QUANTUM CODES FROM AFFINE VARIETY CODES AND THEIR ...

QUANTUM CODES FROM AFFINE VARIETY CODES AND THEIR SUBFIELD-SUBCODES

arXiv:1403.4060v1 [cs.IT] 17 Mar 2014

C. GALINDO, F. HERNANDO Abstract. We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction. With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum GilbertVarshamov bound given by Feng and Ma.

Introduction The interest of using computers based on the principles of quantum mechanics is clearly represented by Shor’s algorithm [25] for factoring integers since it could be used to break some cryptographical systems used nowadays. The model for a quantum computer is an isolated from its environment closed system. However quantum mechanical systems are very sensitive to disturbances and, at the beginning, the fact that arbitrary quantum states cannot be replicated seemed to suggest that error correction could not be used on them [28]. However, this is not true as it was showed in [26]. The initially considered and most studied family of quantum error correcting codes is that of the so-called binary stabilizer codes. In this case, the state of the basic unit of quantum information is the quantum bit or qubit, represented by a normalized linear combination |qi = α|0i + |1i, where α, β ∈ C, C being the complex numbers and |α|2 + |β|2 = 1. Notice that |0i and |1i represent the ground and excited states and that a system can be simultaneously in different states. The values |α|2 and |β|2 represent the probability for the states 0 and 1 when extracting classical information. Quantum registers of length n are normalized complex linear combinations of the 2n mutually orthogonal basis states represented with a tensor product |b1 i ⊗ |b2 i ⊗ · · · ⊗ |bn i, where bi ∈ {0, 1}. For these codes, it is enough to correct certain elementary errors which are modelled by the so-called identity and Pauli matrices, usually denoted σx , σy and σz [19]. There exists a lot of papers which study binary stabilizer codes. For simplicity, we only cite [5, 13] as seminal works. An interesting way to get this type of codes is based in considering linear binary block codes C1 and C2 such that C2⊥ ⊆ C1 , C2⊥ meaning dual code of C2 . In this paper, we are interested in more general stabilizer codes defined over finite fields and constructed by using a class of linear error correcting codes called affine variety ones. For us q = pr will be a positive integer power of a prime number p and Cq the q-dimensional complex vector space representing the states of a quantum mechanical system. |xi will be the vectors of a distinguished orthonormal basis of Cq , where x ∈ Fq , Fq being the finite field with q elements. Then, by definition a quantum error-correcting code will be a n s-dimensional subspace of Cq = Cq ⊗ Cq ⊗ · · · ⊗ Cq . We also consider a basis ǫn of the vector space of complex square matrices of size q n to represent a discrete set of errors. Let Supported by Spain Ministry of Economy MTM2012-36917-C03-03 and University Jaume I: PB11B2012-04. 1

2

C. GALINDO, F. HERNANDO

a, b ∈ Fq be, then the unitary operators on Cq , X(a)|xi = |x + ai and Z(b)|xi = β tr(bx) |xi, where tr : Fq → Fp is the trace map and β a primitive pth root of unity, allow us to consider the set ε = {X(a)Z(b)|a, b ∈ Fq } of error operators. For a = (a1 , a2 , . . . , an ) ∈ Fnq , define X(a) := X(a1 )⊗X(a2 )⊗· · · X(an ) and Z(a) analogously and write εn = {X(a)Z(b)|a, b ∈ n Fnq } a nice error basis on the complex space Cq . Then, if Gn denotes the group spanned by εn , it holds that Gn = {β tr(c) X(a)Z(b)|a, b ∈ Fnq and c ∈ Fp } which is a finite group of order pq 2n , usually called the error group attached with εn . n A stabilizer code C is a non-vanishing subspace of Cq such that n

C = ∩H∈∆ {v ∈ Cq |Hv = v}, n

for some subgroup ∆ of Gn . Notice that if V is the lattice of subspaces of Cq and G that of subgroups of Gn (both lattices with respect to sum and intersection), stabilizer codes are elements in V than remain unchanged when mapped to G and back. A stabilizer code C with stabilizer ∆ has minimum distance d if, and only if, it can detect all errors in Gn with weight less than d, where the weight w(E) of E = β tr(c) X(a)Z(b) is the number of nonidentity tensor components, and, in addition, some error of weight d cannot be detected. We will say that a code C as above is an ((n, s, d))q -code. When the code is an ((n, q k , d))q -code, we will simply say that it is an [[n, k, d]]q -code. Note that if a quantum error-correcting code is able to detect a set E of errors, then it detects all errors in the linear span hEi of E. Codes with distance d can correct errors of weight less than or equal to ⌊(d − 1)/2⌋, where ⌊·⌋, means the round-down function. A code C as above is said to be pure to t whenever the group ∆ does not contain non-scalar matrices whose weight is less than t and C is called pure whenever it is pure to its minimum distance. As in the binary case, classical codes can be used to provide quantum codes. For a start, P2 ¯i yi , where one can consider the Hermitian product defined over Fnq2 as hx, yi1 := qi=1 x x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ) ∈ Fnq2 and for z ∈ Fq , z¯ means its conjugate. The dual space of a linear code over Fnq2 with respect to the inner product h·i1 is denoted by C ⊥1 . A first link between quantum and classical error-correcting codes is given by the following result which can be found in [18, Corollary 19] (see also [5, 4, 1]). Proposition 1. Assume the existence of an [n, k, d] linear code E over Fq2 and such that E ⊥1 ⊆ E. Then, there exists an [[n, 2k − n, ≥ d]]q -quantum code over Fq which is pure to d. As customary in classical coding theory, we prefer to use the standard Euclidean inner product to provide quantum codes. The symbol ⊥ will be used to represent dual spaces with respect to that inner product. So, in this paper we will use the so called CSS code construction after the papers [6] and [27]. We summarize the idea in the two next results which can be found as Lemma 20 and Corollary 21 in [18]. Theorem 1. Let C1 and C2 two linear error-correcting block codes with parameters [n, k1 , d1 ] and [n, k2 , d2 ] over the field Fq and such that C2⊥ ⊆ C1 . Then, there exists an [[n, k1 + k2 − n, d]]q stabilizer code with minimum distance n o d = min wt(c)|c ∈ (C1 \ C2⊥ ) ∪ (C2 \ C1⊥ ) , which is pure to min{d1 , d2 }.

QUANTUM CODES FROM AFFINE VARIETY CODES

3

For an explanation of how to get the codes, we refer to [18, Theorem 13] where the additive code C1⊥ × C2⊥ is used. Corollary 1. Let C be a linear [n, k, d] error-correcting block code over Fq such that C ⊥ ⊆ C. Then, there exists an [[n, 2k − n, d]]q stabilizer code which is pure to d. Using either the CSS code construction or other techniques, a number of families of quantum codes has been derived. Many quantum codes coming from the most classical ones can be seen in [18] and, among other techniques, one can also use twisted BCH codes to do it [4]. It is worthwhile to mention that several bounds for the parameters of an [[n, k, d]]q -code are known. One of the most popular bounds is the quantum singleton bound, n ≥ k + 2d − 2. When this bound is attained, one gets MDS quantum codes. It is known that the length n of any nontrivial MDS stabilizer code satisfies 4 ≤ n ≤ 2q 2 −2 and for its minimum distance, one gets max{3, n−q 2 +2} ≤ d ≤ min{n−1, q 2 } [23, 2, 14, 9, 18]. This bound and others as the Hamming bound give necessary conditions for the existence of quantum codes. With the same philosophy of the classical Gilbert-Varshamov bound, a sufficient condition for the existence of pure stabilizer codes is given by Feng and Ma in [9]. That condition is d−1 X n q n−k+2 , (q 2 − 1)i−1 (1) < 2 q −1 i i=1

where k, d ≥ 2 and n ≡ k (mod 2). It is also worth to mention that affine variety codes are, in some cases, a particular case of multivariable abelian codes. These codes have been studied in [21], where for the case q = 4, self-orthogonal and self-dual codes with respect to the trace inner product are characterized. In this paper, we use affine variety codes to get stabilizer codes by using the CSS code construction. We use Corollary 1 for this purpose. The direct application of this procedure yields, in general, codes whose supporting field can be large. Notwithstanding, our families of codes admit subfield-subcodes, which are well suited for providing stabilizer codes with good parameters. Our goal consists of consider suitable affine variety codes over a field Fpr whose attached polynomials produce stabilizer codes over Fps for some s that divides r. This gives rise to better parameters since one can use codes over large fields to provide stabilizer codes over smaller finite fields. In this way, we get several examples of stabilizer codes improving some of those ones given in [8]. In addition, using our procedure we are able to obtain a pure stabilizer code that does not satisfy the analogue to the classical Gilbert-Varshamov sufficient condition (1). Section 1 of the paper introduces affine variety codes over a finite field Fpr , where p is a prime number and r a positive integer. The codes are given by evaluating the polynomials in several variables belonging to vector spaces generated by monomials whose exponents are in some subsets U of the cartesian product {0, . . . , N1 − 1} × {0, . . . , N2 − 1} × · · · × {0, . . . , Nm − 1}, where the integer Ni divides pr − 1 for all i, 1 ≤ i ≤ m. We also devote Section 1 to show that good choices of sets U give rise to stabilizer codes over Fpr . However, the supporting field of these codes is, in general, large. To pass over this difficulty, in Section 2, we introduce and study subfield-subcodes of our affine variety codes providing, in Theorem 3, a basis for the vector space of polynomials associated with affine variety codes but evaluating to a subfield Fps of Fpr . Note that similar techniques were already used in [15]

4


and [16] for different families of codes. Dual codes of the mentioned subfield-subcodes are treated in Section 3. They are useful for proving our main result, Theorem 6, where we give conditions on the above sets U for obtaining good stabilizer codes and parameters for them. Finally, Section 4 shows parameters of the above mentioned stabilizer codes constructed with our procedure which improve some of the quantum codes included in [8]. Among them, it can be found the one exceeding the Gilbert-Varshamov bound. 1. Affine variety codes We devote this section to introduce the class of classical error-correcting codes that we will use to yield stabilizer codes via the CSS code construction. In this section, we will consider a finite field Fpr where r is a positive integer and in the following sections we will use subfields Fps of Fpr where s|r. Set Fpr [X1 , X2 , . . . , Xm ] the ring of polynomials with m variables over the field Fpr and pick m positive integer numbers N1 , N2 , . . . , Nm such that Ni | pr − 1 for i = 1, 2, . . . , m. Consider the ideal IN1 ,N2 ,...,Nm of Fpr [X1 , X2 , . . . , Xm ] spanned by the set of polynomials Nm − 1}, which will be denoted by I for simplicity. {X1N1 − 1, X2N2 − 1, . . . , Xm Let Z(I) be the set of zeroes of the ideal I which belong to the simultaneous splitting field of the polynomials X1N1 − 1, X2N2 − 1, . . . , X Nm − 1. Since we have chosen Ni | pr − 1 for i = 1, 2, . . . , m, that splitting field is Fpr . The cardinality of Z(I), card(Z(I)), satisfies card(Z(I)) = N1 N2 · · · Nm := n, and we set Z(I) = {P1 , P2 , . . . , Pn }. Now, write R := Fpr [X1 , X2 , . . . , Xm ]/I and consider the evaluation map ev : R → Fnpr which maps any function f ∈ R to ev(f ) = (f (P1 ), f (P2 ), . . . , f (Pn )), where, here, f also denotes any representative of its class. The family of codes we are going to define will be determined by certain linear subspaces of R. Since the polynomials XiNi − 1 have no multiple roots, our codes are of semi-simple type [22]. It is worthwhile to mention that G = {X1N1 − 1, X2N2 − 1, . . . , X Nm − 1} is a Gr¨ obner basis of the ideal I with respect to (say, some fixed) lexicographical ordering. Hence, we can choose a canonical representative of each class given by a polynomial f which will be its reduction module G. Frequently, we will use the same notation for expressing a class in R and its canonical representative. The following result will be useful. Proposition 2. The above introduced evaluation map, ev, is an isomorphism of Fpr -vector spaces. Proof. R and Fnpr have the same cardinality, therefore it is enough to show that the evaluation map is surjective. Let (β1 , . . . , βn ) be an element in Fnpr and assume that Pi = (pi1 , . . . , pin ). Consider the following polynomials, Qi (X1 , . . . , Xn ) =

n Y

n Y

(Xj − pkj ),

j=1 k=1,k6=i

where 1 ≤ i ≤ n. It turns out that Qi (X1 , . . . , Xn )/Qi (Pi ) evaluates to zero at every point of Z(I) except at Pi where evaluates to one. Therefore, Q(X1 , . . . , Xn ) =

n X i=1

evaluates to (β1 , . . . , βn ).

βi

Qi (X1 , . . . , Xn ) Qi (Pi )


5

Notice that ev is also an isomorphism of Fp -algebras, where the product in Fnpr is the coordinate-wise multiplication [17]. Throughout this paper, H will denote be the hypercube H := {0, . . . , N1 − 1} × {0, . . . , N2 − 1} × · · · × {0, . . . , Nm − 1} and our codes will be attached with suitable subsets U of the hypercube H. U gives rise to um |(u , u , . . . , u ) ∈ U } of monomials in F r [X , X , . . . , X ] that the set {X1u1 X2u2 · · · Xm 1 2 m p 1 2 m provide elements in R which generate a vector space over Fpr that will be denoted FU pr . In H particular R = Fpr as vector spaces. Now, we introduce the concept of affine variety code. For a more general definition see [10] or [11]. Definition 1. Let U be a set as above. The image of the restriction of the evaluation map ev to FU pr is denoted by CU and constitutes the affine variety code associated to U over Fpr . As a consequence of the previous proposition, it holds that the restriction map ev : n FU pr → Fpr is injective and therefore dim(CU ) = card(U ). ˆ ∈ H the element u ˆ = Consider an element u = (u1 , u2 , . . . , un ) ∈ H and define u (ˆ u1 , u ˆ2 , . . . , u ˆn ) defined by u î = 0 if ui = 0 and u î = Ni − ui otherwise. Next result generalizes a close one for toric codes. It can be found in [3] and [24]. Proposition 3. Let CU be the affine variety code defined by U ⊂ H, then CU⊥ is the affine variety code defined by U ⊥ = H \ {ˆ u|u ∈ U } . The following result will be useful for obtaining stabilizer (quantum) codes. Theorem 2. With the above notations, the inclusion CU ⊂ CU⊥ happens if, and only if, ˆ∈ u / U for all u ∈ U . ⊥

⊥ Proof. Proposition 3 shows that the code CU⊥ is the evaluation by ev of FU pr , where U = ˆ ∈ H \ {ˆ u|u ∈ U }, hence U ⊂ U ⊥ if and only if u / U for all u ∈ U , what concludes the proof.

By using the CSS code construction, one can deduce the following Corollary 2. Let N1 , N2 , · · · , Nm positive integers such that Ni divides pr −1 for all index i and some prime number p and positive integer r. Let U be some nonempty subset of ˆ ∈ the hypercube H satisfying that u / U for all u ∈ U . Then, from the affine variety code CU , a stabilizer code can be obtained. The parameters for this code are [[n, k, d]]pr , where n = N1 N2 · · · Nm , k = n − 2 card(U ) and d = d(CU⊥ ). Example 1. With the above notation, set R the ring F4 [x, y, z]/ < x3 − 1, y 3 − 1, z 3 − 1 >. This means that the corresponding codes will have length n = 27. Our set U will be U = {(0, 0, 1), (1, 1, 0), (0, 1, 1), (1, 1, 1), (1, 2, 0)(1, 0, 2), (0, 1, 2), (1, 1, 2), (1, 2, 1), (2, 1, 1)} . Since U satisfies the conditions in Corollary 2, U yields a [[27, 7, 6]]4 stabilizer code. The distance has been computed with the computational algebra system Magma [20]. Example 2. Let us show other example. Consider the ring R = F8 [x, y]/ < x7 −1, y 7 −1 > which produces codes of length n = 49. Here, U = {(1, 3), (4, 4), (1, 6), (5, 0), (1, 2)} , that gives a [[49, 39, 4]]8 stabilizer code. The main weakness of this construction is that the supporting field of the obtained codes is too big. Next section considers subfield-subcodes in order to improve it.

6


2. Subfield-Subcodes of Affine variety Codes As above, we write R = Fpr [X1 , X2 , . . . , Xm ]/I, where I is the ideal of the polynomial ring Fpr [X1 , X2 , . . . , Xm ] spanned by the set of polynomials {X1N1 − 1, X2N2 − 1, . . . , X Nm − 1}. In addition, consider a positive integer s which divides r. We say that an element f ∈ R evaluates to Fps whenever f (α) ∈ Fps for all α ∈ Z(I). Now, define T : R → R s

the map given by T (f ) = f + f p + · · · + f p s

r ps( s −1)

s( r s −1)

. Also, set trsr : Fpr → Fps defined

trsr (x) = x + xp + · · · + x and extend it to tr : Fnpr → Fnps by applying trsr to each coordinate. Afterwards, we will need the following results. Proposition 4. With the above notations and for any f ∈ R, it holds: (1) T (af ) = aT (f ), for all element a ∈ Fps . s s (2) T (f )p = T (f p ) = T (f ). (3) ev(T (f )) = tr(ev(f )). (4) ev(T (f )) = 0 happens if, and only if, T (f ) = 0. Proof. Items (1), (2), and (3) follow from the definition of T , properties of finite fields, and the fact that we are working modulo I. Item (4) follows from the fact that the map ev is an isomorphism. Proposition 5. Let g ∈ R. Then, the following statements are equivalent. (1) g = T (h) for some h ∈ R. s (2) gp = g. (3) g evaluates to Fps . Proof. For a start, suppose that for some h ∈ R, g = T (h). Then s

s

gp = T (h)p = T (h) = g, s

where the second equality follows from Proposition 4. If g p = g, then for any α ∈ Fm pr , s , g(α) ∈ F g(α)p = g(α) and so g(α) ∈ Fps . Lastly, suppose that for every α ∈ Fm ps . pr n Since tr is surjective, we can consider y ∈ Fpr such that tr(y) = ev(g). Now, consider the class h of a interpolating polynomial satisfying ev(h) = y, then, ev(T (h)) = tr(ev(h)) = ev(g) and the proof is concluded since ev is an isomorphism.

Given a positive integer t, Zt will stand for the quotient ring Z/tZ. Definition 2. A subset I of the cartesian product ZN1 × ZN2 × · · · × ZNm is called to be a cyclotomic coset if I = p · I = {p · α | α ∈ I}. I will be a minimal cyclotomic coset if there is α ∈ ZN1 × ZN2 × · · · × ZNm such that every element of I can be written psi · α for some integer i. Fix a monomial ordering on Zm ≥0 , where Z≥0 denotes the nonnegative integers. A minimal cyclotomic coset I will be represented by that element a ∈ I with smallest coordinates with respect to the above fixed monomial ordering. Thus, we will set Ia := I and ia := card(Ia ). Finally, the set of elements a representing minimal cyclotomic cosets is denoted by A. For every a = (a1 , a2 , . . . , am ) ∈ A, ia is a divisor of r and it holds ai psia ≡ ai (mod Ni ), 1 ≤ i ≤ m. Every cyclotomic coset is a union of minimal cyclotomic cosets and the minimal cyclotomic cosets constitute a partition of ZN1 × ZN2 × · · · × ZNm . Recall that we denote by f an element in R and its canonical representative and set supp(f ) the support of the


7

canonical representative. Any element f ∈ R may be decomposed in a unique way as a sum of classesP of polynomials with support included in minimal cyclotomic cosets. That is to say, f = a∈A fa , where supp(fa ) ⊆ Ia . We also notice that supp(T (fa )) ⊆ Ia . s(ia −1) s and set Now define the function Ta : R → R as Ta (f ) = f + f p + · · · + f p am . Then, we are ready to state and prove Theorem 3 which gives a X a = X1a1 X2a2 · · · Xm basis for the vector space of elements in R evaluating to the field Fps . First, we provide two results that help us in our purpose. Proposition 6. Let f be an element in R that evaluates to Fps with supp(f ) ⊆ Ia and consider a primitive element β of Fpsia . Then, f can be expressed as a linear combination with coefficients in Fps of the elements in R given by Saβ := Ta (X a ), Ta (βX a ), . . . , Ta (β ia −1 X a ) . P a −1 js s (αX a )p . Proof. Since supp(f ) ⊆ Ia and f p = f , there is some α ∈ Fpr such that f = ij=0 sia

Moreover αp = α, which implies that α ∈ Fpsia . We know that {1, β, . . . , β (ia −1) } is a basis of Fpsia over Fps , so α = a0 + a1 β + · · · + aia −1 β (ia −1) , with ai ∈ Fps for all i. Therefore, !pjs iX iX iX a −1 a −1 a −1 js js js Xp a al β l αp X p a = f= j=0

j=0

=

iX a −1 l=0



iX a −1

al 

β

lpjs

X

j=0

l=0



pjs a 

=

iX a −1

al Ta (β l X a ).

l=0

Saβ

are linearly independent Proposition 7. The vectors in the previous considered set over Fps . P a −1 Proof. Reasoning by contradiction, assume that il=0 al Ta (β l X a ) = 0. Then, the term a whose attached monomial is X and appears in the left hand side of the above equality is (a0 + a1 β + · · · + aia −1 β (ia −1) )X a and it must vanish. This is true only if β is a root of the univariate polynomial a0 + a1 Z + · · · + aia −1 Z (ia −1) . This gives the desired contradiction because the minimal polynomial of β has degree ia . Next, we state the mentioned theorem. Theorem 3. The following set o [n l a ΩR := T (β X ) | 0 ≤ l ≤ i − 1 and β is a primitive element of F si a a a p s a∈A

constitutes a basis for the vector space over Fps of elements in R evaluating to Fps .

Proof. We start by proving that the classes in ΩR s are linearly independent. This holds, on the one hand, because there is no linear dependence between the elements in ΩR s supported on different minimal cyclotomic cosets. Indeed, any monomial of any element supported on Ia is different from that of any other supported on Ia′ with a 6= a′ . On the other hand, Proposition 7 proves the independence of elements supported on the same set Ia which shows our statement. To conclude the proof, we are going to show that ΩR s is a generating set for the vector space of elements in R evaluating to Fps . Recall that we are using canonical polynomials for representing its corresponding classes in R. Consider the term in f with smallest order

8


k1 a 1 k1 a 1 for the above mentioned monomial ordering on Zm ≥0 , say β X , then Ta1 (β X ) = Pia1 −1 k1 a1 pls must appear in f because f evaluates to Fps . Since β k1 X a1 has the l=0 (β X ) smallest order in f , a1 must be one of the elements in A. Assume, without loss of generality that these elements are {a1 , a2 , . . . , at }. Set f1 = f − Ta1 (β k1 X a1 ) and pick its monomial with smallest order, say β k2 X a2 . Again, the polynomial Ta2 (β k2 X a2 ) must appear in f1 . We can repeat the above procedure and consider f2 = f1 − Ta2 (β k2 X a2 ). We will finish in at most t steps and this will provide the desired expression of f as a linear combination of elements in ΩR s , which concludes the proof.

We have just provided a constructive way of obtaining all classes in R that evaluate to Fps . In particular, if we restrict to those ones with support in U , we have a formula for the dimension of an affine variety subfield-subcode. Theorem 4. Let U ⊆ {0, . . . , N1 − 1} × {0, . . . , N2 − 1} × · · · × {0, . . . , Nm − 1}, n = N1 N2 · · · Nm and define CUs := CU ∩ Fnps . Then, U CUs = ev T (FH pr ) ∩ Fpr , CUs is spanned by the images under the evaluation map ev of the following elements in R o [ n Ta (β l X a ) | 0 ≤ l ≤ ia − 1 and β is a primitive element of Fpsia Ia |Ia ⊆U

and dim CUs =

X

ia .

Ia |Ia ⊆U

3. Quantum codes from subfield-subcodes of affine variety codes We devote this section to explain which of our affine variety subfield-subcodes yield stabilizer codes. Notice that our quantum codes will be defined over a small field Fps with the advantage that the original code is supported over a large field Fpr . Previously to state our main result, we will need to describe how dual codes of subfieldsubcodes are. From Proposition 3, we know that CU⊥ is the affine variety code defined by U ⊥ . Then, we get ⊥ U⊥ ) , ) = ev T (F (CUs )⊥ = tr (CU ⊥ ) = tr ev(FU pr pr where the first equality follows from Delsarte’s Theorem [7] and the last one from the fact that, by Proposition 5, the following map composition equality ev ◦ T = tr ◦ ev holds. Notice that, as above, we are identifying classes in R with canonical representatives. Now, ⊥ a ⊥ and γ ∈ F r . If one fixes a and varies γ over T (FU p pr ) is spanned by T (γX ) for a ∈ U l a the field, then the set {Ta (β X )}0≤l≤ia −1 , β primitive, is obtained. Thus we have proved the following result. Theorem 5. Let U ⊆ H be as in Section 1. The dual code (CUs )⊥ of the code CUs is spanned by the evaluation by ev the following elements in R: o n [ l a Ta (β X ) | 0 ≤ l ≤ ia − 1 and β is a primitive element of Fpsia . Ia |Ia ∩U ⊥ 6=∅

As a consequence, it holds that dim(CUs )⊥ =

X

Ia |Ia ∩U ⊥ 6=∅

ia .


9

We conclude this section with our main result which allows us to construct good stabilizer codes. We will need the concept of complementary of a minimal cyclotomic coset Ia . That set is the subset of ZN1 × ZN2 × · · · × ZNm defined as Iâ := {ˆ u | u ∈ Ia }. Theorem 6. Let p be a prime number, r and s positive integers such that s|r. Let H := {0, . . . , N1 − 1} × {0, . . . , N2 − 1} × · · · × {0, . . . , Nm − 1} be the hypercube defined in Section 1, where Ni divides pr − 1 for all index i and consider a nonempty subset U of H. Then, (1) The codes inclusion CUs ⊆ (CUs )⊥ happens if, and only if, Iâ is not contained in U whenever Ia is. (2) Assume that U satisfies the conditions in the previous item. Then, from the affine variety code CUs a stabilizer code can be obtained. The parameters for that codeare P s ⊥ . [[n, k, d]]ps , where n = N1 N2 · · · Nm , k = n − 2 Ia |Ia ⊆U ia and d = d (CU )

Proof. Theorem 5 proves that the dual code (CUs )⊥ is given by the evaluation of the elements Ta (β l X a ), 0 ≤ l ≤ ia − 1 and β being a primitive element of Fpsia , whenever Ia ∩ U ⊥ 6= ∅. Therefore, CUs ⊆ (CUs )⊥ if and only if Ia ∩ U ⊥ 6= ∅ when Ia ⊆ U and this happens if, and only if, the complementary set of Ia satisfies Iâ 6⊆ U for all minimal cyclotomic coset Ia ⊆ U . This proves our first assertion. Second one follows from CSS code construction. It would be interesting to know an explicit formula or tight bound for the value d((CUs )⊥ ). s ⊥ s Reasoning as in Theorem 6, when the inclusion P (CU ) ⊆ CU holds, one scan get an [[n, k, d]]ps code where n is as above, k = 2 Ia |Ia ⊆U ia − n and d = d(CU ). In this case a lower bound for the distance d can be described, since d(CUs ) ≥ d(CU ) and a lower bound for the distance of the code CU can be computed following the procedure given in [12]. Codes in this paper are constructed by applying the CSS code construction to suitable linear codes. However, to get quantum codes, one can also apply different bilinear pairings as the mentioned in the introduction trace inner product, for the case q = 4, or the trace alternating one for the general case [18]. As a referee of this paper pointed out, to get quantum codes, as we have done, with respect to these last inner products is an interesting future work.

4. Examples From the practical point of view it makes sense to choose U to be the union of different minimal cyclotomic cosets, since otherwise the evaluation will not be in Fnps . Next, we will show a table containing parameters corresponding with some quantum codes coming from subfield-subcodes of affine variety codes. These codes either improve or add new parameters with respect to those ones given in [8]. In addition, the first stabilizer code appearing in our table exceed the CSS quantum Gilbert-Varshamov bound [9, Theorem 1.4]. Computations has been done by writing a Magma [20] function. After the table, the reader can find those subsets U and values Ni , p, r and s providing the codes. For simplicity, a code given by U is also called U .

10


Code / Subset U1 U3 U5 U7 U9 U11 U13 U15 U17 U19 U21 U23 U25

n 200 147 189 189 217 245 441 21 45 27 15 147 64

k 184 123 159 141 171 179 423 7 25 19 9 137 52

d q = ps 4 3 4 2 4 2 6 2 6 2 6 2 3 2 5 4 6 4 3 4 3 4 3 4 3 3

Code / Subset U2 U4 U6 U8 U10 U12 U14 U16 U18 U20 U22 U24 -

n 147 147 189 189 245 441 21 35 75 75 45 64 -

k 127 105 147 129 209 411 13 5 67 67 33 48 -

d q = ps 3 2 6 2 5 2 7 2 4 2 4 2 3 4 7 4 3 4 3 4 4 4 4 3 -

Code U1 . Subset: U1 = {(3, 3, 3), (4, 4, 1), (2, 2, 3), (1, 1, 1), (3, 0, 6), (4, 0, 2), (2, 0, 6), (1, 0, 2)}. Values: p = 3, r = 4, s = 1, N1 = 5, N2 = 5, N3 = 8. Code U2 . Subset: U2 = {(2, 4, 2), (4, 1, 1), (1, 2, 2), (2, 4, 1), (4, 1, 2), (1, 2, 1), (2, 0, 0), (4, 0, 0), (1, 0, 0), (2, 5, 0), (4, 3, 0), (1, 6, 0)}. Values: p = 2, r = 6, s = 1, N1 = 7, N2 = 7, N3 = 3. Code U3 . Subset: U3 = {(6, 2, 2), (5, 4, 1), (3, 1, 2), (6, 2, 1), (5, 4, 2), (3, 1, 1), (2, 3, 0), (4, 6, 0), (1, 5, 0), (6, 0, 0), (5, 0, 0), (3, 0, 0)}. Values: p = 2, r = 6, s = 1, N1 = 7, N2 = 7, N3 = 3. Code U4 . Subset: U4 = {(2, 4, 0), (4, 1, 0), (1, 2, 0), (6, 0, 2), (5, 0, 1), (3, 0, 2), (6, 0, 1), (5, 0, 2), (3, 0, 1), (6, 2, 0), (5, 4, 0), (3, 1, 0), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 0, 0) (4, 0, 0), (1, 0, 0)}. Values: p = 2, r = 6, s = 1, N1 = 7, N2 = 7, N3 = 3. Code U5 . Subset: U5 = {(2, 4, 2), (4, 1, 1), (8, 2, 2), (7, 4, 1), (5, 1, 2), (1, 2, 1), (0, 6, 0), (0, 5, 0), (0, 3, 0), (2, 2, 0), (4, 4, 0), (8, 1, 0), (7, 2, 0), (5, 4, 0), (1, 1, 0)}. Values: p = 2, r = 6, s = 1, N1 = 9, N2 = 7, N3 = 3. Code U6 . Subset: U6 = {(0, 2, 0), (0, 4, 0), (0, 1, 0), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 6, 2), (4, 5, 1), (8, 3, 2), (7, 6, 1), (5, 5, 2), (1, 3, 1), (2, 1, 0), (4, 2, 0), (8, 4, 0), (7, 1, 0), (5, 2, 0), (1, 4, 0)}. Values: p = 2, r = 6, s = 1, N1 = 9, N2 = 7, N3 = 3. Code U7 . Subset: U7 = {(2, 4, 1), (4, 1, 2), (8, 2, 1), (7, 4, 2), (5, 1, 1), (1, 2, 2), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 6, 2), (4, 5, 1), (8, 3, 2), (7, 6, 1), (5, 5, 2), (1, 3, 1), (2, 2, 0), (4, 4, 0), (8, 1, 0), (7, 2, 0), (5, 4, 0), (1, 1, 0)}. Values: p = 2, r = 6, s = 1, N1 = 9, N2 = 7, N3 = 3. Code U8 . Subset: U8 = {(2, 3, 0), (4, 6, 0), (8, 5, 0), (7, 3, 0), (5, 6, 0), (1, 5, 0), (6, 6, 0), (3, 5, 0), (6, 3, 0), (3, 6, 0), (6, 5, 0), (3, 3, 0), (2, 3, 1), (4, 6, 2), (8, 5, 1), (7, 3, 2), (5, 6, 1), (1, 5, 2), (2, 4, 2), (4, 1, 1), (8, 2, 2), (7, 4, 1), (5, 1, 2), (1, 2, 1), (6, 2, 2), (3, 4, 1), (6, 1, 2), (3, 2, 1), (6, 4, 2), (3, 1, 1)}. Values: p = 2, r = 6, s = 1, N1 = 9, N2 = 7, N3 = 3. Code U9 . Subset: U9 = {(0, 2), (0, 4), (0, 1), (14, 0), (28, 0), (25, 0), (19, 0), (7, 0), (22, 6), (13, 5), (26, 3), (21, 6), (11, 5), (22, 3), (13, 6), (26, 5), (21, 3), (11, 6), (22, 5), (13, 3), (26, 6), (21, 5), (11, 3)}. Values: p = 2, r = 15, s = 1, N1 = 31, N2 = 7. Code U10 . Subset: U10 = {(2, 4, 0), (4, 1, 0), (1, 2, 0), (6, 0, 2), (5, 0, 4), (3, 0, 3), (6, 0, 1), (5, 0, 2), (3, 0, 4), (6, 0, 3), (5, 0, 1), (3, 0, 2), (6, 0, 4), (5, 0, 3), (3, 0, 1), (2, 0, 0), (4, 0, 0), (1, 0, 0)}. Values: p = 2, r = 12, s = 1, N1 = 7, N2 = 7, N3 = 5. Code U11 . Subset: U11 = {(2, 4, 2), (4, 1, 4), (1, 2, 3), (2, 4, 1), (4, 1, 2), (1, 2, 4), (2, 4, 3), (4, 1, 1), (1, 2, 2), (2, 4, 4), (4, 1, 3), (1, 2, 1), (6, 2, 0), (5, 4, 0), (3, 1, 0), (6, 4, 2), (5, 1, 4), (3, 2, 3),


11

(6, 4, 1), (5, 1, 2), (3, 2, 4), (6, 4, 3), (5, 1, 1), (3, 2, 2), (6, 4, 4), (5, 1, 3), (3, 2, 1), (6, 6, 0), (5, 5, 0), (3, 3, 0), (6, 0, 0), (5, 0, 0), (3, 0, 0)}. Values: p = 2, r = 12, s = 1, N1 = 7, N2 = 7, N3 = 5. Code U12 . Subset: U12 = {(0, 6, 2), (0, 5, 4), (0, 3, 1), (6, 2, 2), (3, 4, 4), (6, 1, 1), (3, 2, 2), (6, 4, 4), (3, 1, 1), (2, 2, 3), (4, 4, 6), (8, 1, 5), (7, 2, 3), (5, 4, 6), (1, 1, 5)}. Values: p = 2, r = 12, s = 2, N1 = 9, N2 = 7, N3 = 7. Code U13 . Subset: U13 = {(2, 0, 1), (4, 0, 2), (8, 0, 4), (7, 0, 1) , (5, 0, 2), (1, 0, 4), (0, 2, 0), (0, 4, 0), (0, 1, 0)}. Values: p = 2, r = 12, s = 2, N1 = 9, N2 = 7, N3 = 7. Code U14 . Subset: U14 = {U = (0, 2), (5, 1), (6, 1), (3, 1)}. Values: p = 2, r = 6, s = 2, N1 = 7, N2 = 3. Code U15 . Subset: U15 = {(0, 2), (4, 1), (2, 1), (1, 1), (5, 0), (6, 0), (3, 0)}. Values: p = 2, r = 6, s = 2, N1 = 7, N2 = 3. Code U16 . Subset: U16 = {(4, 4), (2, 1), (1, 4), (4, 1), (2, 4), (1, 1), (5, 0), (6, 0), (3, 0), (5, 3), (6, 2), (3, 3), (5, 2), (6, 3), (3, 2)}. Values: p = 2, r = 12, s = 2, N1 = 7, N2 = 5. Code U17 . Subset: U17 = {(3, 1, 1), (2, 1, 1), (0, 0, 2), (4, 1, 2), (1, 1, 2), (4, 2, 0), (0, 2, 2), (3, 0, 1), (2, 0, 1)}. Values: p = 2, r = 4, s = 2, N1 = 5, N2 = 3, N3 = 3. Code U18 . Subset: U18 = {(3, 0, 1), (2, 0, 1), (3, 1, 2), (2, 4, 2)}. Values: p = 2, r = 4, s = 2, N1 = 5, N2 = 5, N3 = 3. Code U19 . Subset: U19 = {(2, 2, 0), (0, 1, 2), (2, 0, 1), (1, 0, 0)}. Values: p = 2, r = 2, s = 2, N1 = 3, N2 = 3, N3 = 3. Code U20 . Subset: U20 = {(3, 0, 1), (2, 0, 1)(3, 1, 2), (2, 4, 2)}. Values: p = 2, r = 4, s = 2, N1 = 5, N2 = 5, N3 = 3. Code U21 . Subset: U21 = {(0, 2), (3, 1), (2, 1)}. Values: p = 2, r = 4, s = 2, N1 = 5, N2 = 3. Code U22 . Subset: U22 = {U = (4, 0, 2), (1, 0, 2), (0, 0, 1), (0, 1, 0), (0, 1, 1), (0, 2, 1)}. Values: p = 2, r = 4, s = 2, N1 = 5, N2 = 3, N3 = 3. Code U23 . Subset: U23 = {(4, 0, 1), (2, 0, 1), (1, 0, 1), (4, 5, 2), (2, 6, 2), (1, 3, 2)}. Values: p = 2, r = 6, s = 2, N1 = 7, N2 = 7, N3 = 3. Code U24 . Subset: U24 = {(3, 2), (1, 6), (0, 3), (0, 1), (7, 0), (5, 0), (7, 3), (5, 1)}. Values: p = 3, r = 2, s = 1, N1 = 8, N2 = 8. Code U25 . Subset: U25 = {(3, 0), (1, 0), (6, 5), (2, 7), (7, 7), (5, 5)}. Values: p = 3, r = 2, s = 1, N1 = 8, N2 = 8.

References [1] Ashikhmin, A., Knill, E. Non-binary quantum stabilizer codes, IEEE Trans. Inf. Theory 47 (2001) 3065-3072. [2] Ashikhmin, A., Litsyn, S. Upper bounds on the size of quantum codes, IEEE Trans. Inf. Theory 45 (1999) 1206-1215. [3] Bras-Amor´ os, M., O’Sullivan, M.E. Duality for some families of correction capability optimized evaluation codes, Adv. Math. Commun. 2 (2008) 15-33. [4] Bierbrauer, J., Edel, Y. Quantum twisted codes, J. Comb. Designs 8 (2000) 174-188. [5] Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A. Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theory 44 (1998) 1369-1387. [6] Calderbank A.R., Shor, P. Good quantum error-correcting codes exist, Phys. Rev. A 54 (1996) 10981105. [7] Delsarte, P. On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inform. Theory IT-21 (1975) 575-576. [8] Edel, Y. Some good quantum twisted codes. Online available at http://www.mathi.uni-heidelberg.de/∼yves/Matritzen/QTBCH/QTBCHIndex.html. [9] Feng, K., Ma, Z. A finite Gilbert-Varshamov bound for pure stabilizer quantum codes, IEEE Trans. Inf. Theory 50 (2004) 3323-3325. [10] Fitzgerald, J., Lax, R.F. Decoding affine variety codes using Gr¨ obner bases, Des. Codes Cryptogr. 13 (1998) 147-158.

12


[11] Geil, O. Evaluation codes from an affine variety code perspective. Advances in algebraic geometry codes, Ser. Coding Theory Cryptol. 5 (2008) 153-180. World Sci. Publ., Hackensack, NJ. Eds.: E. Martinez-Moro, C. Munuera, D. Ruano. [12] Geil, O., Martin, S. An improvement of the Feng-Rao bound for primary codes, arXiv:1307.3107. [13] Gottesman, D. A class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A 54 (1996) 1862-1868. [14] Grassl, M., Beth, T., R¨ otteler, M. On optimal quantum codes, Int. J. Quantum Inform. 2 (2004) 757-775. [15] Hernando, F., Marshall, K., O’Sullivan, M.E. The dimension of subcode-subfields of shortened generalized Reed-Solomon codes, Des. Codes Cryptogr. 69 (2013) 131-142. [16] Hernando, F., O’Sullivan, M.E., Popovici, E., Srivastava, S. Subfield-subcodes of generalized toric codes. In 2010 IEEE International Symposium on Information Theory, 1125-1129. [17] Høholdt, T., van Lint, J.H., Pellikaan, R. Algebraic geometry codes. Handbook of coding theory, vol. I (1998) 871-961. Elsevier, Amsterdam. Eds.: V.S. Pless, W.C. Huffman, R.A. Brualdi. [18] Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K. Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory 52 (2006) 4892-4914. [19] Knill, E., Laflamme, R. Theory of quantum error-correcting codes, Phys. Rev. A, 55 (1997) 900-911. [20] Magma Computational Algebra System. http://magma.maths.usyd.edu.au/magma/. [21] Mart´ınez-Moro, E., Pi˜ nera-Nicol´ as A., R´ ua. A.F. Additive semisimple multivariable codes over F4 , Des. Codes Cryptogr. 69 (2013) 161-180. [22] Mart´ınez-Moro, E. On semisimple algebra codes: generator theory, Algebra Discr. Math. 3 (2007) 99-112. [23] Rains, E.M. Nonbinary quantum codes, IEEE Trans. Inf. Theory 45 (1999) 1827-1832. [24] Ruano, D. On the structure of generalized toric codes, J. Symbolic Comput. 44 (2009) 499-506. [25] Shor, P.W. Algorithms for quantum computation: discrete logarithm and factoring, in Proc. 35th Ann. Symp. Foundations of Computer Science, IEEE Computer Society Press 1994, 124-134. [26] Shor, P.W. Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52 (1995) 2493-2496. [27] Steane, A.M. Simple quantum error correcting codes, Phys. Rev. Lett. 77 (1966) 793-797. [28] Wootters W.K., Zurek, W.H. A single quantum cannot be cloned, Nature, 299 (1982) 802-803. Current address: Carlos Galindo and Fernando Hernando: Instituto Universitario de Matem´ aticas y Aplicaciones de Castell´ on and Departamento de Matem´ aticas, Universitat Jaume I, Campus de Riu Sec. 12071 Castell´ o (Spain). E-mail address: Galindo: [email protected]; Hernando: [email protected]