New Quantum MDS Codes over Finite Fields
aa r X i v : . [ c s . I T ] S e p New Quantum MDS Codes over Finite Fields
Xiaolei Fang Jinquan Luo ∗ Abstract : In this paper, we present three new classes of q -ary quantum MDS codes utilizing gener-alized Reed-Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, theminimum distance of some q -ary quantum MDS codes can be bigger than q + 1. Comparing to previousknown constructions, the lengths of codes in our constructions are more flexible. Key words : Quantum MDS code, Generalized Reed-Solomon code, Hermitian construction, Hermi-tian self-orthogonal
Quantum error-correcting codes play an important role in quantum information transmission andquantum computation. Due to the establishment of the connections between quantum codes and classicalcodes (see [2,4,23]), great progress has been made in the study of quantum error-correcting codes. Oneof these connections shows that quantum codes can be constructed from classical linear error-correctingcodes satisfying symplectic, Euclidean or Hermitian self-orthogonal properties (see [1,13,24]).Let q be a prime power. We use [[ n, k, d ]] q to denote a q -ary quantum code of length n , dimension q k and minimum distance d . Similar to the classical counterparts, quantum codes have to satisfy thequantum Singleton bound: k ≤ n − d + 2. The quantum code attaching this bound is called quantummaximum-distance-separable(MDS) code.In the past few decades, quantum MDS codes have been extensively studied. The construction of q -aryquantum MDS codes with length n ≤ q +1 has been investigated from classical Euclidean orthogonal codes(see [7,20]). On the other hand, some quantum MDS codes with length n ≥ q + 1 have been investigated,most of which have minimum distances less than q + 1 (see [11]). So it is a challenging and valuabletask to construct quantum MDS codes with minimal distances larger than q + 1. Recently, researchershave constructed some of such quantum MDS codes utilizing constacyclic codes, negacyclic codes and ∗ The authors are with School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, CentralChina Normal University, Wuhan China 430079.E-mails: [email protected](X.Fang), [email protected](J.Luo). q -ary quantum MDScodes with minimal distances bigger than q + 1 are far from complete.There are dozens of papers on the construction of [[ n, n − d, d + 1]] q quantum MDS codes withrelatively large minimum distances. Most of the known [[ n, n − d, d + 1]] q quantum MDS codes withminimum distances larger than q + 1 have lengths n ≡ , q + 1) (see [3,5,7,9,11,14,15,21,22,28]) or n ≡ , q −
1) (see [5,7,9-12,14,21,22,25,28]), except for the following cases.(i). n = q − l and d ≤ q − l − ≤ l ≤ q − n = mq − l and d ≤ m − l for 0 ≤ l < m and 1 < m < q (see [17] and also [6] for l = 0).(iii). n = t ( q + 1) + 2 and 1 ≤ d ≤ t + 1 for 1 ≤ t ≤ q − p, t, d ) = (2 , q − , q ) (see [6] and also[17] for t = q − q + 1 via generalized Reed-Solomon codes and Hermitian construction. Their lengths aredifferent from the above three cases and also in most cases, are not of the form n ≡ , q ± n, n − d, d + 1]] q quantum MDS codes are as follows:(i). n = 1 + lh + mr − q − st · hr and 1 ≤ d ≤ min { s + h · q +1 s − , q +12 + q − t − } , for odd s | q + 1,even t | q − t ≥ l = q − s , m = q − t , odd h ≤ s − r ≤ t and q − > q − st · hr (see Theorem 3);(ii). n = lh + mr − q − st · hr and 1 ≤ d ≤ min {⌊ s + h ⌋ · q +1 s − , q +12 + q − t − } , for odd s | q + 1, even t | q − t ≥ l = q − s , m = q − t , h ≤ s − r ≤ t and q − > q − st · hr (see Theorem 4);(iii). n = lh + mr and 1 ≤ d ≤ min {⌊ s + h ⌋ · q +1 s − , q +12 + q − t − } , for even s | q + 1, even t | q − t ≥ l = q − s , m = q − t , h ≤ s and r ≤ t (see Theorem 5).This paper is organized as follows. In Section 2, we will introduce some basic knowledge and usefulresults on Hermitian self-orthogonality and generalized Reed-Solomon codes, which will be utilized in theproof of main results. In Sections 3-5, we will present our main results on the constructions of quantumMDS codes. In Section 6, we will make a conclusion. In this section, we introduce some basic notations and useful results on Hermitian self-orthogonalityand generalized Reed-Solomon codes (or GRS codes for short).Let F q be the finite field with q elements and F ∗ q = F q \{ } , where q is a prime power. Obviously, F q is a subfield of F q with q elements and denote by F ∗ q = F q \{ } . For any two vectors −→ x = ( x , . . . , x n )2nd −→ y = ( y , . . . , y n ) ∈ F q , the Euclidean and Hermitian inner products are defined as h−→ x , −→ y i = n X i =1 x i y i and h−→ x , −→ y i H = n X i =1 x i y qi respectively.For a linear code C of length n over F q , the Euclidean dual code of C is defined as C ⊥ := {−→ x ∈ F nq : h−→ x , −→ y i = 0 , for all −→ y ∈ C } , and the Hermitian dual code of C is defined as C ⊥ H := {−→ x ∈ F nq : h−→ x , −→ y i H = 0 , for all −→ y ∈ C } . If C ⊆ C ⊥ H , the code C is called Hermitian self-orthogonal.In 2001, Ashikhmin and Knill [2] proposed the Hermitian Construction of quantum codes, which is avery important technique for constructing quantum codes from classical codes. Theorem 1. ([2, Corollary 1]) A q -ary quantum [[ n, n − d, d + 1]] q MDS code exists provided that an [ n, d, n − d + 1] q MDS Hermitian self-orthogonal code exists.
Choose two vectors −→ v = ( v , v , . . . , v n ) and −→ a = ( a , a , . . . , a n ), where v i ∈ F ∗ q ( v i may not bedistinct) and a i are distinct elements in F q . For an integer d with 1 ≤ d ≤ n , the GRS code of length n associated with −→ v and −→ a is defined as follows: GRS d ( −→ a , −→ v ) = { ( v f ( a ) , . . . , v n f ( a n )) : f ( x ) ∈ F q [ x ] , deg( f ( x )) ≤ d − } . (1)The generator matrix of the code GRS d ( −→ a , −→ v ) is G d ( −→ a , −→ v ) = v v · · · v n v a v a · · · v n a n ... ... . . . ... v a d − v a d − · · · v n a d − n . (2)It is well known that the code GRS d ( −→ a , −→ v ) is a q -ary [ n, d, n − d + 1] MDS code [18, Chapter 11]. Thefollowing theorem will be useful and it has been shown in [19,28]. Theorem 2. ([19,28]) The two vectors −→ a = ( a , . . . , a n ) and −→ v = ( v , . . . , v n ) are defined above. Then GRS d ( −→ a , −→ v ) is Hermitian self-orthogonal if and only if h−→ a qi + j , −→ v q +1 i = 0 , for all ≤ i, j ≤ d − .
3f there are no specific statements, the following notations are fixed throughout this paper. • Let s | q + 1 and t | q − t even. • Let l = q − s and m = q − t . • Let g be a primitive element of F q , δ = g s and θ = g t . Lemma 2.1.
Suppose gcd( s, t ) = 1 . For any α, β ∈ Z q − , the number of ( i, j ) of the equation α + si ≡ β + tj (mod q − satisfying ≤ i < q − s and ≤ j < q − t is q − st .Proof. Let β − α = γ . From α + si ≡ β + tj (mod q − si − tj ≡ γ (mod q − ≤ i < q − s and 0 ≤ j < q − t , si − tj mod q − q − st times through every element of Z q − .Indeed, for any γ ∈ Z q − , we have si − tj ≡ γ (mod q − ⇔ s | tj + γ ⇔ tj ≡ − γ (mod s ). Sincegcd( s, t ) = 1, then j mod s is unique. So when 0 ≤ j < q − t , the number of j satisfying the equationis q − st . The values of γ and i will be determined after fixing j . So the number of ( i, j ) of the equation α + si ≡ β + tj (mod q −
1) is q − st satisfying 0 ≤ i < q − s and 0 ≤ j < q − t is q − st .The following two lemmas have been shown in [5] and [9]. In order to make the paper self completeness,we will give proofs. Lemma 2.2. ([5, Lemmas 5 and 11]) Assume that h ≤ s − .(i). For any ≤ i, j ≤ ⌊ s + h ⌋ · q +1 s − , l | ( qi + j + q + 1) if and only if qi + j + q + 1 = µ · l , with ⌈ s − h ⌉ + 1 ≤ µ ≤ ⌊ s + h ⌋ − .(ii). For any ≤ i, j ≤ ⌊ s + h ⌋ · q +1 s − with ( i, j ) = (0 , , l | ( qi + j ) if and only if qi + j = µ · l , with ⌈ s − h ⌉ + 1 ≤ µ ≤ ⌊ s + h ⌋ − .Proof. (i). When s ≡ h (mod 2), it implies ⌊ s + h ⌋ = s + h and ⌈ s − h ⌉ = s − h . Since 0 ≤ i, j ≤ s + h · q +1 s − 2, then 0 < qi + j + q +1 < q − 1, that is 0 < µ < s . From qi + j + q +1 = q (cid:16) µ · ( q +1) s − (cid:17) + (cid:16) q − µ · ( q +1) s (cid:17) ,it follows that i = µ · ( q + 1) s − , j = q − µ · ( q + 1) s − . By i < s + h · q +1 s − j < s + h · q +1 s − 2, it implies s − h < µ < s + h . So l | ( qi + j ) if and only if qi + j = µ · l , with s − h + 1 ≤ µ ≤ s + h − s h (mod 2), it implies ⌊ s + h ⌋ = s + h − and ⌈ s − h ⌉ = s − h +12 . Then the proof can be completedby proceeding as the situation that s ≡ h (mod 2).(ii). In a similar way, we can complete the proof. So we omit the details. Lemma 2.3. ([9, Lemma 3.1]) The identity m − P ν =0 θ ν ( qi + j + q +12 ) = 0 holds for all ≤ i, j ≤ q +12 + q − t − ,with even t ≥ . roof. It is easy to check that the identity holds if and only if m ∤ qi + j + q +12 . On the contrary, assumethat m | qi + j + q +12 . Let qi + j + q + 12 = µ · m = q · µ ( q − t + µ ( q − t (3)with µ ∈ Z . By t ≥ 2, we have qi + j + q +12 < q − 1, which implies 0 < µ < t . • If j + q +12 ≤ q − 1, comparing remainder and quotient of module q on both sides of (3), we candeduce i = j + q +12 = µ · q − t . Since t is even, then q − t | q − . From q − t | j + 1 + q − , we candeduce that q − t | j + 1. Since j + 1 ≥ 1, then j + 1 ≥ q − t . So i = j + q +12 ≥ q +12 + q − t − 1, whichis a contradiction. • When j + q +12 ≥ q , it takes qi + j + q +12 = q ( i + 1) + (cid:0) j − q − (cid:1) = q · µ ( q − t + µ ( q − t . In a similarway, j − q − = i + 1 = µ · q − t which implies q − t | i + 1. Since i + 1 ≥ 1, then i + 1 ≥ q − t . Therefore, j = i + 1 + q − ≥ q +12 + q − t − 1, which is a contradiction.As a result, m ∤ qi + j + q +12 which yields m − P ν =0 θ ν ( qi + j + q +12 ) = 0 for all 0 ≤ i, j ≤ q +12 + q − t − n = 1 + lh + mr − q − st · hr In this section, we assume that s is odd , h ≤ s − with h odd and r ≤ t . Quantum MDS codesof length n = 1 + lh + mr − q − st · hr will be constructed. The construction is based on [5] and [9].Firstly, we choose elements in F ∗ q / h δ i as the first part of coordinates in the vector −→ a . Secondly, wechoose elements from cosets of F ∗ q / h θ i as the second part of coordinates in −→ a . Finally, we consider theduplicating elements between these two parts. We construct the vector −→ v in a similar way. Then we canconstruct quantum MDS codes of length n = 1 + lh + mr − q − st · hr , whose minimum distances can bebigger than q + 1.The next lemma has been shown in [5]. We give a new proof by Cramer’s Rule, which is shorter than[5]. Lemma 3.1. ([5, Lemma 7]) For s − h + 1 ≤ µ ≤ s + h − , there exists a solution in ( F ∗ q ) h of the followingsystem of equations u + u + · · · + u h − = 1 h − P k =0 g kµl u k = 0 (4)5 roof. Denote by ξ = g l and c = s − h + 1. For any 0 ≤ ν = ν ′ ≤ h − < s − 2, the elements ξ c + ν , ξ c + ν ′ and 1 are distinct. The system of equations (4) can be expressed in the matrix form A −→ u T = (1 , , · · · , T , (5)where A = · · · ξ c · · · ξ ( h − c ... ... . . . ...1 ξ c + h − · · · ξ ( h − c + h − h × h and −→ u = ( u , u , . . . , u h − ) . We will show that u k ∈ F ∗ q for any 0 ≤ k ≤ h − A ) = 0. By Cramer’s Rule, u k = ( − k · det( D k )det( A ) , where D k = ξ c · · · ξ ( k − c ξ ( k +1) c · · · ξ ( h − c ξ c +1 · · · ξ ( k − c +1) ξ ( k +1)( c +1) · · · ξ ( h − c +1) ... ... . . . ... ... . . . ...1 ξ c + h − · · · ξ ( k − c + h − ξ ( k +1)( c + h − · · · ξ ( h − c + h − (6)is an ( h − × ( h − 1) matrix obtained from A by deleting 1-st row and ( k + 1)-th column with 0 ≤ k ≤ h − 1. It is easy to see det( D k ) is equal to non-zero constant times of a Vandermonde determinant. Sodet( D k ) = 0, which implies u k = 0.It remains to show u k ∈ F q , for any 0 ≤ k ≤ h − 1. Since s | q + 1 and ξ s = 1, then ξ k ( c + ν ) q = ξ − k ( s − h +1+ ν ) = ξ k ( s + h − − ν ) = ξ k ( c + h − − ν ) , for any 0 ≤ k ≤ h − ≤ ν ≤ h − 2. So (det( A )) q = ( − h − · det( A ) and det( D k ) q = ( − h − · det( D k ). It follows that u qk = ( − qk · det( D k ) q (det( A )) q = ( − k · det( D k )det( A ) = u k , which implies u k ∈ F ∗ q with 0 ≤ k ≤ h − 1. This completes the proof.Now we let −→ u = ( u , u , . . . , u h − ) satisfy the system of equations (4). Choose −→ a = (0 , , δ, . . . , δ l − , g, gδ, . . . , gδ l − , . . . , g h − , g h − δ, . . . , g h − δ l − )6nd −→ v = ( e, v , . . . , v | {z } l times , . . . , v h − , . . . , v h − | {z } l times ) , where v q +1 k = u k (0 ≤ k ≤ h − 1) and e q +1 = − l . Then we have the following lemma, which has beenshown in [5]. We give proof in order to make the paper self completeness. Lemma 3.2. ([5, Theorem 3]) The identity h−→ a qi + j , −→ v q +11 i = 0 holds for all ≤ i, j ≤ s + h · q +1 s − .Proof. When ( i, j ) = (0 , h−→ a , −→ v q +11 i = e q +1 + l ( v q +10 + · · · + v q +1 h − ) = − l + l ( u + · · · + u h − ) = 0 . When ( i, j ) = (0 , δ is of order l , then h−→ a qi + j , −→ v q +11 i = h − X k =0 g k ( qi + j ) v q +1 k l − X ν =0 δ ν ( qi + j ) = , l ∤ qi + j , l · h − P k =0 g k ( qi + j ) v q +1 k , l | qi + j .We consider the case l | qi + j . According to Lemma 2.2 (ii) and Lemma 3.1, h−→ a qi + j , −→ v q +11 i = h−→ a µl , −→ v q +11 i = l · h − X k =0 g kµl v q +1 k = l · h − X k =0 g kµl u k = 0 . Therefore, the result holds.For the second part of −→ a and −→ v , we choose −→ a = (1 , θ, . . . , θ m − , g, gθ, . . . , gθ m − , . . . , g r − , g r − θ, . . . , g r − θ m − )and −→ v = (1 , g t , . . . , g ( m − · t , , g t , . . . , g ( m − · t , . . . , , g t , . . . , g ( m − · t ) . Then the following lemma can be obtained. Lemma 3.3. The identity h−→ a qi + j , −→ v q +12 i = 0 holds for all ≤ i, j ≤ q +12 + q − t − . roof. By Lemma 2.3, we can calculate directly, h−→ a qi + j , −→ v q +12 i = r − X k =0 m − X ν =0 ( g k θ ν ) qi + j · θ ν · q +12 = r − X k =0 g k ( qi + j ) m − X ν =0 θ ν ( qi + j + q +12 ) = 0 . (7)Now, we give our first construction. Theorem 3. Let n = 1+ lh + mr − q − st · hr , where odd s | q +1 , even t | q − , t ≥ , l = q − s , m = q − t ,odd h ≤ s − and r ≤ t . If q − > q − st · hr , then for any ≤ d ≤ min { s + h · q +1 s − , q +12 + q − t − } ,there exists an [[ n, n − d, d + 1]] q quantum MDS code.Proof. Denote by A = { g α δ i | ≤ α ≤ h − , ≤ i ≤ l − } and B = { g β θ j | ≤ β ≤ r − , ≤ j ≤ m − } . From Lemma 2.1, we know | A ∩ B | = q − st · hr . Let A = A − B and B = B − A . Define f : A ∪ { } → F ∗ q , f ( g α δ i ) = v q +1 α and f (0) = − l,f : B → F ∗ q , f ( g β θ j ) = θ j · q +12 . Let −→ a = (0 , −→ a A , −→ a B , −→ a A ∩ B ) , where −→ a S = ( a , . . . , a k ) for S = { a , . . . , a k } and −→ v q +1 = ( − l, f ( −→ a A ) , λf ( −→ a B ) , f ( −→ a A ∩ B ) + λf ( −→ a A ∩ B )) , where λ ∈ F ∗ q and f j ( −→ a S ) = ( f j ( a ) , . . . , f j ( a k )) with S = { a , . . . , a k } and j = 1 , q − > q − st · hr = | A ∩ B | , then there exists λ ∈ F ∗ q such that all coordinates of f ( −→ a A ∩ B ) + λf ( −→ a A ∩ B ) are nonzero.According to Lemmas 3.2 and 3.3, it takes h−→ a qi + j , −→ v q +1 i = h−→ a qi + j , −→ v q +11 i + λ h−→ a qi + j , −→ v q +12 i = 0 , ≤ i, j ≤ d − 1. As a consequence, by Theorem 2, GRS d ( −→ a , −→ v ) is Hermitian self-orthogonal.Therefore, by Theorem 1, there exists an [[ n, n − d, d + 1]] q quantum MDS code, where n = 1 + lh + mr − q − st · hr and 1 ≤ d ≤ min { s + h · q +1 s − , q +12 + q − t − } . Remark 3.1. We try to choose s, h, t such that s + h · q +1 s − ≈ q +12 + q − t − . For large q , we take s ≈ p q + 1) · h and t ≈ p q + 1) . Then it follows that s + h · q + 1 s − ≈ q r q and q + 12 + q − t − ≈ q r q . This indicates that the minimum distance of the quantum MDS code in Theorem 3 can reach q + p q approximately. Example 3.1. Let q = 641 . Choose s = 107 , t = 32 , h = 5 and r = 1 . In this case, one has s + h s · ( q +1) − and q +12 + q − t − ≈ q + p q = 338 . . The length is n = 1 + lh + mr − q − st · hr = 16081 .There exists [[16081 , , quantum MDS code, which has not been covered in any previous work. n = lh + mr − q − st · hr In this section, we assume s is odd , h ≤ s − and r ≤ t . Now, we consider the first part of coordinatesin vectors −→ a and −→ v . Firstly, we give two useful lemmas, that are Lemmas 4.1 and 4.2, which generalizeLemma 13 and Theorem 5 in [5], respectively. Lemma 4.1. There exists a solution in ( F ∗ q ) h of the following system of equations h − X k =0 g k ( µl − q − u k = 0 (8) for ⌈ s − h ⌉ + 1 ≤ µ ≤ ⌊ s + h ⌋ − .Proof. Let ξ = g l , η = g − q − ∈ F ∗ q and c = ⌈ s − h ⌉ + 1. It is clear that ξ c + ν = ξ c + ν ′ for any 0 ≤ ν = ν ′ ≤ h − < s − 2. We discuss in two cases. Case 1 : h is odd. In this case, ⌈ s − h ⌉ = s − h and ⌊ s + h ⌋ = s + h . The system of equations (8) can beexpressed in the matrix form A −→ u T = (0 , , . . . , T , (9)where A = ξ c η ξ c η · · · ξ ( h − c η h − ξ c +1 η ξ c +1) η · · · ξ ( h − c +1) η h − ... ... ... . . . ...1 ξ c + h − η ξ c + h − η · · · ξ ( h − c + h − η h − 9s an ( h − × h matrix over F q and −→ u = ( u , u , . . . , u h − ) . It is obvious that rank( A ) = h − 1. We will show that u k ∈ F ∗ q for any 0 ≤ k ≤ h − A ′ = · · · ξ c η ξ c η · · · ξ ( h − c η h − ξ c +1 η ξ c +1) η · · · ξ ( h − c +1) η h − ... ... ... . . . ...1 ξ c + h − η ξ c + h − η · · · ξ ( h − c + h − η h − . We consider the equations A ′ −→ u T = (1 , , , . . . , T . (10)It is easy to check that A ′ is row equivalent to A ′ ( q ) and det( A ′ ) = 0. Similarly as the proof of Lemma3.1, we obtain (10) has a solution −→ u = ( u , u , . . . , u h − ) ∈ ( F ∗ q ) h . Since the solution of (10) is also thesolution of (9), the result has been proved. Case 2 : h is even. In this case, ⌈ s − h ⌉ = s − h +12 and ⌊ s + h ⌋ = s + h − . The system of equations (8)can be expressed in the matrix form A −→ u T = (0 , , . . . , T , (11)where A = ξ c η ξ c η · · · ξ ( h − c η h − ξ c +1 η ξ c +1) η · · · ξ ( h − c +1) η h − ... ... ... . . . ...1 ξ c + h − η ξ c + h − η · · · ξ ( h − c + h − η h − is an ( h − × h matrix over F q . By s | q + 1 and ξ s = 1, it takes (cid:16) ξ k ( c + ν ) η k (cid:17) q = ξ − k ( s − h +12 +1+ ν ) η k = ξ k ( s + h − − − ν ) η k = ξ k ( c + h − − ν ) η k , for any 0 ≤ k ≤ h − ≤ ν ≤ h − 3. Therefore, A and A ( q ) are row equivalent. By deleting the first(resp. the last) column of A and we obtain an ( h − × ( h − 1) matrix denote by A (resp. A h − ). Then A (resp. A h − ) is row equivalent to A ( q )0 (resp. A ( q ) h − ). Obviously, rank( A ) = rank( A h − ) = h − A −→ x T = (0 , . . . , T , A h − −→ y T = (0 , . . . , T −→ x = ( x , x , . . . , x h − ) , −→ y = ( y , y , . . . , y h − ) ∈ ( F ∗ q ) h − . From h < q + 1, there exists λ ∈ F ∗ q \ { x y , . . . , x h − y h − } such that −→ u = (0 , −→ x ) − λ ( −→ y , ∈ ( F ∗ q ) h . Then it implies A −→ u T = (cid:18) A −→ x T (cid:19) − λ (cid:18) A h − −→ y T (cid:19) = (0 , , . . . , T . Therefore, the result has been proved.We choose −→ a = (1 , δ, . . . , δ l − , g, gδ, . . . , gδ l − , . . . , g h − , g h − δ, . . . , g h − δ l − )and −→ v = ( v , v δ, . . . , v δ l − , v , v δ, . . . , v δ l − , . . . , v h − , v h − δ, . . . , v h − δ l − ) , where v q +1 k = u k (0 ≤ k ≤ h − 1) and −→ u = ( u , u , . . . , u h − ) satisfy (8). Lemma 4.2. The identity h−→ a qi + j , −→ v q +11 i = 0 holds for all ≤ i, j ≤ ⌊ s + h ⌋ · q +1 s − .Proof. Similarly as Lemma 3.2, we only need to consider the case l | qi + j + q + 1. From Lemma 2.2 (i)and Lemma 4.1, we deduce that h−→ a qi + j , −→ v q +11 i = h−→ a µl − q − , −→ v q +11 i = l · h − X k =0 g k ( µl − q − v q +1 k = 0 . Therefore, for all 0 ≤ i, j ≤ ⌊ s + h ⌋ · q +1 s − h−→ a qi + j , −→ v q +11 i = 0 . The vectors −→ a and −→ v are the same as in Section 3. Theorem 4. Let n = lh + mr − q − st · hr , where odd s | q + 1 , even t | q − , t ≥ , l = q − s , m = q − t , h ≤ s − and r ≤ t . Assume that q − > q − st · hr , then for any ≤ d ≤ min {⌊ s + h ⌋· q +1 s − , q +12 + q − t − } ,there exists an [[ n, n − d, d + 1]] q quantum MDS code.Proof. Similarly as Theorem 3, we also let A = { g α δ i | ≤ α ≤ h − , ≤ i ≤ l − } , B = { g β θ j | ≤ β ≤ r − , ≤ j ≤ m − } , A = A − B and B = B − A . Define f : A → F ∗ q , f ( g α δ i ) = ( v α δ i ) q +1 ,f : B → F ∗ q , f ( g β θ j ) = θ j · q +12 . −→ a = ( −→ a A , −→ a B , −→ a A ∩ B ) , where −→ a S = ( a , . . . , a k ) for S = { a , . . . , a k } and −→ v q +1 = ( f ( −→ a A ) , λf ( −→ a B ) , f ( −→ a A ∩ B ) + λf ( −→ a A ∩ B )) , where λ ∈ F ∗ q is chosen such that all the coordinates of f ( −→ a A ∩ B ) + λf ( −→ a A ∩ B ) are nonezero and f j ( −→ a S ) = ( f j ( a ) , . . . , f j ( a k )) with S = { a , . . . , a k } for j = 1 , GRS d ( −→ a , −→ v ) is Hermitianself-orthogonal. As a consequence, by Theorem 1, there exists [[ n, n − d, d + 1]] q quantum MDS code,where n = lh + mr − q − st · hr with odd h and 1 ≤ d ≤ min {⌊ s + h ⌋ · q +1 s − , q +12 + q − t − } . Remark 4.1. Similarly as Remark 3.1, the minimum distance can reach q + p q approximately. n = lh + mr In this section, s is even , h ≤ s2 and r ≤ t2 and quantum MDS codes with length n = lh + mr will beconstructed. Similarly as the previous constructions, we also divide the vectors −→ a and −→ v into two parts.However, in this case, coordinates of these two parts in the vector −→ a have no duplication. Therefore, thequantum MDS codes in this section have larger minimum distances than the codes in previous sections.The proof of the next result is similar to that of Lemma 4.1 and we omit the details. Lemma 5.1. The following system of equations h − X k =0 g (2 k +1)( µl − q − u k = 0 (12) has a solution denote by −→ u = ( u , u , . . . , u h − ) ∈ ( F ∗ q ) h for all ⌈ s − h ⌉ + 1 ≤ µ ≤ ⌊ s + h ⌋ − . Here we choose −→ a = ( g, gδ, . . . , gδ l − , g , g δ, . . . , g δ l − , . . . , g h − , g h − δ, . . . , g h − δ l − )and −→ v = ( v , v δ, . . . , v δ l − , v , v δ, . . . , v δ l − , . . . , v h − , v h − δ, . . . , v h − δ l − ) , where v q +1 k = u k (0 ≤ k ≤ h − 1) and −→ u = ( u , u , . . . , u h − ) is a solution of (12).12 emma 5.2. The identity h−→ a qi + j , −→ v q +11 i = 0 holds for all ≤ i, j ≤ ⌊ s + h ⌋ · q +1 s − .Proof. The result follows from Lemmas 2.2 (i) and 5.1.Now we construct the second part of coordinates in −→ a and −→ v . We choose −→ a = (1 , θ, . . . , θ m − , g , g θ, . . . , g θ m − , . . . , g r − , g r − θ, . . . , g r − θ m − )and −→ v = (1 , g t , . . . , g ( m − · t , , g t , . . . , g ( m − · t , . . . , , g t , . . . , g ( m − · t ) . Then we have the following lemma. Lemma 5.3. The identity h−→ a qi + j , −→ v q +12 i = 0 holds for all ≤ i, j ≤ q +12 + q − t − .Proof. By Lemma 2.3, h−→ a qi + j , −→ v q +12 i = r − X k =0 m − X ν =0 ( g k θ ν ) qi + j · θ ν · q +12 = r − X k =0 g k ( qi + j ) m − X ν =0 θ ν ( qi + j + q +12 ) = 0 . (13)Since both s and t are even, it is clear that all coordinates of −→ a are nonsquares and all coordinatesof −→ a are squares. Thus there exists no duplication between these two parts. Choose −→ a = ( −→ a , −→ a )and −→ v = ( −→ v , −→ v ). Theorem 5. Let n = lh + mr , where even s | q + 1 , even t | q − , t ≥ , l = q − s , m = q − t , h ≤ s and r ≤ t . Then for any ≤ d ≤ min {⌊ s + h ⌋ · q +1 s − , q +12 + q − t − } , there exists an [[ n, n − d, d + 1]] q quantum MDS code.Proof. The vectors −→ a and −→ v are defined as above. According to Lemmas 5.2 and 5.3, it takes h−→ a qi + j , −→ v q +1 i = h−→ a qi + j , −→ v q +11 i + h−→ a qi + j , −→ v q +12 i = 0 , ≤ i, j ≤ d − 1. Therefore, by Theorem 2, the code GRS d ( −→ a , −→ v ) is Hermitian self-orthogonal.By Theorem 1, there exists an [[ n, n − d, d + 1]] q quantum MDS code, where n = lh + mr and 1 ≤ d ≤ min {⌊ s + h ⌋ · q +1 s − , q +12 + q − t − } . Remark 5.1. When h approaches to s and t = 4 , both ⌊ s + h ⌋ · q +1 s − and q +12 + q − t − approach to q . So the minimum distance of the quantum MDS code can approach to q . Example 5.1. When q ≡ , applying Theorem 5 with ( s, h, t, r ) = (10 , , , , there exists q -ary quantum MDS codes with parameters (cid:20)(cid:20) q − , q − q + 7920 , q − (cid:21)(cid:21) q whose minimal distance is approximately . q when q is large. In general, the length satisfies ( q − , q ± . Therefore this class of quantum MDS codes are new. Example 5.2. When q ≡ 29 (mod 60) , applying Theorem 5 with ( s, h, t, r ) = (30 , , , , there existsquantum MDS codes with parameters (cid:20)(cid:20) q − , q − q + 22960 , q − (cid:21)(cid:21) q whose minimal distance is approximately q/ ≈ . q when q is large. Also the length satisfies ( q − , q ± and these quantum MDS codes are new. Applying Hermitian construction and GRS codes, we construct several new classes of quantum MDScodes over F q through Hermitian self-orthogonal GRS codes. Some of these quantum MDS codes canhave minimum distance bigger than q + 1. Since the lengths are chosen up to two variables h and r .This makes their lengths more flexible than previous constructions. Using our results, we can producemany new quantum MDS codes with new lengths which have not appeared in previous works. We givean example. Example 6.1. Choose q = 37 . Utilizing the results in this paper, there are 438 new [[ n, n − d, d + 1]] quantum MDS codes with minimum distance d + 1 ≥ q + 1 , which were not reported in previous papers.We list some of new [[ n, n − d, d + 1]] quantum MDS codes in Table 1. For a fixed q , it is expected to have [[ n, n − d, d + 1]] q quantum MDS codes for any length of q + 1 < n ≤ q + 1 and minimum distance q + 1 ≤ d + 1 ≤ min { n , q + 1 } . But sum up all the results,14uch quantum MDS codes is still very sparse. It is expected that more quantum MDS codes with largeminimal distance will be explored.Table 1: Some of New [[ n, n − d, d + 1]] Quantum MDS Codes n n − d d + 1588 544 23624 580 23660 614 24696 650 24702 658 23732 684 25738 694 23768 720 25774 728 24804 756 25810 764 24816 772 23840 792 25846 798 25852 808 23882 834 25918 868 26954 904 26 This research is supported by National Natural Science Foundation of China under Grant 11471008 andGrant 11871025 and the self-determined research funds of CCNU from the colleges’ basic research andoperation of MOE(Grant No. CCNU18TS028). References [1] Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans.Inf. Theory (3), 1183-1188 (2007)[2] Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory (7),3065-3072 (2001) 153] Chen, B., Ling, S., Zhang, G.: Apllication of constacyclic coodes to quantum MDS codes. IEEETrans. Inf. 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