On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound

Li, Ruihu; Li, Xueliang; Guo, Luobin

doi:10.1007/s11128-015-1143-5

Cited by 19 publications

(13 citation statements)

References 30 publications

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“…We can also consider the case of imperfect ebits, which should be similar to the study in [26]. Finally, these results could be strengthened in the case of linear EA stabilizer codes [58], [59].…”

Section: Discussionmentioning

confidence: 76%

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Lai

Ashikhmin

2018

IEEE Trans. Inform. Theory

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Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large.On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.

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Section: Discussionmentioning

confidence: 76%

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Lai

Ashikhmin

2018

IEEE Trans. Inform. Theory

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“…By the Griesmer bound (1) and [12, 2 ⌋ errors acting on the n channel qubits (see e.g. [11] and [12]). 1 (0, 0, 1, 2, 3, 3, 0, 1, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2) C 43,2 (0, 0, 0, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2) C 43,3 (0, 1, 1, 2, 2, 3, 1, 0, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2) C 43,4 (0, 0, 1, 2, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2) C 43,5 (0, 0, 1, 2, 3, 3, 0, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2) C 43,6 (0, 1, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2) C 43,7 (0, 0, 0, 3, 3, 3, 1, 0, 2, 2, 3, 1, 3, 3, 3, 1, 3, 3, 2, 0, 2, 2) C 43,8 (0, 0, 0, 3, 3, 3, 0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 1, 1, 2, 2) C 43,9 (0, 0, 1, 2, 3, 3, 0, 2, 2, 1, 2, 3, 2, 3, 2, 2, 3, 3, 0, 2, 2, 2) C 43,10 (1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2) C 48,1 (0, 0, 1, 3, 3, 3, 1, 3, 3, 3, 0, 1, 1, 1, 2, 3, 3, 3, 2, 3, 3, 3) C 48,2 (0, 0, 1, 3, 3, 3, 0, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3) C 48,3 (0, 1, 1, 2, 3, 3, 1, 3, 2, 3, 2, 3, 3, 2, 0, 2, 2, 2, 2, 2, 3, 3) C 48,4 (0, 1, 1, 2, 3, 3, 1, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2) C 48,5 (0, 2, 2, 0, 3, 3, 2, 0, 3, 3, 0, 3, 3, 3, 3, 3, 3, 0, 3, 3, 0, 3) C 52,1 (0, 0, 2, 3, 3, 3, 1, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2) C 52,2 (0, 0, 1, 3, 3, 3, 0, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3) C 52,3 (0, 0, 2, 3, 3, 3, 0, 0, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) C 52,4 (0, 0, 2, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3) C 52,5 (0, 1, 1, 3, 3, 3, 1, 3, 3, 3, 0, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3) C 56,1 (0, 0, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3) C 56,2 (0, 0, 1, 3, 3, 3, 1, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3) C 56,3 (0, 1, 2, 3, 3, 3, 1, 0, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) C 56,4 (0, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3) C 56,5 (0, 0, 2, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3) C 56,6 (0, 1, 1, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3) C 64,1 (0, 0, 2, 4, 4, 4, 1, 3, 3, 3, 3, 4, 4, 4, 2, 3, 3, 3, 2, 3, 3, 3) C 64,2 (0, 1, 1, 4, 4, 4, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4) C 64,3 (0, 2, 2, 3, 3, 4, 2, 3, 3, 4, 0, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3) C 64,4 (0, 2, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) C 64,5 (0, 0, 2, 4, 4, 4, 0, 0, 1, 1, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4) C 64,6 (0, 0, 2, 4, 4, 4, 2, 4, 4, 4, 2, 3, 3, 3, 3, 4, 4, 4, 1, 2, 2, 2) C 64,7 (0, 1, 2, 3, 4, 4, 2, 4, 4, 3, 3, 4, 3, 4, 3, 3, 4, 4, 0, 2, 2, 2) C 64,8 (0, 1, 2, 3, 4, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) C 64,9 (0, 0, 2, 4, 4, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 for a nonnegative integer s. In the remainder of this section, we consider the remaining cases.…”

Section: Determination Of D 4 (N 3)mentioning

confidence: 99%

“…As a special case of Theorem 3.4 (ii), we have the following: From Table 1, it is known that there is no quaternary Hermitian LCD [11,3,8] code. This completes the proof.…”

Section: Determination Of D 4 (N 3)mentioning

confidence: 99%

Quaternary Hermitian Linear Complementary Dual Codes

Araya

Harada

Saito

2020

IEEE Trans. Inform. Theory

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The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s+5, 3, 16s+3; 21s+2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26,3, 19] code. Recently, much work has been done concerning LCD codes for both theoretical and practical reasons (see e.g. [1], [5], [6], [8], [11], [12] and the references given therein). For example, if there is a quaternary Hermitian LCD [n, k, d] code, then there is a maximal entanglement [[n, k, d; n − k]] entanglementassisted quantum error-correcting code (EAQECC for short) (see e.g. [11] and [12]). From this point of view, quaternary Hermitian LCD codes play an important role in the study of maximal entanglement EAQECC's.It is a fundamental problem to determine the largest minimum weight d 4 (n, k) among all quaternary Hermitian LCD [n, k] codes for a given pair (n, k). It was shown that d 4 (n, 2) = ⌊ 4n 5 ⌋ if n ≡ 1, 2, 3 (mod 5) and d 4 (n, 2) = ⌊ 4n 5 ⌋ − 1 if n ≡ 0, 4 (mod 5) for n ≥ 3 in [10] and [12]. In this paper, we give some conditions on the nonexistence of quaternary Hermitian LCD codes with large minimum weights. We give a classification of (unrestricted) quaternary [4r, 3, 3r] codes for r = 9, 10, 12, 13, 14, 16 and quaternary [43, 3, 32] codes. Using the above classification and the classification in [3], we completely determine the largest minimum weight among all quaternary Hermitian LCD codes of dimension 3. In addition, for a positive integer s, a maximal entanglement [[21s + 5, 3, 16s + 3; 21s + 2]] EAQECC is constructed for the first time from a quaternary Hermitian LCD [26,3, 19] code. This paper is organized as follows. In Section 2, we prepare some definitions, notations and basic results used in this paper. In Section 3, we give characterizations of quaternary Hermitian LCD codes. It is shown that there is no quaternary Hermitian LCD [ 4 k −1 3 s, k, 4 k−1 s] code for k ≥ 3 and s ≥ 1 (Theorem 3.3). In addition, if 4(4 k−1 n − 4 k −1 3 α) < k, where k ≥ 3 and 4α − 3n ≥ 1, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2, where d(C) denotes the minimum (Hamming) weight of a quaternary code C and C ⊥ H denotes the Hermitian dual code of C. If 4(4 k−1 n − 4 k −1 3 α) ≥ k ≥ 3, where 4α − 3n ≥ 1 and there is no quaternary Hermitian LCD [4(4 k−1 n − 4 k −1 3 α), k, 3(4 k−1 n − 4 k −1 3 α)] code C 0 with d(C ⊥ H 0 ) ≥ 2, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2 (Theorem 3.4). In Section 4, from the classification of quaternary codes of dimension 3 by Bouyukliev, Grassl and Varbanov [3], we determine...

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“…[28, 6,12], [28, 13,8], [29,13,8], [30,7,12], [30,14,8], [32,19,6], [32,20,6], [33,22,6], [34,10,12], [34,21,6], [34,22,6], [36,6,16], [36,23,6], [40,6,18] and [40,8,16].…”

Section: Introductionmentioning

confidence: 99%

“…These codes determine the largest minimum weights as follows: [21,8,9], [21,10,8], [21,11,7], [22,8,10], [23,18,4], [24,16,6], [25,18,4], [25,19,4], [26,16,6], [26,17,6], [26,20,4], [26,21,4], [27,21,4], [27,22,4], [28,21,4], [28,22,4], [28,23,4], [29,22,4],…”

Section: Introductionmentioning

confidence: 99%

Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes

Harada

2021

Preprint

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On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound

Cited by 19 publications

References 30 publications

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Quaternary Hermitian Linear Complementary Dual Codes

Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes

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