Constructions of Pure Asymmetric Quantum Alternant Codes Based on Subclasses of Alternant Codes

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by  Jihao Fan

Constructions of Pure Asymmetric Quantum
Alternant Codes Based on Subclasses of Alternant
Codes
Jihao Fan Hanwu Chen
Department of Computer Science and Engineering Department of Computer Science and Engineering
Southeast University Southeast University
Nanjing, Jiangsu 211189, China Nanjing, Jiangsu 211189, China
4 Email: [email protected] Email: hw [email protected]
1
0
2

n
a Abstract—In this paper, we construct asymmetric quantum have been widely used to construct QECs [12] and AQCs
J error-correcting codes(AQCs) based on subclasses of Alternant [16], [18]. However,other subclasses of Alternantcodes have
codes. Firstly, We propose a new subclass of Alternant codes
6 receivedlessattention.Andthereisanimportantproblemthat
which can attain the classical Gilbert-Varshamov bound to
1 whetherexistingasymptoticallygoodquantumAlternantcodes
construct AQCs. It is shown that when dx = 2, Z-parts of the
AQCscanattaintheclassicalGilbert-Varshamovbound.Thenwe could attain the quantum Gilbert-Varshamov bound. Inspired
]
T construct AQCs based on a famous subclass of Alternant codes bythese,wecarryouttheconstructionofasymmetricquantum
I called Goppa codes. As an illustrative example, we get three Alternant codes.
. [[55,6,19/4]],[[55,10,19/3]],[[55,15,19/2]] AQCs from thewell
s
c known [55,16,19] binary Goppa code. At last, we get asymp- II. PRELIMINARIES
[ totically good binary expansions of asymmetric quantum GRS Let p be a prime number and q a power of p, i.e., q = pr
codes, which are quantum generalizations of Retter’s classical
2 results. All the AQCs constructed in this paper are pure. forsomer >0.LetFq denotethefinitefieldwithq elements.
v The finite field Fqm is a field extension of degree m of the
5 I. INTRODUCTION field Fq. The trace mapping Tr : Fqm → Fq is given by
21 In many quantum mechanical systems the mechanisms for Tr(a)=a+aq+...+aqm−1, for a∈Fqm.
3 the occurrence of bit flip and phase flip errors are quite A. Classical Codes
. different. Recently, several papers argue that in most of the
1 We review some basic results of GRS codes and Alternant
0 known quantum computing models, the phase-flip errors (Z-
codes firstly.
4 typeerrors)happenmorefrequentlythanthebit-fliperrors(X-
The Reed-Solomon code of length n=qm−1(denoted by
1 type errors) and other types of errors. And the asymmetry is
v: largeingeneral[7].Motivatedbythisphenomena,asymmetric RS(n,δ))isacycliccodeoverFqm withroots1,α,...,αδ−2,
whereδ is an integer,2≤δ ≤n−1,α is a primitiveelement
i quantumerror-correctingcodes(AQCs)aredesignedtoadjust
X of F . The parameters of RS(n,δ) are [n,k,d] , where
this asymmetry, which may have more flexbility than general qm qm
r k = n−δ+1, d = δ. The parity check matrix of RS(n,δ)
a quantum error-correcting codes (QECs). is given by
Steane first stated the importance of AQCs in [20]. Some
recent progress is given in [1], [5], [7]. Sarvepalli et al. 1 1 ··· 1
constructed AQCs using a combination of BCH and finite  1 α ··· αn−1 
H = . . . . . (1)
geometryLDPCcodesin[18].Amorecomprehensivecharac- RS(n,δ) . . . .
 . . . . 
terizationofAQCswasgivenbyWangetal.whichunifiedthe  
 1 αδ−2 ··· α(n−1)(δ−2) 
nonadditive AQCs as well [24]. Ezerman et al. [8] proposed  
so-called CSS-like constructions based on pairs of nested GRS codes are obtained by a further generalization of RS
subfield linear codes. They also used nested codes (such as codes. Let a = (α1,α2,...,αn) where the αi are distinct
BCH codes, circulant codes, etc.) over F4 to construct AQCs elements of Fqm, and let v = (v1,v2,...,vn) where the vi
in their earlier work [9]. The asymmetry was introduced into are nonzero elements of Fqm. For any 1 ≤ k ≤ n−1, the
topological quantum codes in [10]. GRS code GRSk(a,v) is defined by
Alternant codes are a very large family of linear error- GRS (a,v)= (v F(α ),v F(α ),...,v F(α )) |
k 1 1 2 2 n n
correcting codes. Many interesting and famous subclasses of
(cid:8) F(x)∈F [x], degF(x)<k . (2)
Alternantcodeshavebeenobtained,forinstance,BCH codes, qm
(cid:9)
Goppa codes, etc. There exist long Alternant codes meeting The parameters of GRS (a,v) are [n,k,n−k+1] . The
k qm
the Gilbert-Varshamov bound. BCH codes and GRS codes dualofa GRScodeisalso aGRScode,i.e.,GRS (a,v)⊥ =
k
GRSn−k(a,y), where y = (y1,y2,...,yn) and yi · vi = III. ASYMPTOTICALLYZ-PARTSGOOD ASYMMETRIC
1/ j6=i(αi−αj), for 1≤i≤n. The parity check matrix of QUANTUMALTERNANTCODES
GRQSk(a,v) is given by We take y = (y ,y ,...,y ) as the encoded codeword of
1 2 n
y1 y2 ··· yn theRScodewithparitycheckmatrixHRS(n,δ).Theelements
 α1y1 α2y2 ··· αnyn  in the codeword must be all nonzero. Then all such codes
H = . . . . (3) consist a subclass of Alternant codes, which we call Sub-
GRSk(a,v)  .. .. .. ..  Alternant codes. The code in the subclass is denoted by
 
 αr1−1y1 αr2−1y2 ··· αrn−1yn  S−Ar(a,y).
where r =n−k. In this section, we only consider the binary primitive
Both RS codes and GRS codes are MDS codes. The Alternant codes, i.e., we take q = 2, n = 2m − 1, αi =
Hamming weight enumerator of any MDS code [n,k,d] αi, 0≤i≤n−1, r =n−k. Then the parity check matrix
q
where d=n−k+1 is completely determined by of the binary primitive Alternant code Ar(a,y) is given by
n w−d w−1 y1 y2 ··· yn
Aw = (q−1) (−1)j qw−d−j (4)  y1 y2α ··· ynα(n−1) 
(cid:18)w(cid:19) Xj=0 (cid:18) j (cid:19) HAr(a,y) = ... ... ... ... . (6)
 
from [15].  y y αr−1 ··· y α(n−1)(r−1) 
 1 2 n 
Alternant codes are obtained as subfield subcodes of GRS
codes. For the notation given above, Alternant code Ar(a,y) It is easy to see that HAr(a,y) =HRS(n,r+1)·diag(y) where
is defined as A (a,y) = GRS (a,v) | F . Therefore diag(y) is a diagonalmatrixwith y asthe diagonalelements.
r k q
A (a,y) has the same parity check matrix as GRS (a,v).
r k
Definition 3.1: For any y = (y ,y ,...,y ) ∈ RS(n,δ)
1 2 n
B. Quantum Error-Correcting codes whose every position is nonzero element, i.e., H yT =
RS(n,δ)
Let C be the complex number field. For a positive integer 0, and yi 6=0 for all 1≤i≤n. Then S−Ar(a,y) is defined
n, let V =(Cq)⊗n =Cqn be the nth tensor product of Cq. as:
n
Definition 2.1: A q-ary asymmetric quantum code of S−Ar(a,y)={c∈Fn2|HAr(a,y)cT =0}
length n, denoted by [[n,k,d /d ]] is a subspace Q of V
z x q n whereH istheparitycheckmatrixin(1)andH
overfinitefieldF withdimensionqk,whichcandetectd −1 RS(n,δ) Ar(a,y)
q x is the one in (6).
qubits of X-errorsand, at the same time, d −1 qubits of Z-
z We have the following asymptotic behavior of these Sub-
errors.
Alternant codes.
Lemma 2.2 (AQCs Constructions[18], [24]): LetC and
1 Lemma 3.2: Let δ/2<r <min{δ,n/2}, there exist long
C denote two classical linear codes with parameters
2 codes S−A (a,y) meeting the Gilbert-Varshamov bound.
[n,k ,d ] and[n,k ,d ] suchthatC⊥ ⊆C .Thenthereex- r
1 1 q 2 2 q 2 1 Proof: Consider any binary word c=(c ,c ,...,c ) of
istsan[[n,k +k −n,d /d ]] AQC,whered =wt(C \C⊥) 1 2 n
1 2 z x q z 1 2 weight t. For c to be a codeword of S−A (a,y), it must
andd =wt(C \C⊥). If d =d andd =d , then thecode r
x 2 1 z 1 x 2 satisfy H cT =0. Then
is pure. Ar(a,y)
For a givenpair (δx,δz) of realnumbersand a family Q= HRS(n,r+1)(y1c1,y2c2,...,yncn)T =0.
[[n(i),k(i),d(i)/d(i)]] ∞ ofasymptoticquantumcodeswith
z x i=1 Letthenonzeroelementsincbe{c ,c ,...,c }where1≤
(cid:8) (cid:9) i1 i2 it
d(xi) d(zi) i1 <i2 <···<it ≤n. Then we have
liminf ≥δ , liminf ≥δ
i→∞ n(i) x i→∞ n(i) z H (...,y c ,...,y c ,...)T =0,
RS(n,r+1) i1 i1 it it
denote the asymptotic quantity as
where“...”denotethezeroelementsifnecessary.Thisimplies
k(i) that H (...,y ,...,y ,...)T = 0 because c is
RQ(δx,δz)=limi→s∞upn(i) binary.RIfSw(ne,r+le1t) i1 it
One of the central asymptotic problems for quantum codes w−(r+1)
is to find families Q of asymptotic quantum codes such that B′ =(2m−1) (−1)j w−1 2m(w−(r+1)−j),
w (cid:18) j (cid:19)
for a fixed pair (δx,δz), the value RQ(δx,δz) is as large as Xj=0
possible. The best known nonconstructive lower bound on
then the Hamming weight enumerator of the RS code with
RQ(δx,δz) can be obtained from [6]: parity check matrix H is B = n B′ . Then the
RS(n,r+1) w w w
′
RQ(δx,δz)≥1−H(δx)−H(δz) (5) number of (...,yi1,...,yit,...) is at most(cid:0)Bt(cid:1).
According to Definition 3.1 and r < δ, we have
where H(x)=−xlog2x−(1−x)log2(1−x) is the binary HRS(n,r+1)(y1,y2,...,yn)T =0. Then
entropy function. It is the quantum Gilbert-Varshamov bound
for AQCs. H (...,y ,...,y ,...)T =0,
RS(n,r+1) j1 j(n−t)
where (...,y ,...,y ,...)T = (y ,y ,...,y )T − where 3≤n≤2m+1, 1<r <δ <n.
j1 j(n−t) 1 2 n
(...,y ,...,y ,...)T, 1 ≤ j < j < ... < y ≤ n, As n → ∞ and δ/2 < r < min{δ,n/2}, there exist a
i1 it 1 2 j(n−t)
“...” denote the zero elements if necessary. Then the number family Q of asymptotically Z-type good AQCs such that
′
of (...,y ,...,y ,...) is at most B . Therefore the
j1 j(n−t) n−t ′ ′ RQ =1−H(δz)−ǫ,
number of y = (y ,y ,...,y ) is at most B B . Notice
1 2 n t n−t 2
that δ = →0,
x
B′ ≤(2m−1)w−r, n
w 1
0<δ < .
then z 2
Bt′Bn′−t ≤(2m−1)n−2r. Proof: Let I =[11 ··· 1] and C1 =[n,n−1,2] with I
Therefore for all codewords of weight t < ω, the number of n
as its parity check matrix. For any C =S−A (a,y) and let
vectors y that include such codewords in the corresponding | {z } 2 r
r <δ, we have
Alternant code S-A(a,y) is at most
H ·IT = H ·diag(y)·IT
ω−1 ω−1 A(a,y) RS(n,r+1)
B′B′ n ≤(2m−1)n−2r n . = H ·yT
t n−t(cid:18)t(cid:19) (cid:18)t(cid:19) RS(n,r+1)
t=Xr+1 t=Xr+1 = 0.
Ontheotherhand,thetotalnumberofsuchAlternantcodes
Therefore C⊥ ⊆ C . By Lemma 2.2 there exists a family of
equal to the number of choices for y, which is 1 2
AQCs with parameters
n−δ
n−1 [[n,≥n−mr−1,≥r+1/2]]
A = (2m−1) (−1)j 2m(n−δ−j) q
n
(cid:18) j (cid:19)
Xj=0 where 3≤n≤qm+1, 1<r <δ <n.
n−1 TheasymptoticresultfollowsfromLemma3.2immediately.
≥ (2m−1)2m(n−δ)(1− )
2m
> (2m−1)n−δ. It shows that when dx = 2, Z-parts of our new AQCs
can attain the classical Gilbert-Varshamovbound, not just the
So if
quantum version.
ω−1
n
(2m−1)n−2r <(2m−1)n−δ IV. AQCS FROM NESTEDGOPPA CODES
(cid:18)t(cid:19)
t=Xr+1 In 1970s, V. D. Goppa introduced a class of linear codes
which can be simplified calledGoppacodesorΓ(L,G)codeswhichformanimportant
subclass of Alternant codes and asymptotically meet the
ω−1
n
<(2m−1)2r−δ, Gilbert-Varshamov bound [15].
t=Xr+1(cid:18)t(cid:19) Definition 4.1: LetG(z)beamonicpolynomialwithcoef-
ficients from F , L= {α ,α ,...,α }⊆ F [z] such that
there exists a [2m,≥ 2m−m(2r−δ),≥ ω] code. Using the ∀i,G(α )6=0.qTmhe Goppa1code2 Γ(L,Gn) of lenqgmth n overF ,
estimates of binomial coefficients in [15, Ch.10. Corollary 9] i q
is the set of codewords c=(c ,c ,...,c )∈Fn such that
and taking the limit as n → ∞, we can write this condition 1 2 n q
as n c
d m(2r−δ) i =0 mod G(z) (9)
H(n)+o(1)< n +o(1). (7) Xi=1 z−αi
G(z) is called the Goppa polynomial, L is the location set.
Let τ = 2r − δ, ǫ = o(1) and choose the values of
We have the following nested Goppa codes which are
parameters properly, then there exists a Sub-Alternant code
similar to nested cyclic codes.
with mτ/n = H(d/n)+ǫ. And by a property of Alternant
Lemma 4.2: Let G(z), F(z) be Goppa polynomials of
codes, the rate R of this code satisfies
q-ary Goppa codes Γ(L,G) and Γ(L,F) respectively. If
mτ
R ≥ 1− F(z)|G(z), then Γ(L,G)⊆Γ(L,F).
n Proof: Let G(z) ∈ F [z] be a monic polynomial of
d qm
> 1−H( )−ǫ. (8) degree r . Then we can decompose the Goppa polynomial
n 1
G(z) into distinct irreducible polynomials Gu(z) over Fqm
HencetheaboveSub-Alternantcodeisasymptoticallyclose as: G(z) = su=1{Gu(z)}du, where du and s are integers
to the Gilbert-Varshamov bound. thatsatisfy Qsu=1du(degGu(z))=r1,degGu(z)≥1.Since
FromDefinition3.1andLemma3.2,wehavethefollowing
the polynomPials Gu(z), u = 1,2,...,s are relatively prime,
result directly.
the defining set (9) for Γ(L,G) can be rewritten as:
Theorem 3.3: There exists a family of AQCs with param-
n
eters ci =0 mod {Gu(z)}du, (10)
[[n,≥n−mr−1,≥r+1/2]] Xi=1 z−αi
TABLEI
GOODBINARYAQCSCONSTRUCTEDFROMNESTEDGOPPACODESUSINGMAGMA
No. Field Γ(L,G) G(z) Γ(L,F)⊥ F(z) [[n,k,dz/dx]]
1 F26 [55,16,19](OPC) z9+1 [55,49,3](OPC) (z−1)6·G(z) [[55,10,19/3]]
2 F26 [56,16,20](OPC) ETC [56,50,3](OPC) DETC [[56,10,20/3]]
3 F26 [54,16,18](OPC) PTC [54,48,3](OPC) DPTC [[54,10,18/3]]
4 F26 [55,16,19](OPC) z9+1 [55,45,4](BKLC) (z−1)2·G(z) [[55,6,19/4]]
5 F26 [55,15,20](OPC) EPC [55,46,3(4)] DEPC [[55,6,20/3]]
6 F26 [56,16,20](OPC) ETC [56,46,4](BKLC) DETC [[56,6,20/4]]
7 F26 [54,15,19](OPC) STC [54,45,3(4)] DSTC [[54,6,19/3]]
8 F26 [54,16,18](OPC) PTC [54,44,4](BKLC) DPTC [[54,6,18/4]]
9 F28 [239,123,35](OPC) z17+1 [239,229,4](BKLC) (z−1)60·G(z) [[239,113,35/4]]
10 F28 [239,122,36](OPC) EPC [239,230,3(4)] DEPC [[239,113,36/3]]
11 F28 [240,123,36](OPC) ETC [240,230,4](BKLC) DETC [[240,113,36/4]]
12 F28 [238,122,35](OPC) STC [238,229,3(4)] DSTC [[238,113,35/3]]
13 F28 [238,123,34](OPC) PTC [238,228,4](BKLC) DPTC [[238,113,34/4]]
14 F28 [239,123,35](OPC) z17+1 [239,218,6](BKLC) (G(z))5 [[239,102,35/6]]
15 F28 [239,122,36](OPC) EPC [239,219,5(6)] DEPC [[239,102,36/5]]
16 F28 [240,123,36](OPC) ETC [238,217,6](BKLC) DETC [[238,102,34/6]]
17 F28 [238,122,35](OPC) STC [240,219,6](BKLC) DSTC [[240,102,36/6]]
18 F28 [238,123,34](OPC) PTC [238,218,5(6)] DPTC [[238,102,35/5]]
19 F28 [239,123,35](OPC) z17+1 [239,208,8](BKLC) (z−1)30·G(z) [[239,92,35/8]]
20 F28 [239,122,36](OPC) EPC [239,209,7(8)] DEPC [[239,92,36/7]]
21 F28 [240,123,36](OPC) ETC [240,209,8](BKLC) DETC [[240,92,36/8]]
22 F28 [238,122,35](OPC) STC [238,208,7(8)] DSTC [[238,92,35/7]]
23 F28 [238,123,34](OPC) PTC [238,207,8](BKLC) DPTC [[238,92,34/8]]
for u=1,2,...,s. (9) and (10) are equivalent for Γ(L,G). the corresponding BKLC’s distance is 4. “Dim” stands for
Since F(z)|G(z), then: dimension of the code. “LB” stands for lower bound of the
code. Firstly we give an explicit example below.
F(z)= {G (z)}fv
v Example 4.3: Loeloeian and Conan gave a Γ(L,G) =
v∈{uY1,...,ut} [55,16,19]binaryGoppacodein[13]whichisaBKLC(Best
where t and f are integers, and {u ,u ,...,u } ⊆ {1, knownlinearcode),aBDLC(Bestdimensionlinearcode)and
v 1 2 t
2,...,s}, 0≤f ≤d , v ∈{u ,u ,...,u }. a BLLC (Best length linear code) over F in the databases of
v v 1 2 t 2
It is easy to see that, for every c = (c ,c ,...,c ) ∈ Magma and [11]. The Goppa polynomialof Γ(L,G) is given
1 2 n
Γ(L,G) which satisfies (10) also satisfies by
n c G(z)=(z−α9)(z−α12)(z−α30)(z−α34)(z−α42)
Xi=1 z−iαi =0 mod {Gv(z)}fv, ·(z−α43)(z−α50)(z−α54)
for v =u ,u ,...,u . where α is a primitive element of F . Take Γ(L,F) with
1 2 t 26
Then, there is c = (c ,c ,...,c ) ∈ Γ(L,F). Therefore Goppa polynomial F(z)=(z−α9)2·G(z), then Γ(L,F)⊆
1 2 n
Γ(L,G)⊆Γ(L,F) Γ(L,G).UsingMagma,weknowthatΓ(L,F)⊥ =[55,45,4].
FromLemma4.2,weknowthatthenestedGoppacodesare Then we get an [[55,6,19/4]] AQC. If F(z) = (z −α9)6 ·
widespread.People have foundthatcertain Goppacodeshave G(z), then Γ(L,F)⊥ = [55,49,3], we get an [[55,10,19/3]]
goodpropertiesand some of these codeshave the bestknown AQC. From Theorem 4.4 below, we get an [[55,15,19/2]]
minimum distance of any known codes with the same length AQC. Fromthe databases,we knowthat[55,45,4],[55,49,3]
and rate. It induces us to identify these codes and investigate and [55,54,2] are all BKLCs. [55,49,3] and [55,54,2]
their nested relationship. And we use Magma to compute the are BDLCs and BLLCs as well. Therefore [[55,10,19/3]]
dualdistance ofnested Goppacodesto some computationally and [[55,15,19/2]] are BDAQCs(Best dimension asymmetric
reasonable length. Some good AQCs are given in TABLE I. quantum code).
Theshorthandsinthetablesareexplainedasfollows.Ifacode In [3], Bezzateev and Shekhunova described a subclass
is both BKLC and BDLC, or achieves the upper bound, we of Goppa codes with minimal distance equal to the design
callit OPC(optimalcode).“EPC” standsforexpurgatedcode, distance. We find that their codes can be used to construct
“ETC” stands for extended code, “STC” stands for shortened AQCs with d =2.
x
code and “PTC” stands for punctured code. “DEPC” stands Theorem 4.4: LetthepolynomialG(z)=zt+A∈F [z],
2m
for the dual of expurgated code, others are the same. “d = where t|(2m −1), i.e., 2m −1 = t·l and A is a tth power
3(4)”, for example, means the minimum distance is 3, and in F \{0}. N = {α ∈ F : G(α) 6= 0}. Denote S =
2m 2m
TABLEII
(2) If S = 0, for punctured Γ(L,G) with G(z) = G(z)
BINARYAQCSCONSTRUCTEDFROMGOPPACODESWITHdx=2
and L = N −{0}, there exists a punctured AQC with
m t S n G(z) Dim LB [[n,k,dz/dx]] Refs. parameters
6 3 0 60 z3+1 43 43 [[60,42,6/2]]
7 0 56 z7+1 17 15 [[56,16,14/2]] [23] [[2m−t−1,≥2m−t−mt−1,≥2t/2]].
9 1 55 z9+1 16 1 [[55,15,19/2]] [19],[23] Proof:WefollowtheproofprocessofTheorem2.1given
– – 56 ETC 16 – [[56,15,20/2]] by Bezzateev & Shekhunova in [3]. For simplicity, we take
– – 54 PTC 16 – [[54,15,18/2]] A = 1. For S = l−1 1/(αµt +1), then S = 1 or 0 as
8 3 0 252 z3+1 229 229 [[252,228,6/2]] µ=1
5 1 251 z5+1 211 211 [[251,210,11/2]] S =S2. P
– – 252 ETC 211 – [[252,210,12/2]] (1) If S = 1. We take G(z) = G(z) = zt + 1, L =
– – 250 PTC 211 – [[250,210,10/2]] N = {α1,α2,...,αn}. For 1 ≤ µ ≤ l − 1, we consider
15 0 240 z15+1 124 121 [[240,123,30/2]] [23] binaryvectorsaµ =(aµ1,aµ2,...,aµn) with Hammingweightt
17 1 239 z17+1 123 103 [[239,122,35/2]] [19],[23] and such that its nonzero components are on positions which
– – 240 ETC 123 – [[240,122,36/2]] correspond to the following subset of L:
– – 238 PTC 123 – [[238,122,34/2]]
51 0 204 z51+1 2 -203 [[204,1,102/2]] {(αl)i·βµ, i=0,1,...,t−1}
9 73 1 439 z73+1 58 -218 [[439,57,147/2]] [4] α is a primitive element of F and β =αµ. Then
– – 440 ETC 58 – [[440,57,148/2]] 2m µ
– – 438 PTC 58 – [[438,57,146/2]] n 1 1
10 31 0 992 z31+1 687 683 [[992,686,62/2]] [23] aµj x−α = βt +1xt−1 mod xt+1
33 1 991 z33+1 686 661 [[991,685,67/2]] [19],[23] Xj=1 j µ
– – 992 ETC 686 – [[992,685,68/2]] for 1≤µ≤l−1.
– – 990 PTC 686 – [[990,685,66/2]] Let the last binary vector a = (al,al,...,al) have only
93 1 931 z93+1 105 1 [[931,104,187/2]] l 1 2 n
one nonzero component on the position which correspond to
– – 932 ETC 105 – [[932,104,188/2]]
{0}. Then for this vector
– – 930 PTC 105 – [[930,104,186/2]]
11 89 1 1959 z89+1 980 980 [[1959,979,179/2]] n 1
– – 1960 ETC 979 – [[1960,979,180/2]] aljx−α =xt−1 mod xt+1.
– – 1958 PTC 979 – [[1958,979,178/2]] Xj=1 j
12 63 0 4032 z63+1 3282 3277 [[4032,3281,126/2]] [23] Now let us consider the sum of vectors a ,a ,...,a
65 1 4031 z65+1 3281 3251 [[4031,3280,131/2]] [19],[23] 1 2 l
– – 4032 ETC 3281 – [[4032,3280,132/2]] n l 1 1 1
– – 4030 PTC 3281 – [[4030,3280,130/2]] aµj x−α =(βt +1 +···+ βt +1 +1)
195 0 3900 z195+1 1759 1561 [[3900,1758,390/2]] Xj=1µX=1 j 1 l−1
273 1 3823 z273+1 1311 547 [[3823,1310,547/2]] [4] ·xt−1 mod xt+1.
– – 3824 ETC 1311 – [[3824,1310,548/2]]
– – 3822 PTC 1311 – [[3822,1310,546/2]] So as S = µl−=11 βt1+1 = µl−=11 αµt1+1 =1, then
315 0 3780 z315+1 474 1 [[3780,473,630/2]] P µ P
455 0 3640 z455+1 197 -1819 [[3640,196,910/2]] n l 1
585 1 3511 z585+1 196 -3509 [[3511,195,1171/2]] aµj x−α =0 mod xt+1.
– – 3512 ETC 196 – [[3512,195,1172/2]] Xj=1µX=1 j
– – 3510 PTC 196 – [[3510,195,1170/2]] Thus vector a = a + a + ··· + a = (1,1,...,1) is a
819 0 3276 z819+1 2 -6551 [[3276,1,1638/2]] codewordoftheGop1papol2ynomialG(zl)=zt+1andL=N
and its Hamming weight is equal to 2m−t. Therefore there
exists an AQC with parameters
l−1 1/(αµt+1), α is a primitive element of F . Then S
mPuµs=t1be 1 or 0. 2m [[2m−t,≥2m−t−mt−1,2t+1/2]],
(1) If S = 1, then for a Goppa code Γ(L,G) with Goppa this code can be extended into
polynomial G(z) = G(z) and L = N, there exists an
[[2m−t+1,≥2m−t−mt−1,2t+2/2]],
AQC with parameters
and can be punctured into
[[2m−t,≥2m−t−mt−1,2t+1/2]],
[[2m−t−1,≥2m−t−mt−1,2t/2]].
this code can be extended to
(2) If S = 0, we take Γ(L,G) with G(z) = G(z) and
[[2m−t+1,≥2m−t−mt−1,2t+2/2]], L=N −{0}, the proof is similar to (1) above. And we can
omit the last binary vector a = (al,al,...,al) as S = 0.
l 1 2 n
and can be punctured to Then there exists a punctured AQC with parameters
[[2m−t−1,≥2m−t−mt−1,2t/2]]. [[2m−t−1,≥2m−t−mt−1,≥2t/2]].
1
Classical GV bound & Z−parts of Theorem 3.3
From the proof of Theorem 4.4, we know that classical 0.9 GV bound for general stabilizer codes
GV bound for CSS codes & Theorem 5.3
codes corresponding to X-parts of AQCs are all [n,n−1,2] 0.8
optimal codes. Therefore the error correction abilities of the
0.7
correspondingGoppacodesarealltransformedintoZ-partsof R
e 0.6
AQCswithonlyoneinformationbitlosseach.Maatouketal. at
n r
[14] found that the classical codes described in Theorem 4.4 atio0.5
m
achievedbetterthantheGVboundwhenthefieldsizeissmall. or0.4
nf
For some “typical” cases, the estimation of the dimension i0.3
is much better than the lower bound [4], [19], [21], and
0.2
sometimes the estimation is the true dimension [22], [23].
0.1
AQCsderivedfromTheorem4.4aregiveninTABLEII.When
the field size is large we only give partial AQCs with loose 0
0 0.040.080.12 0.16 0.2 0.240.28 0.320.36 0.4 0.440.48
lower bound(LB). relative minimum distance d
Fig.1. Comparisonofdifferent versionsofbinaryGVbound.
V. ASYMPTOTICALLY GOOD BINARY EXPANSION OF
QUANTUM GRS CODES
In [17], Retter showedthatmostbinaryexpansionsof GRS δ ≥H−1(α ),
x 1
codes are asymptotically good.
δ ≥H−1(α ).
Theorem 5.1 ([17, Theorem 1]): For any small ǫ > 0, z 2
thereexistsannsuchthatthebinaryexpansionsofmostGRS Proof: For the asymmetric quantum GRS codes (11), it
codes of any length greater than n satisfy follows from the CSS constructionsLemma 2.2 and Theorem
d k 5.1 that there exist a family Q of AQCs with parameters
H( )>1− −ǫ
n n
[[n,k +k −n,d /d ]]
1 2 z x 2
From [2], we have the following result.
Corollary 5.2: Let C1 and C2 be codes over F2m and where n = mN,k1 = mK1,k2 = mK2, dx ≥ d1, and dz ≥
C2⊥ ⊆ C1. Let αi,i = 1,...,m, be self-dual basis of F2m d2, the corresponding classical codes are D1 = [n,k1,d1]2
over F2, i.e., and D2 =[n,k2,d2]2 which satisfy
Tr(αiαj)=δij. k1 k2
=1−α , =1−α ,
1 2
Let D and D⊥ be codes obtained by the symbolwise binary n n
1 2
expansionofcodesC1andC2⊥inthebasisαi.ThenD2⊥ ⊆D1 δ = d1 ≥H−1(α ),
and D⊥ is the binary dual of C . 1 n 1
2 2
LetN =2m−1,N/2≤K1 ≤K2 ≤N−1beintegers,for δ = d2 ≥H−1(α ).
aGRScodeGRS (a,v)oflengthN.Itfollowsimmediately 2 2
K1 n
that GRS (a,v)⊥ = GRS (a,y) ⊆ GRS (a,y) ⊆
K1 N−K1 K1 Then we have
GRS (a,y),wherey ·v =1/ (α −α )=α ,1≤i≤
K2 i i j6=i j i i k k
N. Then there exists a corresponQding AQC with parameters: R = 1 + 2 −1=1−α −α ,
Q 1 2
n n
[[N,K +K −N,N −K +1/N −K +1]] . (11)
1 2 1 2 2m δ = dx ≥δ ≥H−1(α ),
x 1 1
Denote C = GRS (a,v) and C = GRS (a,y) of n
1 K1 2 K2
length N. Then C⊥ ⊆C . The binary expansions of C and d
2 1 1 δ = z ≥δ ≥H−1(α ).
C2 with respect to a self-dual basis give D2⊥ ⊆D1 of binary z n 2 2
codes with parameters n=mN, k =mK , k =mK .
1 1 2 2
From Theorem 5.1, we can choose suitable y to make sure Theorem5.3isalsoavailableforQECs. Thecomparisonof
D2 is asymptotically good. Because yi ·vi = 1/ j6=i(αj − classical GV bound and two versions of quantum GV bound
αi)=αi,1≤i≤N,thendifferenty givesdiffereQntv. Since is given in Fig. 1.
the binary expansions of most GRS codes are asymptotically
good when n is large, there always exist the correspondingv VI. CONCLUSION AND DISCUSSION
which also give asymptotically good D . In this paper, we have constructed several classes of pure
1
Summing up, we have the following theorem. asymmetricquantumAlternantcodes(AQACs)basedontheir
Theorem 5.3: For a pair of (α ,α ) real numbers satisfy- nestedrelationships.Asaspecialcase,Z-partsofourAQACs
1 2
ing 0 < α ≤ α < 1/2, there exists a family Q of AQCs can attain the classical Gilbert-Varshamov bound when d =
1 2 x
which can attain the asymmetric quantum Gilbert-Varshamov 2. We have identified the nested Goppa codes and computed
bound with the dual distance of some special Goppa codes. When d =
x
R =1−α −α , 2, a famous subclass of Goppa codes with fixed minimum
Q 1 2
distance are convertedto AQCs with only one informationbit [23] ——,“ProofofconjecturesonthetruedimensionofsomebinaryGoppa
loss each. Some AQACs with good parameters are listed. At codes,”Designs,CodesandCryptography,vol.36,no.3,pp.317–325,
2005.
last, Retter’s classical results about the asymptotically good
[24] L.Wang,K.Feng,S.Ling,andC.Xing,“Asymmetricquantumcodes:
binaryexpansionsof GRS codeshave been generalizedto the characterization and constructions,” IEEE Trans. Inf. Theory, vol. 56,
quantum situation. no.6,pp.2938–2945,2010.
TheasymptoticproblemforgeneralAQACsandsymmetric
quantum Alternant codes is still unsolved. How to construct
quantumcodesusingbinaryAlternantcodesespeciallybinary
Goppa codes is an interesting problem which need further
exploring.
REFERENCES
[1] P.Aliferis andJ.Preskill, “Fault-tolerant quantum computation against
biasednoise,”Phys.Rev.A,vol.78,p.052331,2008.
[2] A. Ashikhmin, S. Litsyn, and M. A. Tsfasman, “Asymptotically good
quantum codes,”Phys.Rev.A,vol.63,no.3,p.032311,2001.
[3] S.V.BezzateevandN.A.Shekhunova,“SubclassofbinaryGoppacodes
with minimal distance equal to the design distance,” IEEE Trans. Inf.
Theory,vol.41,no.2,pp.554–555, 1995.
[4] ——, “Asubclass ofbinary Goppacodes withimproved estimation of
thecodedimension,”Designs,CodesandCryptography, vol.14,no.1,
pp.23–38,1998.
[5] P. Brooks and J. Preskill, “Fault-tolerant quantum computation with
asymmetricBacon-Shorcodes,”Phys.Rev.A,vol.87,no.3,p.032310,
2013.
[6] A. R. Calderbank, E. M. Rains, P. Shor, and N. J. Sloane, “Quantum
errorcorrectionviacodesoverGF(4),”IEEETrans.Inf.Theory,vol.44,
no.4,pp.1369–1387,1998.
[7] Z. Evans, A. Stephens, J. Cole, and L. Hollenberg, “Error correc-
tion optimisation in the presence of x/z asymmetry,” arXiv preprint
arXiv:0709.3875, 2007.
[8] M. F. Ezerman, S. Jitman, S. Ling, and D. V. Pasechnik, “CSS-like
constructions ofasymmetric quantum codes,” IEEETrans.Inf.Theory,
vol.59,no.10,pp.6732–6754,2013.
[9] M. F. Ezerman, S. Ling, and P. Sole, “Additive asymmetric quantum
codes,”IEEETrans.Inf.Theory,vol.57,no.8,pp.5536–5550, 2011.
[10] A.G.Fowler, “Analytic asymptoticperformance oftopological codes,”
Phys.Rev.A,vol.87,no.4,p.040301,2013.
[11] M. Grassl, “Bounds on the minimum distance of linear codes and
quantum codes,” Online available at http://www.codetables.de, 2007,
accessed on2013.
[12] A.Ketkar,A.Klappenecker,S.Kumar,andP.K.Sarvepalli,“Nonbinary
stabilizer codes over finite fields,” IEEE Trans. Inf. Theory, vol. 52,
no.11,pp.4892–4914, 2006.
[13] M. Loeloeian and J. Conan, “A [55,16,19] binary Goppa code (cor-
resp.),”IEEETrans.Inf.Theory,vol.30,no.5,pp.773–773, 1984.
[14] G. Maatouk, A. Shokrollahi, and M. Cheraghchi, “Good ensembles of
Goppacodes,”Semester Project2006-2007, 2007.
[15] F.J.MacWilliamsandN.J.A.Sloane,TheTheoryofError-Correcting
Codes. Amsterdam:TheNetherlands: North-Holland, 1981.
[16] Y. Meng Chee, M. Frederic Ezerman, S. Jitman, H. Mao Kiah, and
S.Ling,“PureasymmetricquantumMDScodesfromCSSconstruction:
A complete characterization,” Int. J. Quantum Inf, vol. 11, no. 3, p.
1350027,2013.
[17] C. T. Retter, “The average binary weight-enumerator for a class of
generalized Reed-Solomon codes,” IEEE Trans. Inf. Theory, vol. 37,
no.2,pp.346–349,1991.
[18] P.K.Sarvepalli, A.Klappenecker, andM.Ro¨tteler, “Asymmetricquan-
tumcodes:constructions, boundsandperformance,” inProc.Roy.Soc.
A,vol.465,2009,pp.1645–1672.
[19] N.A.Shekhunova,S.V.Bezzateev,andE.T.Mironchikov,“Asubclass
ofbinaryGoppacodes,”Probl.PeredachiInf,vol.25,no.3,pp.98–102,
1989.
[20] A.Steane,“Multiple-particleinterferenceandquantumerrorcorrection,”
inProc.Roy.Soc.A,vol.452,1996,pp.2551–2577.
[21] P. Ve´ron, “Goppa codes and trace operator,” IEEE Trans. Inf. Theory,
vol.44,no.1,pp.290–294, 1998.
[22] ——, “True dimension of some binary quadratic trace Goppa codes,”
Designs,CodesandCryptography, vol.24,no.1,pp.81–97,2001.