IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 11, NOVEMBER 2015
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Decoding of Dual-Containing Codes From Hermitian Tower and Applications Lingfei Jin and Haibin Kan
Abstract— In this paper, we study the decoding of dual-containing codes from Hermitian tower and applications to quantum codes. The contribution of this paper is threefold. First, we construct the quantum stabilizer codes from the Hermitian tower. Second, we provide a deterministic decoding algorithm with decoding radius that almost achieves the optimal decoding radius, i.e., (1− R)/4, where R is the rate. Last and most importantly, we present a Monte Carlo algorithm with decoding radius roughly equal to (1 − R)/3, which is beyond the optimal decoding radius (1 − R)/4. There are several features in this paper. First of all, we employ a differential for the Hermitian tower. This differential plays a crucial role for decoding. We also extend our decoding by passing to the constant field extension. This constant field extension makes the decoding work perfectly. Index Terms— Decoding radius, places, algebraic geometry codes, rate.
I. I NTRODUCTION HERE are various constructions of quantum stabilizer codes in the literature (see [1], [3]–[5]) since the seminar paper [2]. However, there are very few papers devoted to study of decoding quantum stabilizer codes. In this paper, we study the decoding of dual-containing codes from Hermitian tower and present the quantum stabilizer codes from the Hermitian tower as well as their decoding.
T
A. Known Results and Facts It is well-known that a quantum stabilizer code Q with minimum distance d can correct up to (d − 1)/2 errors (see [2], [6]). On the other hand, for a q-ary [[n, k, d]]quantum stabilizer code Q, it satisfies the Singleton bound, i.e., k +2d n +2. Thus, one can correct at most (n −k −1)/4 errors, i.e., the decoding radius is at most (1 − R)/4, where R is the rate k/n. The question is how to design an efficient algorithm to decode about (d − 1)/2 errors for a quantum stabilizer code Q with minimum distance d. So far, not many results are known in this direction. Manuscript received December 26, 2014; revised August 10, 2015; accepted August 14, 2015. Date of publication September 1, 2015; date of current version October 16, 2015. L. Jin was supported in part by the Shanghai Sailing Program under Grant 15YF1401200. H. Kan was supported in part by the National Natural Science Foundation of China under Grant 61170208 and in part by the Shanghai Key Program of Basic Research under Grant 12JC1401400. L. Jin is with the Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China, and also with the State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China (e-mail:
[email protected]). H. Kan is with the Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China (e-mail:
[email protected]). Communicated by C. Xing, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2015.2475269
B. Our Results and Techniques Our contribution is three-fold. First, we give a construction of the quantum stabilizer codes from the Hermitian tower via standard algebraic geometry codes by Goppa. One feature for this construction is that an algebraic geometry code from the Hermitian tower is either Euclidean self-orthogonal or dual-containing. This feature plays a key role in our decoding. We then consider a deterministic decoding algorithm of our quantum stabilizer codes with decoding radius that almost achieves the optimal bound (1 − R)/4, where R is the rate of the quantum code. The algorithm only involves solving an equation system and thus complexity is bounded by O(q 4.5 ), where q is the alphabet size of the quantum code. Last and most importantly, by passing functions to the function field with constant field extension, we are able to design a Monte Carlo algorithm that decodes beyond the optimal decoding radius (1 − R)/4. C. Organization In Section 2, we introduce some basic results on the Hermitian tower and function fields. Section 3 presents the construction of quantum stabilizer codes via standard algebraic geometry codes by Goppa based on the Hermitian tower. Two decoding algorithms are given in the last section. II. P RELIMINARIES ON THE H ERMITIAN T OWER The Hermitian tower was first studied in [9]. The reader may refer to [9] for the detailed background on the Hermitian function tower, and Stichtenoth’s book [10] for general background on algebraic function fields. Let r be a prime power and let q = r 2 . Let Fq be the finite field with q elements. The Hermitian tower is defined by the following recursive equations , i = 1, 2, . . . , − 1. z ri+1 + z i+1 = z r+1 i Put H = Fq (z 1 , z 2 , . . . , z ) for 2. We will assume that r 2. A. Rational Places The function field H has r +1 + 1 rational places. One of these is the “point at infinity” which is the unique pole P∞ of z 1 (and is fully ramified). The other r +1 come from the rational places lying over the unique zero of z 1 − α for each α ∈ Fq . Note that for every α ∈ Fq , the zero of z 1 − α splits completely in H. Hence, there are r −1 rational places lying
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 11, NOVEMBER 2015
over the zero of z 1 − α. In fact, the “finite” rational places of H are given by -tuples (α1 , α2 , . . . , α ) ∈ Fq satisfying r αi+1 + αi+1 = αir+1 for i = 1, 2, . . . , − 1. For each value of α ∈ Fq , there are precisely r solutions β ∈ Fq satisfying β r +β = αr+1 , so the number of such -tuples is r +1 (q = r 2 choices for α1 , and then r choices for each successive αi , 2 i ).
splits completely. Thus, one has f (z 1 )(P) = f (α) for any z 1 −α 1 −α) function f (z 1 ) ∈ Fq (z 1 ). Write ω := d(z z 1 −α × z q −z 1 . Then, 1 the residue of ω at P is given by z 1 − α res P (ω) = q (5) z 1 =α = 1. z1 − z1
B. Riemann-Roch Spaces
A. Algebraic Geometry Codes
For a place P of H, we denote by ν P the discrete valuation of P. For an integer m, we consider the Riemann-Roch space defined by
Let us introduce algebraic geometry codes based on the Hermitian tower given in the previous section. For the general results on algebraic geometry codes and their duals, the reader may refer to Stichtenoth’s book [10]. For 0 < m < N + 2g − 2, denote by C(m) the q-ary linear code defined by
L(m P∞ ) := {h ∈ H \ {0} : ν P∞ (h) −m} ∪ {0}. Then the dimension dimFq L(m P∞ ) is at least m − g + 1, where g is the genus of H . Furthermore, dimFq L(m P∞ ) = m − g + 1 if m 2g − 1. A basis over Fq of L(m P∞ ) can be explicitly constructed as follows {z 11 · · · z : ( j1 , . . . , j ) ∈ Z0 , j
j
ji r −i (r + 1)i−1 m,
i=1
ji ≤ r − 1, i = 2, 3, . . . , }.
(1)
For any integer e 2, we consider the field extension Fq e and the constant field extension Fq e · H /H. Define a Riemann-Roch space in Fq e · H by Le (m P∞ ) := {h ∈ Fq e · H \ {0} : ν P∞ (h) −m} ∪ {0}. Then Le (m P∞ ) is an F -vector space. Furthermore, by [8] dimFq e Le (m P∞ ) = dimFq Le (m P∞ ) and Le (m P∞ ) has an Fq e -basis in L(m P∞ ). qe
C. Genus The genus g of the function field H is given by −1 1 1 i−1 r 1+ − (r + 1)−1 + 1 g = 2 r i=1 r r i−1 1 r 2 2 r i r i−1 i=1
(2)
i=1
where the last step used r 2. D. A Differential To construct our code with good decodibility, the differential introduced here plays a very important role. Consider the differential dz 1 ω := q . (3) z1 − z1 We label all “finite” points of H by {P1 , P2 , . . . , PN }, where N = r +1 . Then, the divisor associated with ω is div(ω) = (2g − 2 + N)P∞ −
N
Pi .
(4)
i=1
Let P be a rational place of H lying over the zero of z 1 − α. Then z 1 − α is a local parameter of P since the zero of z 1 − α
III. Q UANTUM C ODES F ROM THE H ERMITIAN T OWER
C(m) := {( f (P1 ), . . . , f (PN )) : f ∈ L(m P∞ )}.
(6)
The dimension of the code is m−g+1 if 2g −1 m N −1. The designed distance of the code is N − m. The Euclidean dual code is given by C(m)⊥ E :=
(res P1 (η), . . . , res PN (η)) : η ∈ m P∞ −
N
Pi
.
(7)
i=1
By using the differential ω and [10, Proposition II.2.10], we have that C(m)⊥ E = C(N + 2g − 2 − m) since div(ω) + D − m P∞ = (N + 2g − 2 − m)P∞ . Next we introduce symplectic inner product. For two vectors u = (u1 |u2 ), v = (v1 |v2 ) ∈ F2n q , the symplectic inner product of u and v is defined by u, v S := u1 , v2 E − u2 , v1 E , where ·, · E is the usual Euclidean inner product, i.e., the dot product. Two vectors u, v ∈ F2n q are called symplectic orthogonal if u, v S = 0. For an Fq -linear code C ⊆ F2n q , the symplectic dual, denoted by C ⊥ S , is defined to be the set of vectors of F2n q that are symplectic orthogonal to all codewords in C. It is clear that dimFq (C) + dimFq (C ⊥ S ) = 2n. An Fq -linear code C is called symplectic self-orthogonal (or symplectic dual-containing, respectively) if C ⊆ C ⊥ S (or C ⊇ C ⊥ S , respectively). From the codes C(m), one can construct symplectic dual-containing codes as follows. Theorem 1: For any integers m 1 m 2 in the interval [2g − 1, N − 1], the code C(m 2 ) × C(N + 2g − 2 − m 1 ) = {(a|b) : a ∈ C(m 2 ); b ∈ C(N + 2g − 2 − m 1 )}
(8)
is symplectic dual-containing with dimension equal to N + m 2 − m 1. Proof: First of all, the dimension of C(m 2 ) × C(N + 2g − 2 − m 1 ) is the sum of dimensions of C(m 2 ) and C(N + 2g − 2 − m 1), i.e., m 2 − g + 1 + N + 2g − 2 − m 1 − g + 1 = N + m 2 − m 1. It is clear that C(m 2 ) × C(N + 2g − 2 − m 1 ) contains the code C(m 1 ) × C(N + 2g − 2 − m 2 ). Thus, it is sufficient
JIN AND KAN: DECODING OF DUAL-CONTAINING CODES FROM HERMITIAN TOWER AND APPLICATIONS
to show that the symplectic dual of C(m 2 ) × C(N + 2g − 2 − m 1 ) is exactly the code C(m 1 ) × C(N + 2g − 2 − m 2 ). To prove it, we note that the sum of dimensions of C(m 2 ) × C(N + 2g − 2 − m 1 ) and C(m 1 ) × C(N + 2g − 2 − m 2 ) is N + m 2 − m 1 + N + m 1 − m 2 = 2N. Thus, it remains to prove that C(m 2 ) × C(N + 2g − 2 − m 1) and C(m 1 ) × C(N + 2g − 2 − m 2 ) are symplectic orthogonal. This is clear since C(N + 2g − 2 − m 2 ) is the Euclidean dual of C(m 2 ) and C(N + 2g − 2 − m 1 ) is the Euclidean dual of C(m 1 ). The proof is completed. B. Quantum Stabilizer Codes First we give a brief introduction to quantum stabilizer codes and their connection with classical symplectic orthogonal codes. This is necessary for our decoding in the next section. The reader may refer to [2] and [6] for the details of quantum stabilizer codes. To simplify our presentation in this subsection, we consider only binary quantum stabilizer codes. Let us briefly describe the background on quantum stabilizer codes and their decoding. The reader may refer to [2], [6], and [7] for the details on decoding of quantum stabilizer codes. The state space of one qubit is actually a 2-dimensional complex space with a basis {|0, |1}. We can simply denote this state space of one qubit by C2 . Let G1 = {±I, ±i I, ±X, ±i X, ±Y, ±i Y, ±Z , ±i Z } be the Pauli group acting on C2 , where i is the imaginary unit, I is the 2 × 2 identity matrix and 0 1 1 0 X= , Z= , Y = i X Z. 1 0 0 −1 The tensor product (C2 )⊗n is called the state space of n qubits. Let Gn denote the Pauli group acting on (C2 )⊗n , i.e., Gn = {i m σ1 ⊗ σ2 ⊗ · · · ⊗ σn : m ∈ {0, 1, 2, 3}, σ j ∈ {I, X, Y, Z }},
(9)
where the action of an element of Gn on a state of n qubits is through the componentwise action of σi on C2 . Quantum stabilizer codes are defined in the following manner. Let S be a subgroup of Gn such that −I ⊗ I ⊗ · · · ⊗ I ∈ S. Then S is a 2-elementary abelian group. Assume that the 2-rank of S is k for some k ∈ [0, n] and S is generated by {g1 , g2 , . . . , gk }. The subgroup S has a fixed subspace Q S of (C2 )⊗n defined by Q S = {v ∈ (C2 )⊗n : g(v) = v for all g ∈ S}. The subspace Q S is called an [[n, n − k]]-quantum stabilizer code and it has dimension 2n−k . The rate of Q S is defined by R := (n − k)/n = 1 − k/n. To connect the quantum stabilizer code Q S with a classical linear code, we define a group epimorphism ψ : Gn → F2n 2 given by ψ(i m σ1 ⊗ σ2 ⊗ · · · ⊗ σn ) = (x 1 , x 2 . . . , x n |z 1 , z 2 . . . , z n ) = (x|z),
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where x j , z j are elements of F2 that are determined as below σj xj zj
I 0 0
X 1 0
Y 1 1
Z 0 1
Furthermore, we define a 2n × 2n matrix over F2 On I n = , I n On where On is the n × n zero matrix and In is the n × n identity matrix. Then it is easy to see that, for two elements g, h ∈ Gn , gh = hg if and only if ψ(g) ψ(h)T = 0, i.e., ψ(g) and ψ(h) are symplectic orthogonal. Through the k generators {g1, g2 , . . . , gk }, we define a k × 2n matrix over F2 ⎛ ⎞ ψ(g1 ) ⎜ · ⎟ ⎜ ⎟ ⎟ H =⎜ ⎜ · ⎟. ⎝ · ⎠ ψ(gk ) It is easy to see that H has rank k. Since S is abelian, we have H H T = 0. Thus, the binary code C with H as a parity-check matrix is symplectic dual-containing. It is a binary [2n, 2n − k]-linear code. For a vector (u|v) ∈ F2n q , the symplectic weight is defined by wt S (u|v) = |{1 i n : (u i , v i ) = (0, 0)}|. Then the minimum distance of Q S is given by min{wt S (u|v) : (u|v) ∈ C \ C ⊥ S }. We will see in the next section that a quantum code Q S with minimum distance d can uniquely correct (d −1)/2 errors. Returning to our symplectic self-orthogonal code C(m 1 ) × C(N + 2g − 2 − m 2 ), we obtain a q-ary [[N, N − (N − m 2 + m 1 )]] = [[N, m 2 − m 1 ]] quantum stabilizer code, denoted by Q(m 1 , m 2 ). It is clear that the minimum distance of Q(m 1 , m 2 ) is at least the minimum distance of C(m 2 ) × C(N +2g −2−m 1) which is min{N −m 2 , m 1 −2g +2}. It is called the designed minimum distance of Q(m 1 , m 2 ). If we set d := N −m 2 = m 1 −2g +2, then m 2 −m 1 = N −2d −2g +2. Thus, we get the following result. Theorem 2: For any integer d in the interval [1, N/2 − g + 1], one can construct a q-ary quantum stabilizer code Q(d) = Q(m 1 , m 2 ) with d = N − m 2 = m 1 − 2g + 2 from the symplectic dual-containing code in Theorem 1 with parameters [[N, N − 2d − 2g + 2, d]]. IV. D ECODING OF Q UANTUM C ODES F ROM THE H ERMITIAN T OWER A. General Framework of Decoding Quantum Stabilizer Codes Consider an [[n, n−k]]-quantum stabilizer code Q S defined above. Assume that a state of n − k quibits is encoded into a coded state |α of n qubits. Let ρ = |αα| be the channel input and let Eρ E † be the channel output with error E ∈ Gn , where E † denotes the Hermitian conjugation of E. By computing the syndrome measurements of the received state, one can determine the binary syndrome s which is equal to ψ(E) H T (see [2], [6]). To decode, i.e., recover the channel input ρ, it is sufficient to determine the error E. On the other hand,
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 11, NOVEMBER 2015
finding E can be reduced to finding ψ(E) (note that the scalar i m does not affect error). Thus, we turn the problem of decoding quantum stabilizer codes into decoding of C (to see this, we notice that H is a parity-check matrix of C). Assume that E has at most t errors, i.e., in the representation E = i m σ1 ⊗ σ2 ⊗ · · · ⊗ σn , there are at most t indices j such that σ j = I . Thus, the corresponding binary vector ψ(E) has symplectic weight at most t. This implies that we have to find T an error e = ψ(E) ∈ F2n 2 such that wt S (e) t and e H = s. T Note that H H = 0. Thus, finding e is in fact equivalent to finding a coset e + C ⊥ S such that wt S (e) t and e H T = s. If the minimum distance of Q S is d, then we claim that there is at most one coset satisfying wt S (e) (d − 1)/2 and e H T = s. Suppose that there are two cosets e1 + C ⊥ S and e2 + C ⊥ S such that wt S (ei ) (d − 1)/2 and ei H T = s for i = 1, 2. Then (e1 − e2 ) H T = 0, i.e., e1 − e2 ∈ C ⊥ S . On the other hand, we have wt S (e1 −e2 ) wt S (e1 )+wt S (e2 ) d −1, we must have e1 − e2 ∈ C ⊥ S . This implies that one can uniquely correct (d − 1)/2 errors. The above decoding of Q S is sightly different from decoding classical dual-containing code C ⊥ S . To decode Q S , we have to find all cosets e + C ⊥ S such that wt S (e) t and e H T = s, while decoding C ⊥ S is to find all words e ∈ F2n q such that wt S (e) t and e H T = s. Thus, decoding C is stronger than decoding Q S in the sense that one can find the quantum error E as long as one can find the classical error e. To decode Q S , we can simply find the coset e + C ⊥ S such that wt S (e) t and e H T = s. It is clear that if we can T find the word e ∈ F2n q such that wt S (e) t and e H = s, then the coset e + C satisfying wt S (e) t and e H T = s is found. Thus, we conclude that decoding the quantum code Q S is turned into decoding of the corresponding symplectic dual-containing code C. In the following section, we will consider decoding of the quantum code Q(d) given in Theorem 2. B. Deterministic Decoding First we give a deterministic decoding algorithm for the quantum stabilizer code Q(d) given in Theorem 2 with decoding radius (d − 1)/2N − g /N, i.e., one can correct (d − 1)/2 − g errors with Q S . Note that d = N − m 2 = m 1 −2g +2. Thus, we have C(m 2 ) = C(N +2g −2 −m 1) = C(N − d). Fix an element α in Fq 2 \ Fq . Assume that we receive a word (u1 |u2 ) ∈ F2N q . The purpose is to find a unique pair ( f 1 | f 2 ) with f 1 , f2 ∈ L((N − d)P∞ ) such that wt S ((c f1 |c f2 ), (u1 |u2 )) (d − 1)/2 − g, where c fi stands for the vector ( fi (P1 ), . . . , f i (PN )). This is equivalent to finding the function f = f 1 + α f2 with f 1 , f 2 ∈ L((N − d)P∞ ) and wt H (c f , u) (d − 1)/2 − g , where wt H stands for the Hamming weight and u is u1 + αu2 . Let h 0 , h 1 , . . . , h N−g be an Fq 2 -basis of L2 (N P∞ ) such that ν P∞ (h 0 ) < ν P∞ (h 1 ) < · · · < ν P∞ (h N−g ). Put D = (d − 1)/2. We can find a0,0 , a0,1 . . . , a0,D+N−d , a1,0 , a1,1 . . . , a1,D ∈ Fq 2 such that not all of them are 0 and A0 (Pk ) + A1 (Pk )u k = 0 for allk = 1, 2, . . . , N, D+N−d where u = (u 1 , u 2 , . . . , u N ), A0 = a0,i h i and i=0
D a1,i h i . Note that this is possible since there are A1 = i=0 2D + N −d +2 unknowns and N equations and 2D + N −d + 2 > N. For those Pk with f (Pk ) = u k , we must have A0 (Pk )+ A1 (Pk ) f (Pk ) = 0. Since there are at least N − ((d − 1)/ 2 − g ) points Pk such that A0 (Pk ) + A1 (Pk ) f (Pk ) = 0 and A0 + A1 f ∈ L2 ((D + N − d + g )P∞ ). This forces that A0 + A1 f = 0 as D + N − d + g < N − ((d − 1)/2 − g ). The function f is solved from the identity A0 + A1 f = 0. Note that A1 = 0, otherwise A0 must be 0 as well. The decoding complexity consists of finding a basis of L2 (N P∞ ) and solving a homogenous equations. Thus the total time complexity is O(N 3 ). Summarizing the above decoding algorithm, we obtain the following result. Theorem 3: One can deterministically decode the stabilizer quantum code Q(d) given in Theorem 2 with decoding radius √ at least (1 − R)/4 − 3/2 q, where R is the rate of Q(d). Proof: The code Q(d) is a q-ary [[N, N − 2d − 2g + 2]]quantum code. Thus, the rate of Q(d) is 1 − 2d/N− (2g − 2)/N. Let τ = (d − 1)/(2N) − g /N be the decoding radius given in the algorithm of this subsection. Then we have R + 4τ = 1 −
6 6g − 2 1− √ . N q
The desired result follows. Remark 1: This is optimal. Note that for any q-ary [[n, k, d]]-quantum stabilizer code Q S , it satisfies the Singleton bound, i.e., d (n − k + 2)/2. Thus, the decoding radius of Q S is (d − 1)/2 which is at most (1 − R)/4. On the other hand, when q is sufficiently large compared with , the decoding radius given in Theorem 3 is roughly (1 − R)/4. This means that our algorithm achieves the optimal decoding. C. Monte Carlo Decoding In the previous section, we give an algorithm to decode the quantum code Q(d) with almost optimal decoding radius τ . This algorithm is deterministic, i.e., as long as the number of errors is at most τ N, we can deterministically find the unique error. In this section, we are going to extend the decoding radius beyond (1 − R)/4. Then the algorithm cannot be deterministic because the radius (1 − R)/4 is already optimal for deterministic unique decoding. The idea is to decode probabilistically. In other words, for a received a word, we design a Monte Carlo decoding algorithm with high probability of success. Use the the same notations as in the previous subsection. For simplicity, let us consider the case where q is odd (for even q, we can do similarly). Fix a primitive element γ of Fq and let α be a root of x 2 − γ . Then α belongs to Fq 2 \ Fq and α q−1 = γ (q−1)/2 = −1. Assume that we receive a word (u1 |u2 ) ∈ F2N q with at most 2(d −2)/3−g errors and we want to find a unique pair ( f 1 | f 2 ) with f 1 , f 2 ∈ L((N − d)P∞ ) such that wt S (c f1 |c f2 ), (u1 |u2 )) τ N for some τ . Put f = f 1 + α f2 . Put D = (d − 2)/3. We can find a0,0 , a0,1 . . . , a0,D+N−d , a1,0 , a1,1 . . . , a1,D , a2,0 , a2,1 . . . , a2,D ∈ Fq 2 , such that not
JIN AND KAN: DECODING OF DUAL-CONTAINING CODES FROM HERMITIAN TOWER AND APPLICATIONS
all of them are 0 and A0 (Pk ) + A1 (Pk )u k +
q A2 (Pk )u k
=0 D+N−d
(10)
for all k = 1, 2, . . . , N, where A0 = a0,i h i and i=0 D a1 j,i h i for j = 1, 2. Note that this is possible A j = i=0 since there are 3D + N − d + 3 unknowns and N equations and 3D + N − d + 3 > N. For those Pk with f (Pk ) = u k , we must have A0 (Pk ) + A1 (Pk ) f (Pk ) + A2 (Pk ) f (Pk )q = 0.
(11)
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probability theoretically. However, we conducted an experiment and it shows that if we randomly choose a received word (u1 |u2 ) ∈ F2N q with at most 2(d − 2)/3 − g errors, then with very high probability one can find a nonzero solution of (10) such that (15) has rank 2. Our experiment data using the software Magma is tabulated below. Due to computation limitation, we choose r = 3 and let distance varies from 12 to 18. For each of these distances, we randomly choose 800 received words with e 2(d −2)/3−g errors.
The above equation is equivalent to A0 (Pk ) + A1 (Pk ) f (Pk ) + A2 (Pk )( f 1 + α q f 2 )(Pk ) = 0. (12) Since there are at least N − ((d − 1)/2 − g) points Pk such that (11) is satisfied and A0 + A1 f + A2 ( f 1 + α q f 2 ) ∈ L2 ((D + N − d + g )P∞ ). This forces that A0 + A1 f + A2 ( f 1 + α q f 2 ) = 0.
(13)
as D + N − d + g < N − ((d − 2)/3 − g ). Write Ai = Ai1 + α Ai2 for i = 0, 1, 2. Then Equation (13) can be written into A01 + (A11 + A21 ) f 1 + γ (A12 − A22 ) f 2 + α(A02 + (A12 + A22 ) f 1 + (A11 − A21 ) f 2 ) = 0.
(14)
The above equation splits into a simultaneous equation system A11 + A21 γ (A12 − A22 ) f1 −A01 = . (15) A12 + A22 A11 − A21 f2 −A02 First of all, the coefficient matrix of the above equation system is not zero. Otherwise, A01 = A02 = 0 and this makes all Ai = 0 for i = 0, 1, 2, which is impossible. Thus, the rank of the coefficient matrix is 1 or 2. If the rank is 2, then f 1 and f 2 can be solved uniquely and the unique decoding algorithm is completed. So the problem is to find the probability that the coefficient matrix of (15) has rank 2. In conclusion, we have the following result. Theorem 4: If the coefficient matrix of (15) has rank 2, one can deterministically decode the stabilizer quantum code Q(d) given in Theorem 2 with decoding radius roughly equal to √ (1 − R)/3 − 5/(3 q), where R is the rate of Q(d). Proof: The rate of the code Q(d) is 1 − 2d/N− (2g − 2)/N. Let τ = 2(d − 2)/(3N) − g /N be the decoding radius given in the algorithm of this subsection. Then we have R + 3τ = 1 −
5 1 5g + 1 1− √ − . N q N
The desired result follows. Remark 2: Theorem 5 tells us that as long as the coefficient matrix of (15) has rank 2, one can deterministically decode the quantum stabilizer code Q(d) given in Theorem 2 with √ decoding radius roughly equal to (1 − R)/3 − 5/(3 q). This radius approaches (1 − R)/3 when q is sufficiently large compared with . Thus, we have a Monte-Carlo algorithm to decode beyond the optimal deterministic decoding radius. Now the question is what is the probability that the coefficient matrix of (15) has rank 2. We could not find this
In view of the above remark, we conclude our paper by the following result. Theorem 5: Assume that q is sufficiently large compared with . With high probability, one can design a Monte Carlo algorithm to decode the quantum stabilizer code Q(d) given in Theorem 2 with decoding radius roughly equal to (1− R)/3, where R is the rate of Q(d). Furthermore, the time complexity of the algorithm is O(q 4.5 ). R EFERENCES [1] A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 3065–3072, Nov. 2001. [2] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1369–1387, Jul. 1998. [3] L. Jin, S. Ling, J. Luo, and C. Xing, “Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4735–4740, Sep. 2010. [4] L. Jin, “Quantum stabilizer codes from maximal curves,” IEEE Trans. Inf. Theory, vol. 60, no. 1, pp. 313–316, Jan. 2014. [5] A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 4892–4914, Nov. 2006. [6] K.-Y. Kuo and C.-C. Lu. (Jun. 2013). “On the hardnesses of several quantum decoding problems.” [Online]. Available: http://arxiv.org/abs/ 1306.5173 [7] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge, U.K.: Cambridge Univ. Press, 2000. [8] J. H. Silverman, The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics), vol. 106. New York, NY, USA: Springer-Verlag, 2009. [9] B.-Z. Shen, “A Justesen construction of binary concatenated codes that asymptotically meet the Zyablov bound for low rate,” IEEE Trans. Inf. Theory, vol. 39, no. 1, pp. 239–242, Jan. 1993. [10] H. Stichtenoth, Algebraic Function Fields and Codes (Universitext). Berlin, Germany: Springer-Verlag, 1993. Lingfei Jin received her Ph.D degree in mathematics from Nanyang Technological University, Singapore in 2013. She is currently a Pre-tenure associate professor at the School of Computer Science, Fudan University, china. Her research interests include classical and quantum coding. Haibin Kan received the Ph.D. degree from Fudan University, Shanghai, China, 1999. After receiving the Ph.D. degree, he became a faculty of Fudan University. From June 2002 to February 2006, he was with the Japan Advanced Institute of Science and Technology as an assistant professor. He went back Fudan University in February 2006, where he is currently a full professor. His research topics include coding theory, Cryptography, and Computation Complexity.
