Abstract
Entanglement-assisted quantum error-correcting codes (EAQECCs) constructed from Reed–Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed–Solomon code, for the Hermitian metric, and from any BCH code with extension degree 2 and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for c, the minimum number of required maximally entangled quantum states.
Similar content being viewed by others
References
Bierbrauer, J.: The theory of cyclic codes and a generalization to additive codes. Des. Codes Cryptogr. 25(2), 189–206 (2002)
Brun, T., Devetak, I., Hsieh, M.-H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)
Brun, T.A., Devetak, I., Hsieh, M.-H.: Catalytic quantum error correction. IEEE Trans. Inform. Theory 60(6), 3073–3089 (2014)
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \({\rm GF}(4)\). IEEE Trans. Inform. Theory 44(4), 1369–1387 (1998)
Cascudo, I.: On squares of cyclic codes. IEEE Trans. Inform. Theory 65(2), 1034–1047 (2019)
Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(12), 22 (2017) (Paper No. 303)
Chen, X., Zhu, S., Kai, X.: Entanglement-assisted quantum negacyclic BCH codes. Int. J. Theor. Phys. 58(5), 1509–1523 (2019)
Du, Z., Li, C., Mesnager, S.: Constructions of self-orthogonal codes from hulls of BCH codes and their parameters. IEEE Trans. Inform. Theory 66, 1 (2020)
Galindo, C., Geil, O., Hernando, F., Ruano, D.: On the distance of stabilizer quantum codes from \(J\)-affine variety codes. Quantum Inf. Process. 16(4), 32 (2017) (Art. 111)
Galindo, C., Hernando, F.: Quantum codes from affine variety codes and their subfield subcodes. Des. Codes Crytogr. 76, 89–100 (2015)
Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted quantum error-correcting codes over arbitrary finite fields. Quantum Inf. Process. 18(4), 18 (2019) (Art. 116)
Galindo, C., Hernando, F., Ruano, D.: Stabilizer quantum codes from \(J\)-affine variety codes and a new Steane-like enlargement. Quantum Inf. Process. 14(9), 3211–3231 (2015)
Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A (3) 54(3), 1862–1868 (1996)
Guo, G., Li, R.: New entanglement-assisted quantum MDS codes derived from generalized Reed-Solomon codes. Int. J. Theor. Phys. 59(4), 1241–1254 (2020)
Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory 52(11), 4892–4914 (2006)
Lai, C.-Y., Ashikhmin, A.: Linear programming bounds for entanglement-assisted quantum error-correcting codes by split weight enumerators. IEEE Trans. Inform. Theory 64(1), 622–639 (2018)
Liu, Y., Li, R., Lv, L., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes. Quantum Inf. Process. 17(8), 19 (2018) (Paper No. 210)
Lü, L.-D., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12(3), 1450015 (2014)
Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18(3), 12 (2019) (Paper No. 89)
Luo, G., Cao, X., Chen, X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inform. Theory 65(5), 2944–2952 (2019)
Lv, L., Li, R., Fu, Q., Li, X.: Maximal entanglement entanglement-assisted quantum codes from quaternary BCH codes. In: 2015 IEEE Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), pp. 709–713 (2015)
Pereira, F.R.F.: Entanglement-assisted quantum codes from cyclic codes. arXiv:1911.06384 [cs.IT] (2019)
Pereira, F.R.F., Pellikaan, R., La Guardia, G.G., de Assis, F.M.: Application of complementary dual AG codes to entanglement-assisted quantum codes. In: 2019 IEEE International Symposium on Information Theory (ISIT), pp. 2559–2563 (2019)
Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86(7), 1565–1572 (2018)
Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS and almost MDS codes. Quantum Inf. Process. 18(3), 12 (2019) (Paper No. 71)
Ruano, D.: The metric structure of linear codes. In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pp. 537–561. Springer, Cham (2018)
Sarıand, M., Kolotoğlu, E.: An application of constacyclic codes to entanglement-assisted quantum MDS codes. Comput. Appl. Math. 38(2), 13 (2019) (Paper No. 75)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the Spanish Grants PGC2018-096446-BC21, PGC2018-096446-B-C22 and RED2018-102583-T (MCI/AEI/FEDER, UE), by the Spanish MINECO Grant RYC-2016-20208 (AEI/FSE/UE), by the Junta de CyL (Spain) Grant VA166G18, by the Generalitat Valenciana (Spain) Grant AICO- 2019-223 and by Universitat Jaume I Grant P1-1B2018-10.
Rights and permissions
About this article
Cite this article
Galindo, C., Hernando, F. & Ruano, D. Entanglement-assisted quantum error-correcting codes from RS codes and BCH codes with extension degree 2. Quantum Inf Process 20, 158 (2021). https://doi.org/10.1007/s11128-021-03101-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-021-03101-4