists an overlap between the
r
which shows that the minimum distance d = d, as claimed. parametersgiven in ii) and in iv), but we list here both, since
r
each code construction has its own merits.
Remark 2: We conjecture that F -linear MDS subsystem Remark 3: ByTheorem5,pureMDSsubsystemcodescan
q
codes are actually optimal among all subsystem codes, but a always be derived from MDS stabilizer codes, see Table I.
4
TABLEI
By Theorem 1, there are two additive codes C and D
OPTIMALPURESUBSYSTEMCODES
associated with an ((n,K,R,d)) Clifford subsystem code
q
such that
SubsystemCodes Parent
Code(RSCode) |C|=qnR/K
[[8,1,5,2]]3 [8,6,3]32
[[8,4,2,2]]3 [8,3,6]32 and
[[8,5,1,2]]3 [8,2,7]32 |D|=|C∩C⊥s|=qn/(KR).
[[9,1,4,3]]3 [9,6,4]†32,δ=3
[[9,4,1,3]]3 [9,3,7]†32,δ=6 We can derive from the code C two new additive codes of
[[15,1,10,3]]4 [15,12,4]42 length2n+2overFq, namelyC′ andD′ =C′∩(C′)⊥s. The
[[15,9,2,3]]4 [15,4,12]42 codes C′ and D′ determine a ((n+1,K′,R′,d′)) Clifford
[[15,10,1,3]]4 [15,3,13]42 q
[[16,1,9,4]]4 [16,12,5]†42,δ=4 subsystem code. Since
[[[[2244,,11,61,72,,44]]]]55 [[2244,,250,2,05]]5522 D′ = C′∩(C′)⊥s =C′∩(C⊥s)′
[[24,17,1,4]]5 [24,4,21]52 = (C∩C⊥s)′,
[[24,19,1,3]]5 [24,3,22]52
[[24,21,1,2]]5 [24,2,23]52 we have |D′| = q|D|. Furthermore, we have |C′| = q|C|. It
[[[[[[422833,,,111,6,31,783,,,633]]]]]]755 [[2233,,[542,801,,944]2]∗5∗5,227,,]δ7δ2==250 fo(lil)owKs′fr=omqnT+h1e/opre|mC′1||Dth′a|t=qn/p|C||D|=K,
*Punctured code (ii) R′ =(|C′|/|D′|)1/2 =(|C|/|D|)1/2 =R,
†Extendedcode (iii) d′ =swt((D′)⊥s \C′)≥swt((D⊥s \C)′)=d.
Since C′ contains a vector (0α|00) of weight 1, the resulting
subsystem code is pure to 1.
Therefore, one can derive in fact all possible parameter sets Corollary 11: If there exists an [[n,k,r,d]] subsystem
q
of pure MDS subsystem codes with the help of Theorem 7. code with k > 0 and 0 ≤ r < k, then there exists an
Remark 4: In the case of stabilizer codes, all MDS codes [[n+1,k,r,≥d]] subsystem code that is pure to 1.
q
must be pure. For subsystem codes this is not true, as the
[[9,1,4,3]]2 subsystem code shows. Finding such impure Fq- We can also shorten the length of a subsystem code in a
simple way as shown in the following Theorem.
linear [[n,k,r,d]] MDS subsystem codes with k+r = n−
q
Theorem 12: If a pure ((n,K,R,d)) subsystem code ex-
2d+2 is a particularly interesting challenge. q
ists,thenthereexistsapure((n−1,qK,R,d−1)) subsystem
Recall that a pure subsystem code is called perfect if q
code.
and only if it attains the Hamming bound with equality.
Proof:By[6,Lemma10],theexistenceofapureClifford
We conclude this section with the following consequence of
subsystem code with parameters ((n,K,R,d)) implies the
Theorem 7: q
existence of a pure ((n,KR,d)) stabilizer code. It follows
Corollary 9: If there exists an F -linear pure [[n,k,d]] q
q q from[14,Lemma70]thatthereexistapure((n−1,qKR,d−
stabilizer code that is perfect, then there exists a pure F -
q 1)) stabilizer code, which can be regarded as a pure ((n−
linear [[n,k−r,r,d]] perfectsubsystem code for all r in the q
q 1,qKR,1,d−1)) subsystem code. Thus, there exists a pure
range 0≤r≤k. q
((n−1,qK,R,d−1)) subsystemcodebyTheorem4,which
q
proves the claim.
V. EXTENDING AND SHORTENING SUBSYSTEMCODES In bracket notation, the previous theorem states that the
existence of a pure [[n,k,r,d]] subsystem code implies the
In Section III, we showed how one can derive new subsys- q
existence of a pure [[n−1,k+1,r,d−1]] subsystem code.
tem codes from known ones by modifying the dimension of q
the subsystem and co-subsystem. In this section, we derive
new subsystem codes from known ones by extending and VI. COMBINING SUBSYSTEMCODES
shortening the length of the code. In this section, we show how one can obtaina new subsys-
Theorem 10: Ifthereexistsan((n,K,R,d)) Cliffordsub- temcodebycombiningtwogivensubsystemcodesin various
q
systemcodewithK >1,thenthereexistsan((n+1,K,R,≥ ways.
d))q subsystem code that is pure to 1. Theorem 13: If there exists a pure [[n1,k1,r1,d1]]2 sub-
Proof: We first note that for any additive subcode X ≤ systemcodeandapure[[n2,k2,r2,d2]]2 subsystemcodesuch
F2qn, we can define an additive code X′ ≤F2qn+2 by that k2 + r2 ≤ n1, then there exist subsystem codes with
parameters
X′ ={(aα|b0)|(a|b)∈X,α∈F }.
q
[[n1+n2−k2−r2,k1+r1−r,r,d]]2
We have |X′| = q|X|. Furthermore, if (c|d) ∈ X⊥s, then
(cα|d0) is contained in (X′)⊥s for all α in Fq, whence for all r in the range 0 ≤ r < k1+r1, where the minimum
(X⊥s)′ ⊆ (X′)⊥s. By comparing cardinalities we find that distance d≥min{d1,d1+d2−k2−r2}.
equality must hold; in other words, we have Proof: Since there exist pure [[n1,k1,r1,d1]]2 and
[[n2,k2,r2,d2]]2subsystemcodeswithk2+r2 ≤n1,itfollows
⊥ ′ ′ ⊥
(X s) =(X ) s. from Theorem 4 that there exist stabilizer codes with the
5
parameters [[n1,k1 + r1,d1]]2 and [[n2,k2 + r2,d2]]2 such APPENDIX
that k2 + r2 ≤ n1. Therefore, there exists an [[n1 + n2 − WerecallthattheHermitianconstructionofstabilizercodes
k2 −r2,k1 +r1,d]]2 stabilizer code with minimum distance yields F -linear stabilizer codes, as can be seen from the
q
d ≥ min{d1,d1 + d2 − k2 − r2} by [3, Theorem 8]. It following reformulation of [15, Corollary 2].
follows from Theorem 2 that there exists [[n1 +n2 −k2 − Lemma 15 ( [15]): If there exists an Fq2-linear code X ⊆
r2,k1+r1−r,r,≥d]]2 subsystemcodesforallr intherange Fnq2 such that X ⊆ X⊥h, then there exists an Fq-linear code
0≤r <k1+r1. C ⊆ F2n such that C ⊆ C⊥s, |C| = |X|, swt(C⊥s −C) =
Theorem 14: LetQ1 andQ2 betwopuresubsystemcodes wt(X⊥qh −X) and swt(C)=wt(X).
with parameters [[n,k1,r1,d1]]q and [[n,k2,r2,d2]]q, respec- Proof:Let{1,β}beabasisofFq2/Fq.Thentrq2/q(β)=
tively. If Q2 ⊆ Q1, then there exists pure subsystem codes β+βq is an element β0 of Fq; hence, βq =−β+β0. Let
with parameters
C ={(u|v)|u,v∈Fn,u+βv ∈X}.
q
[[2n,k1+k2+r1+r2−r,r,d]]q
It follows from this definition that |X| = |C| and that
for all r in the range 0 ≤ r ≤ k1+k2+r1+r2, where the wt(X) = swt(C). Furthermore, if u+βv and u′ +βv′ are
minimum distance d≥min{d1,2d2}. elements of X with u,v,u′,v′ in Fn, then
q
Proof: By assumption,there exists a pure [[n,k ,r ,d ]]
i i i q 0 = (u+βv)q ·(u′+βv′)
subsystemcode,whichimpliestheexistenceofapure[[n,k +
r ,d ]] stabilizer code by Theorem 4, where i ∈ {1,2i}. = u·u′+βq+1v·v′+β0v·u′+β(u·v′−v·u′).
i i q
By [14, Lemma 74], there exists a pure stabilizer code On the right hand side, all terms but the last are in F ;
q
with parameters [[2n,k1 +k2 +r1 +r2,d]]q such that d ≥ hence we must have (u·v′ −v·u′) = 0, which shows that
min{2d2,d1}. By Theorem 2, there exist a pure subsystem (u|v)⊥s(u′|v′), whence C ⊆ C⊥s. Expanding X⊥h in the
code with parameters[[2n,k1+k2+r1+r2−r,r,d]]q for all basis {1β} yields a code C′ ⊆ C⊥s, and we must have
r in the range 0 ≤ r ≤ k1+k2+r1+r2, which proves the equality by a dimension argument. Since the basis expansion
claim. is isometric, it follows that swt(C⊥s −C)=wt(X⊥h −X).
TheF -linearityofC isadirectconsequenceofthedefinition
q
VII. CONCLUSIONS AND OPEN PROBLEMS of C.
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