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Published in: Applicable Algebra in Engineering, Communication and Computing 3/2022

14-08-2020 | Original Paper

On some binary symplectic self-orthogonal codes

Authors: Heqian Xu, Wei Du

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 3/2022

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Abstract

Symplectic self-orthogonal codes over finite fields are an important class of linear codes in coding theory, which can be used to construct quantum codes. In this paper, characterizations of symplectic self-orthogonal codes over finite fields \(F_{q}\) are given. A necessary and sufficient condition for determining symplectic self-orthogonal codes is obtained. Several classes of symplectic self-orthogonal codes are constructed. Furthermore, the symplectic weight distributions of some new classes of binary symplectic self-orthogonal codes are completely determined.
Literature
1.
go back to reference Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007) MathSciNetCrossRef Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007) MathSciNetCrossRef
2.
go back to reference Ashikhmin, A., Knill, E.: Nonbinary quantum stablizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001) CrossRef Ashikhmin, A., Knill, E.: Nonbinary quantum stablizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001) CrossRef
6.
go back to reference 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(4), 1369–1387 (1998) MathSciNetCrossRef 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(4), 1369–1387 (1998) MathSciNetCrossRef
7.
go back to reference Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side—channel attacks. Coding Theory and Applications, pp. 97–105. Springer, Cham (2015) CrossRef Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side—channel attacks. Coding Theory and Applications, pp. 97–105. Springer, Cham (2015) CrossRef
8.
go back to reference Chau, H.F.: Five quantum register error correction code for higher spin systems. Phys. Rev. A 56, 1–4 (1997) CrossRef Chau, H.F.: Five quantum register error correction code for higher spin systems. Phys. Rev. A 56, 1–4 (1997) CrossRef
9.
go back to reference Chen, H., Ling, S., Xing, C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005) MathSciNetCrossRef Chen, H., Ling, S., Xing, C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005) MathSciNetCrossRef
10.
go back to reference Cohen, G., Encheva, S., Litsyn, S.: On binary constructions of quantum codes. IEEE Trans. Inf. Theory 45(7), 2495–2498 (1999) MathSciNetCrossRef Cohen, G., Encheva, S., Litsyn, S.: On binary constructions of quantum codes. IEEE Trans. Inf. Theory 45(7), 2495–2498 (1999) MathSciNetCrossRef
11.
go back to reference Du, Y., Ma, Y.: \([[2n, n]]\) additive quantum error correcting codes. J. Electron. 25(004), 519–522 (2008) Du, Y., Ma, Y.: \([[2n, n]]\) additive quantum error correcting codes. J. Electron. 25(004), 519–522 (2008)
12.
go back to reference Feng, K.Q., Ma, Z.: A finite Gilbert–Varshmov Bound for pure stabilizer quantum codes. IEEE. Trans. Inf. Theory 50(12), 3323–3325 (2004) CrossRef Feng, K.Q., Ma, Z.: A finite Gilbert–Varshmov Bound for pure stabilizer quantum codes. IEEE. Trans. Inf. Theory 50(12), 3323–3325 (2004) CrossRef
13.
go back to reference Grassl, M., Beth, T.: Quantum BCH codes. In: International Symposium on Theoretical Electrical Engineering, Magdeburg pp. 207–212 (1999) Grassl, M., Beth, T.: Quantum BCH codes. In: International Symposium on Theoretical Electrical Engineering, Magdeburg pp. 207–212 (1999)
14.
go back to reference Jin, L., Xing, C., Zhang, X.: On the list-decodability of random self-orthogonal codes. IEEE. Trans. Inf. Theory 61(2), 820–828 (2015) MathSciNetCrossRef Jin, L., Xing, C., Zhang, X.: On the list-decodability of random self-orthogonal codes. IEEE. Trans. Inf. Theory 61(2), 820–828 (2015) MathSciNetCrossRef
15.
16.
go back to reference Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: Nonbinary stablizer codes over finite fields. IEEE. Trans. Inf. Theory 52(11), 4892–4914 (2006) CrossRef Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: Nonbinary stablizer codes over finite fields. IEEE. Trans. Inf. Theory 52(11), 4892–4914 (2006) CrossRef
17.
go back to reference Li, R., Xu, Z., Li, X.: Standard forms of stabilizer and normalizer matrices for additive quantum codes. IEEE. Trans. Inf. Theory 54(8), 3775–3778 (2008) MathSciNetCrossRef Li, R., Xu, Z., Li, X.: Standard forms of stabilizer and normalizer matrices for additive quantum codes. IEEE. Trans. Inf. Theory 54(8), 3775–3778 (2008) MathSciNetCrossRef
18.
go back to reference Li, Z., Xing, L.J., Wang, X.M.: A family of asymptotically good quantum codes based on code concatenation. IEEE. Trans. Inf. Theory 55(8), 3821–3824 (2009) MathSciNetCrossRef Li, Z., Xing, L.J., Wang, X.M.: A family of asymptotically good quantum codes based on code concatenation. IEEE. Trans. Inf. Theory 55(8), 3821–3824 (2009) MathSciNetCrossRef
19.
go back to reference Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016) MathSciNetCrossRef Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016) MathSciNetCrossRef
20.
go back to reference Shi, M., Qian, L., Sok, L., Solé, P.: On constacyclic codes over \(Z_{4}[u]/\langle u^2-1\rangle\) and their Gray images. Finite Fields Appl. 45, 86–95 (2017) MathSciNetCrossRef Shi, M., Qian, L., Sok, L., Solé, P.: On constacyclic codes over \(Z_{4}[u]/\langle u^2-1\rangle\) and their Gray images. Finite Fields Appl. 45, 86–95 (2017) MathSciNetCrossRef
21.
go back to reference Shi, M., Sok, L., Solé, P.: Self-dual codes and orthogonal matrices over large finite fields. Finite Fields Appl. 54, 297–314 (2018) MathSciNetCrossRef Shi, M., Sok, L., Solé, P.: Self-dual codes and orthogonal matrices over large finite fields. Finite Fields Appl. 54, 297–314 (2018) MathSciNetCrossRef
22.
go back to reference Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995) CrossRef Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995) CrossRef
23.
go back to reference Zhou, Z., Tang, C., Li, X., Ding, C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory 65(1), 16–27 (2018) MathSciNetCrossRef Zhou, Z., Tang, C., Li, X., Ding, C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory 65(1), 16–27 (2018) MathSciNetCrossRef
Metadata
Title
On some binary symplectic self-orthogonal codes
Authors
Heqian Xu
Wei Du
Publication date
14-08-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 3/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00455-7

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