On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes
aa r X i v : . [ c s . I T ] D ec A GENERALIZATION OF THE CONSTRUCTION OF QUANTUMCODES FROM HERMITIAN SELF-ORTHOGONAL CODES
CARLOS GALINDO AND FERNANDO HERNANDO
Abstract.
An important strength of the q -ary stabilizer quantum codes is that they canbe constructed from Hermitian self-orthogonal q -ary linear codes. We prove that thisresult can be extended to q ℓ -ary linear codes, ℓ >
1, and give a result for easily obtainingcodes of the last type. As a consequence we provide several new binary stabilizer quantumcodes which are records according to [23] and new 2 = q -ary ones improving others inthe literature. Introduction
A clear sample of the relevance of efficient quantum computation is the Shor algorithm[34] for obtaining discrete logarithms and prime factorization since it works in polynomialtime on quantum computers. Nowadays there are some evidences of the existence of quan-tum processors capable of executing certain tasks greatly improving classical processors[1] which increases the interest in tools for the proper functioning of quantum computersas the quantum error-correcting codes (QECCs). QECCs are mainly designed for pro-tecting quantum information from quantum noise and decoherence. Notice that, despitequantum information cannot be cloned [14, 38], quantum error correction works [35, 37].These reasons explain why many researchers are interested in obtaining QECCs with goodparameters (which measure the behaviour of the codes) and the literature contains a largequantity of papers in order to find QECCs with better parameters than others previouslyobtained.Let q be a power prime, a q -ary QECC of length n and dimension k is a subspaceof the Hilbert space H = C q n of dimension q k . The most used class of quantum codesare stabilizer quantum codes. These are obtained as the intersection of the eigenspaces,corresponding to the eigenvalue 1, of the elements of some subgroup of the error groupgenerated by a suitable error basis of the Hilbert space H . The parameters of a QECC, C :length, dimension and minimum distance are usually written as [[ n, k, d ]] q , where errorswith weight less than d either can be detected or have no effect on C but some error withweight d cannot be detected.QECCs were firstly introduced in the binary case which contains the seminal papers inthe subject [9, 22, 7, 8, 25, 3, 4]. Later QECCs were studied for the general q -ary case (see[6, 33, 5, 16, 24, 28, 2, 29, 31, 27, 30, 36, 19, 21, 20, 10, 11] among many others articles).This general case is particularly interesting for fault-tolerant computing [32].One of the main advantages of stabilizer codes is that their existence is equivalent tothat of self-orthogonal additive codes with respect to certain trace-symplectic form (see [5]or [28, Theorem 13]). This trace-symplectic form is not very friendly but the above result Key words and phrases.
Stabilizer quantum codes; Hermitian duality.Partially supported by the Spanish Government MICINN/FEDER/AEI/UE, grants PGC2018-096446-B-C22, RED2018-102583-T, as well as by Generalitat Valenciana, grant AICO-2019-223 and UniversitatJaume I, grant UJI-2018-10. allows us to deduce that many stabilizer quantum codes can be derived from self-orthogonalclassical codes with respect to the Hermitian or the Euclidean inner product. Usually onefinds good stabilizer codes over F q by considering Hermitian self-orthogonal codes over F q . The specific and widely used result, Theorem 1.1, shows that an [[ n, n − k, ≥ d ⊥ h ]] q quantum code can be constructed from a Hermitian self-orthogonal [ n, k ] q linear code C over F q , where d ⊥ h stands for the minimum distance of the dual Hermitian code C ⊥ h . Thisresult has been extensively used in many papers to give many good QECCs [29, 30, 27, 39].In this paper we prove that Theorem 1.1 can be regarded as a particular case of a moregeneral result by considering linear codes over extensions of F q larger than F q . The specificresult, Theorem 1.2, states that if C is a linear code over F q ℓ , ℓ ≥
1, with parameters[ n, k ] q ℓ which is self-orthogonal with respect to the Hermitian inner product, then thereexists a [[2 ℓ − n, ℓ − n − ℓ k, ≥ d ⊥ h ]] q stabilizer quantum code. That is, the classical resultis our result specialized for ℓ = 1.We think that many good quantum codes will be improved by this result. To givesome evidence for it, we state (and prove) Theorem 2.2 which combined with Theorem 1.2gives rise to a number of stabilizer quantum codes with good parameters. Theorem 2.2is supported in [27] and shows an easy way to find Hermitian self-orthogonal codes. Thecombination above mentioned produces, in the binary case, new QECCs which are recordsaccording to [23]. There is no collection of tables as [23] for non-binary QECCs but onecan found many papers in the literature about them. Most of these papers are devotedto quantum MDS codes which have relatively small length [12, 26, 39]. Since we are ableto construct QECCs of large length, we use two recent articles [36, 11] for comparing andshow that with our results we can improve the parameters of a number of codes thereintroduced.Section 1 of the paper is devoted to state and prove our main result for obtainingQECCs from linear codes while Theorem 2.2 and parameters (some of them displayed intables) of new QECCs can be seen in Section 2. As mentioned all the provided parameterscorrespond to QECCs obtained by applying Theorems 2.2 and 1.2. In the binary case, ourresults together with lengthening (a propagation rule) determine 26 new QECCs whichare records according to [23].1. A new construction of stabilizer quantum codes
The result.
Let q = p r a power of a prime number p , r ≥
1. Many good stabilizerquantum codes over the finite field with q elements F q are obtained from linear codes overthe finite field F q which are self-orthogonal for the Hermitian inner-product. Recall nowthat given two vectors x = ( x , x , . . . , x n ) and y = ( y , y , . . . , y n ) in F nq , n ≥
1, their
Hermitian inner product is defined to be x · h y := n X i =1 x i y qi ∈ F q , and the specific result to construct stabilizer quantum codes is the following one provedin [28, Corollary 16 and Lemma 18]. Theorem 1.1.
Let C be an F q -linear code of length n and dimension k . Assume that C is Hermitian self-orthogonal, i.e. C ⊆ C ⊥ h := n x ∈ F nq | x · h y = 0 for all y in C o . GENERALIZATION OF THE CONSTRUCTION OF QUANTUM CODES 3
Then, there exists a stabilizer quantum code over F q with parameters [[ n, n − k, ≥ d ⊥ h ]] q ,where d ⊥ h stands for the minimum distance of the code C ⊥ h . The aim of this section is to extend Theorem 1.1 allowing the use of codes over extensionfields of F q . We will prove that one can obtain long stabilizer codes with good parametersover F q by considering linear codes over fields F q ℓ which are self-orthogonal with respectto the Hermitian inner product. Our Section 2 will show that stabilizer codes with verygood parameters can be derived from our next result; in fact, according to [23], we are ableto get 26 new binary quantum codes which are records. Non-binary stabilizer quantumcodes improving some others in the literature are also provided.Now we state the main result in this section and in the paper. Theorem 1.2.
Let C be an F q ℓ -linear code of length n and dimension k . Suppose that C ⊆ C ⊥ h , where C ⊥ h := ( x ∈ (cid:16) F q ℓ (cid:17) n | x · h y := n X i =1 x i y q ℓ − i = 0 for all y in C ) . Then, there exists an F q -stabilizer quantum code with parameters (cid:2)(cid:2) ℓ − n, ℓ − n − ℓ k, ≥ d ⊥ h (cid:3)(cid:3) q , where d ⊥ h is the minimum distance of the code C ⊥ h . Proof of Theorem 1.2.
We devote this sub-section to give a proof of Theorem 1.2.For simplicity’ sake, we perform our proof for the case ℓ = 2. Our arguments are easilyextended to any value ℓ > Definition 1.3.
The symplectic weight of a vector ( x | y ) in the vector space F nq , n > x | y ) := { i ∈ { , , . . . , n } : ( x i , y i ) = (0 , } , where Definition 1.4.
Given two vectors ( x | y ) and ( u | v ) in F nq , we define their trace-symplecticform as ( x | y ) · trs ( u | v ) = tr q | p ( x · v − y · u ) , where the products x · v and y · u are inner Euclidean and tr q | p : F q → F p the standardtrace map.The following result, proved in [28, Theorem 13], gives a necessary and sufficient con-dition for the existence of a stabilizer quantum code over the field F q . Theorem 1.5.
There exists a stabilizer code over F q with parameters (( n, K, d )) q if andonly if there exists an additive code C included in F nq with cardinality q n /K satisfyingthat C ⊆ C ⊥ trs , and d = swt (cid:0) C ⊥ trs \ C (cid:1) whenever K > and d = swt (cid:0) C ⊥ trs (cid:1) in case K = 1 . In view of the above result, for proving Theorem 1.2, we start with an F q -linear codeof length n and dimension k such that C ⊆ C ⊥ h . Set d ⊥ h the minimum distance of the F q -linear code C ⊥ h . Then we will find an F q -linear code D satisfying D ⊆ F nq × F nq ofsize q k such that D ⊆ D ⊥ trs and swt (cid:0) D ⊥ trs (cid:1) ≥ d ⊥ h . We devote the remaining of this CARLOS GALINDO AND FERNANDO HERNANDO section to describe such a D .Let ω, ω q be a normal basis of the field F q over the field F q . Consider the bijectivemap(1) ϕ : F nq −→ F nq × F nq , defined as ϕ ( a ) := ( a (1) , a (2)) , where for elements a = ( a , a , . . . , a n ) ∈ F nq , a i ∈ F q , 1 ≤ i ≤ n , if a i = a i (1) ω + a i (2) ω q with a i ( j ) ∈ F q , 1 ≤ j ≤
2, then a ( j ) := ( a ( j ) , a ( j ) , . . . , a n ( j )) ∈ F nq .Consider now the trace-alternating form over F nq :(2) a · taf b := tr q | q a · b q − a q · b ω − ω q ! , where a q := ( a q , a q , . . . , a q n ) ∈ F nq .Notice that a · b q − a q · b = n X i =1 (cid:16) a i (1) ω + a i (2) ω q (cid:17) (cid:16) b i (2) ω + b i (1) ω q (cid:17) − (cid:16) a i (2) ω + a i (1) ω q (cid:17) (cid:16) b i (1) ω + b i (2) ω q (cid:17) = n X i =1 (cid:0) a i (1) b i (2) − a i (2) b i (1) (cid:1) (cid:16) ω − ω q (cid:17) . Therefore a · b q − a q · b ω − ω q ∈ F q and a · taf b is well-defined.In our next step we consider the standard trace (over F q ) symplectic form of F nq × F nq ,which is defined to be(3) ( x | y ) · tsf ( z | t ) := tr q | q ( x · t − y · z )and allow us to state and prove our first partial result. Lemma 1.6.
Let C be an F q -linear code of length n and dimension k such that C ⊆ C ⊥ h .Let ϕ be the map defined in (1). Then, ϕ ( C ) is an F q -linear code of dimension k suchthat ϕ ( C ) ⊆ ϕ ( C ) ⊥ tsf .Proof. We start by proving that ϕ ( C ) ⊆ ϕ ( C ) ⊥ tsf . Indeed, C ⊆ C ⊥ taf because a · b q = 0for a , b ∈ C since C ⊆ C ⊥ h and then a · taf b = 0. This concludes this part of the proofafter noticing that(4) a · taf b = ϕ ( a ) · tsf ϕ ( b ) . Now, set { a , a , . . . , a k } a basis of C as F q -vector space. C is a vector space over F q which has B = { ω a , ω q a , . . . , ω a k , ω q a k } as a basis. Let us show it. On the one GENERALIZATION OF THE CONSTRUCTION OF QUANTUM CODES 5 hand, B generates C as vector space over F q because if c ∈ C , c = P ki =1 α i a i , with α i = α i (1) ω + α i (2) ω q ∈ F q and α i ( j ) ∈ F q , j = 1 ,
2, and then c = k X i =1 α i (1) ω a i + k X i =1 α i (2) ω q a i . On the other hand, elements in B are free over F q . In fact, P ki =1 α i (1) ω a i + α i (2) ω q a i = 0for α i ( j ) ∈ F q , j = 1 ,
2, can be expressed as P ki =1 ( α i (1) ω + α i (2) ω q ) a i = 0 which implies α i (1) = α i (2) = 0 for all 1 ≤ i ≤ r because the a i ’s are a basis over F q .This concludes the proof since ϕ is a morphism of F q -vector spaces and bijective. (cid:3) Set E = ϕ ( C ). E ⊆ F nq × F nq is an F q -linear code of dimension 2 k which satisfies E ⊆ E ⊥ tsf . To continue with the proof we divide our reasoning in two cases according tothe characteristic of the field F q is, or not, 2.1.2.1. Ch( F q ) = 2 . Consider a basis { γ, γ q } of the vector space F q over F q , where γ is aprimitive element of F q such that tr q | q ( γ ) = 1 [13]. Define(5) ψ : F nq × F nq −→ F nq × F nq as ψ ( x | y ) = (cid:0) ( x (1) , x (2)) | ( y (1) , y (2)) (cid:1) , where if x = ( x , x , . . . , x n ) ∈ F nq , x i ∈ F q , 1 ≤ i ≤ n and x i = x i (1) γ + x i (2) γ q , x i ( j ) ∈ F q , 1 ≤ j ≤
2, then x ( j ) = (cid:0) x ( j ) , x ( j ) , . . . , x n ( j ) (cid:1) , j ∈ { , } . Lemma 1.7.
Recall that F q is a finite field of characteristic . Denote · s the standardsymplectic product in F nq × F nq . Consider elements ( x | y ) and ( z | t ) in F nq × F nq . Then,the following equality in F q holds: (6) ( x | y ) · tsf ( z | t ) = ψ ( x | y ) · s ψ ( z | t ) . Proof.
We start with the left hand of our equality( x | y ) · tsf ( z | t ) = tr q | q ( x · t − y · z ) = tr q | q n X i =1 x i t i − y i z i ! = tr q | q n X i =1 ( x i (1) γ + x i (2) γ q )( t i (1) γ + t i (2) γ q ) − ( y i (1) γ + y i (2) γ q )( z i (1) γ + z i (2) γ q ) ! (7) = tr q | q (cid:18) n X i =1 (cid:0) x i (1) t i (1) − y i (1) z i (1) (cid:1) γ + (cid:0) x i (2) t i (2) − y i (2) z i (2) (cid:1) γ q + (cid:0) x i (1) t i (2) + x i (2) t i (1) − y i (1) z i (2) − y i (2) z i (1) (cid:1) γ q +1 (cid:19) . Taking into account that tr q | q ( γ ) = γ + γ q = tr q | q ( γ q ) andtr q | q ( γ q +1 ) = γ q +1 + ( γ q +1 ) q = γ q +1 + γ q +1 = 0 CARLOS GALINDO AND FERNANDO HERNANDO because ( γ q +1 ) q = ( γ q +1 ) q − γ q +1 = γ q − γ q +1 = γ q +1 , we deduce that (7) equals n X i =1 (cid:0) x i (1) t i (1) + x i (2) t i (2) − y i (1) z i (1) − y i (2) z i (2) (cid:1) tr q | q ( γ )= (cid:0) ψ ( x | y ) · s ψ ( z | t ) (cid:1) tr q | q ( γ ) = ψ ( x | y ) · s ψ ( z | t ) , because tr q | q ( γ ) = γ + γ q = ( γ + γ q ) = (cid:0) tr q | q ( γ ) (cid:1) = 1. (cid:3) From the above results, we deduce the following proposition.
Proposition 1.8.
With the previous notation and assuming that the characteristic of thefield F q is , if C is an additive code included in F nq , then ψ ◦ ϕ ( C ⊥ taf ) = ( ψ ◦ ϕ ) ⊥ s . In addition, if C is an F q -linear code of length n and dimension k such that C ⊆ C ⊥ h ,then D := ψ ◦ ϕ ( C ) is an F q -linear code included in F nq of cardinality q k such that D ⊆ D ⊥ s ⊆ D ⊥ trs .Proof. Our first statement follows from (4) and (6). The fact that C ⊆ C ⊥ h implies C ⊆ C ⊥ taf and our first statement proves D ⊆ D ⊥ s . Finally, the inclusion D ⊥ s ⊆ D ⊥ trs isobvious and the dimensionality can be proved in two stages as we did in Lemma 1.6. (cid:3) F q ) = 2 . Let β be a primitive element of F q , notice that f ( x ) = x − β is anirreducible polynomial in the polynomial ring in one variable F q [ x ] because otherwise β i ,for some 0 ≤ i ≤ q − f ( x ) and then β i = β which is not possiblesince q is odd. Set ξ = x + ( f ) ∈ F q [ x ] / ( f ) = F q and consider the basis { , ξ } of F q asvector space over F q .Like in the characteristic two case, we use a map ψ : F nq × F nq −→ F nq × F nq defined in a slightly different way :(8) ψ ( x | y ) = (cid:0) ( x (1) , x (2)) | ( y (2) , y (1)) (cid:1) , where if x = ( x , x , . . . , x n ) ∈ F nq , x i ∈ F q , 1 ≤ i ≤ n and x i = x i (1) + x i (2) ξ , x i ( j ) ∈ F q ,1 ≤ j ≤
2, then x ( j ) = ( x ( j ) , x ( j ) , . . . , x n ( j )) ; j ∈ { , } . In this case where the characteristic is odd, we are going to change our alternating andinner products. We define them analogously but without using traces. More specifically,with the same notation as in (2), we define the alternating form in F nq as(9) a · af b := a · b q − a q · b ω − ω q . And with the notation of (3) the symplectic form in F nq × F nq is defined to be( x | y ) · sf ( z | t ) := x · t − y · z . It is clear that replacing ⊥ tsf by ⊥ sf and considering · af instead of · taf , Lemma 1.6 alsoholds and if C is a code included in F nq of dimension k and such that C ⊆ C ⊥ h , then ϕ ( C )is an F q -linear code of dimension 2 k and ϕ ( C ) ⊆ ( ϕ ( C )) ⊥ sf . GENERALIZATION OF THE CONSTRUCTION OF QUANTUM CODES 7
Our next result will help to complete the proof.
Lemma 1.9.
Let E be an F q -linear code of dimension k included in F nq × F nq and suchthat E ⊆ E ⊥ sf . Then ψ ( E ) is an F q -linear code of dimension k included in F nq × F nq such that ψ ( E ) ⊆ ( ψ ( E )) ⊥ s .Proof. The map ψ is an isomorphism of F q -vector spaces and reasoning as in Lemma 1.6, ψ ( E ) has dimension 4 k .Let us see that ψ ( E ) ⊆ ( ψ ( E )) ⊥ s . Consider ( x | y ) , ( z | t ) ∈ E . Then0 = ( x | y ) · sf ( z | t ) = n X i =1 (cid:0) x i (1) + x i (2) ξ (cid:1)(cid:0) t i (1) + t i (2) ξ (cid:1) − (cid:0) y i (1) + y i (2) ξ (cid:1)(cid:0) z i (1) + z i (2) ξ (cid:1) = n X i =1 (cid:0) x i (1) t i (1) − y i (1) z i (1) (cid:1) + h(cid:0) x i (1) t i (2) + x i (2) t i (1) (cid:1) − (cid:0) y i (2) z i (1) + y i (1) z i (2) (cid:1)i ξ + (cid:0) x i (2) t i (2) − y i (2) z i (2) (cid:1) ξ . Noticing that ξ = β ∈ F q and ξ ∈ F q \ F q , we deduce that n X i =1 (cid:0) x i (1) t i (2) + x i (2) t i (1) (cid:1) − (cid:0) y i (2) z i (1) + y i (1) z i (2) (cid:1) = 0and therefore ψ ( x | y ) · s ψ ( z | t ) = 0 , which proves that ψ ( E ) ⊆ ( ψ ( E )) ⊥ s . (cid:3) As a consequence we have proved the following result.
Proposition 1.10.
Assume that the characteristic of the field F q is odd. Let C be an F q -linear code of dimension k such that C ⊆ C ⊥ h . Then ψ ◦ ϕ ( C ) is an F q -linear code includedin F nq of cardinality q k which satisfies that ψ ◦ ϕ ( C ) ⊆ ( ψ ◦ ϕ ( C )) ⊥ s and therefore ψ ◦ ϕ ( C ) ⊆ (cid:0) ψ ◦ ϕ ( C ) (cid:1) ⊥ trs . To finish our proof, we notice that C ⊥ h is a linear space of dimension n − k over F q ,which shows that ψ ◦ ϕ ( C ⊥ h ) is an F q -linear code included in F nq of dimension 4 n − k .We have also proved that ψ ◦ ϕ ( C ⊥ h ) ⊆ (cid:0) ψ ◦ ϕ ( C ) (cid:1) ⊥ trs . This fact, together with that ofboth spaces have the same dimension over F q proves that ψ ◦ ϕ ( C ⊥ h ) = ( ψ ◦ ϕ ( C )) ⊥ trs , which, recalling the construction of ϕ and ψ proves thatswt ( ψ ◦ ϕ ( C )) ⊥ trs ≥ d ⊥ h , where d ⊥ h is the minimum distance of the code C ⊥ h .This last fact, together with Proposition 1.8 in characteristic 2 and Proposition 1.10otherwise, proves our result for ℓ = 2 by Theorem 1.5.The case ℓ > ϕ : ( F q ℓ ) n −→ ( F q ℓ − ) n × ( F q ℓ − ) n together with maps ψ r : ( F q ℓ − − r ) r n × ( F q ℓ − − r ) r n −→ ( F q ℓ − − ( r +1) ) r +1 n × ( F q ℓ − − ( r +1) ) r +1 n , CARLOS GALINDO AND FERNANDO HERNANDO where 0 ≤ r ≤ ℓ − Hermitian self-orthogonal codes and examples
We devote this section to show that some very good stabilizer quantum codes can bederived from Theorem 1.2. We think that, generally speaking, to combine Theorem 1.2with suitable good families of Hermitian self-orthogonal codes will give rise to stabilizercodes with good parameters.2.1.
A useful result.
Next, we state and prove a result derived from [27, Theorem 2.5]which allows us provide linear codes to which we will apply Theorem 1.2. This proceduregive some binary stabilizer quantum codes which are records according to [23] and alsosome q -ary stabilizer quantum codes, q = 2, improving some recent ones appearing in theliterature.We start by recalling Theorem 2.5 in [27]. Theorem 2.1.
Let e be a power prime and set Q = e . For any positive integer ≤ n ≤ Q ,write n = n + n + · · · + n t , where ≤ t ≤ e and ≤ n i ≤ e for all ≤ i ≤ t . Then, forany positive integer ≤ k ≤ min { n , n , . . . , n t } , there exists an [ n, k ] Q linear code C over the field F Q which is Hermitian self-orthogonaland the minimum distance of C ⊥ h is k + 1 . Now we state a new result derived from Theorem 2.1 which will be useful.
Theorem 2.2.
Let e > be a prime power and set Q := e . Consider a positive integer ≤ n ≤ Q and write n as n = ae + b , where ≤ a < e and ≤ b < e , i.e., the e -adicexpression of n and include the case a = e and b = 0 .Define K n as follows: K n := ⌊ e/ ⌋ when b = 0 . K n := ⌊ n/ ⌋ when a = 0 . K n := ⌊ ( e − / ⌋ when a = 0 , b = 0 and a + b ≥ e . Otherwise, K n := (cid:22) max {⌊ n/ ( a + 1) ⌋ , a + b } (cid:23) . Then, for each ≤ k ≤ K n , there exist an [ n, k ] Q linear code C which is self-orthogonal forthe Hermitian inner product and such that the minimum distance of the Hermitian dualcode C ⊥ h is k + 1 .Proof. Assume b = 0, then the result holds by setting n = n = · · · = n a = e andapplying Theorem 2.1. Whenever a = 0, the same theorem with n = n proves the result.Suppose a = 0 = b and a + b ≥ e . Let us see that there exist non-negative integers i, j such that i + j = a and positive integers n = · · · = n i = e , n i +1 = · · · = n i + j = e − n a +1 = e − ie + j ( e −
1) + e − i + j ) e + e − − j = ae + e − − j = n, for some j because the fact that a + b ≥ e proves the existence of such a j with 0 ≤ j ≤ e − a + b < e . Then, on the one hand, setting n = n = · · · = n a = e − n a +1 = a + b , we find the second bound by Theorem 2.1. With respect to the firstone, it is clear that ( a + 1) (cid:22) na + 1 (cid:23) ≤ n ≤ ( a + 1) (cid:18)(cid:22) na + 1 (cid:23) + 1 (cid:19) . GENERALIZATION OF THE CONSTRUCTION OF QUANTUM CODES 9 n k ≥ d n k ≥ d n k ≥ d
252 * 204 7 252 * 196 8 248 * 200 7248 * 192 8 249 200 7 250 200 7251 200 7 249 192 8 250 192 8251 192 8 244 * 196 7 244 * 188 8245 196 7 246 196 7 247 196 7245 188 8 246 188 8 247 188 8240 * 192 7 240 * 184 8 241 192 7242 192 7 243 192 7 241 184 8242 184 8 243 184 8 - - -
Table 1.
Stabilizer quantum binary recordsThis implies that a set { n i } a +1 i =1 as in Theorem 2.1 can be constructed for values n i whichare either j na +1 k or j na +1 k + 1. This concludes the proof because we can choose the bestbound. (cid:3) Examples.
In this subsection we determine parameters for (constructible) stabilizerquantum codes over finite fields of small cardinality.2.2.1.
Binary stabilizer quantum codes.
We start with binary codes. In this case we givea number of quantum codes which are records according to [23], that is we give codes forentries in [23] whose constructions were missing. We explain in detail how we get our firstbinary code.We start with the values e = 16 and Q = 16 = 256. By Theorem 2.2, we can considerthe value n = 63 because 2 ≤ ≤
256 and 63 = n = 3 ·
16 + 15. Thus a = 3 and b = 15.Since a + b ≥ K = ⌊ / ⌋ = 7. Then, by Theorem 2.2, there exist a linear code C over F Q with length n = 63, dimension k = 6 and d ( C ⊥ h ) = 7. Now Q = q ℓ for q = 2 and ℓ = 3. Applying Theorem 1.2, we get a [[ , , ≥ ]] quantum code over F which isa record. If we pick n = 62 and k = 7, then a new record is obtained: [[ , , ≥ ]] .With an analogous procedure we obtain new records according to [23]. By lengthening(a propagation rule) [17] we get more records. All of them are grouped in Table 1, wherethe parameters obtained without lengthening are marked with a *.2.2.2. Non-binary stabilizer quantum codes.
As we did in the binary case, we explain indetail our first construction of a family of good stabilizer quantum codes over F . We willonly show the parameters for the remaining stabilizer codes which can be obtained in asimilar way.Set e = 16, Q = 16 = 256 and, applying Theorem 2.2, pick n = 76, which accomplishes2 ≤ ≤ n = 76 = 4 ·
16 + 12, then under the notation of that theorem a = 4and b = 12. Since a + b ≥ K = 7 and then, there exists a linear code over F Q whichis self-orthogonal for the Hermitian inner product, where Q = q ℓ for q = 4 and ℓ = 2.Applying Theorem 1.2, we find the stabilizer quantum codes over F showed in Table 2.By lengthening, we obtain quantum codes with parameters[[153 , , ≥ , [[153 , , ≥ , [[153 , , ≥ [[153 , , ≥ improving some codes in (and adding a new one to) [36, Table 3].Similarly, considering n = 234, we get stabilizer quantum codes over F with parametersas in Table 3. Notice that these codes make a great improvement to some ones in [36, n k ≥ d n k ≥ d
152 148 2 152 144 3152 140 4 152 136 5152 132 6 152 128 7152 124 8 - - -
Table 2.
Stabilizer quantum codes over F n k ≥ d n k ≥ d
468 452 5 468 448 6468 444 7 468 440 8468 436 9 468 432 10468 428 11 468 424 12
Table 3.
Stabilizer quantum codes over F Table 4].We conclude this section by giving some more families of stabilizer quantum codesobtained with our procedure.We start with the parameters of a family of QECCs over F . Consider Q = 2401 = 7 and n = 196, by Theorems 2.2 and 1.2 we get a family of stabilizer quantum codes withparameters n [[392 , − j, ≥ j ]] o j =3 . Comparing with [11, Table 3], we obtain many more 7-ary quantum codes of length 392.For j = 3 , , , ≥ codewhich improves the [[392 , , ≥ code in that paper.Our next family corresponds to the field F . Theorems 2.2 and 1.2 for Q = 8 = 4096and n = 283 give rise to a new family of QECCs with parameters n [[566 , − j, ≥ j ]] o j =5 . After lengthening, one gets a set of QECCs with parameters n [[567 , − j, ≥ j ]] o j =5 . As before, we add many new codes to those 8-ary ones in [11, Table 1] of length 567 andobtain a code with parameters [[567 , , ≥ improving the [[567 , , ≥ code in[11].To end, set Q = 6561 = 9 . As above • Picking n = 200, we get a family of stabilizer quantum codes over F with param-eters: n [[400 , − j, ≥ j ]] o j =3 . • Setting n = 400, we obtain: n [[800 , − j, ≥ j ]] o j =3 . GENERALIZATION OF THE CONSTRUCTION OF QUANTUM CODES 11 • With n = 405, we obtain: n [[810 , − j, ≥ j ]] o j =3 . • Finally, with n = 162, we get: n [[324 , − j, ≥ j ]] o = j =7 . Again comparing with Tables 1, 3, 5 and 8 in [11] we are able to provide quite a few newcodes. In addition, the following ones improve codes in [11]: [[400 , , ≥ , [[800 , , ≥ , [[800 , , ≥ , [[810 , , ≥ , [[810 , , ≥ , [[810 , , ≥ , [[324 , , ≥ and [[324 , , ≥ .Notice that, when providing our families of codes over F , F and F , we have considereddifferent values for the indices j in order to get parameters which are either new or betterthan or equal to those in [11]. Finally it is worth pointing out that, when comparison ispossible, our codes improve (in general, a lot) those in [15]. References [1] Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconductingprocessor,
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Des. Codes Cryptogr. (2017)503-517. Current address : Instituto Universitario de Matem´aticas y Aplicaciones de Castell´on and Departamentode Matem´aticas, Universitat Jaume I, Campus de Riu Sec. 12071 Castell´o (Spain).
Email address : Galindo: [email protected];