Abstract
In this paper, we investigate a family of q2-ary narrow-sense and non-narrow-sense negacyclic BCH codes with length \(n=\frac {q^{2m}-1}{2}\), where q is an odd prime power and m ≥ 3 is odd. We propose Hermitian dual-containing conditions for narrow-sense and non-narrow-sense negacyclic BCH codes, and precisely compute the dimensions of these negacyclic BCH codes whose maximal designed distance can achieve \(\delta _{max}^{neg}\). Consequently, many new q-ary quantum codes can be derived from these dual-containing negacyclic BCH codes. Moreover, these new quantum codes are presented either with parameters better than or equal to the ones available in the literature, and also have larger designed distance than those from classical BCH codes.
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References
Shor, P.W.: Scheme for reducing decoherence in quantum computing memory. Phys. Rev. A 52(4), R2493–R2496 (1995)
Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)
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)
Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)
Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)
Li, R., Li, X.: Quantum codes constructed from binary cyclic codes. Int. J. Quantum Inf. 2, 265–272 (2004)
Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. Proc. Int. Symp. Inf. Theory ISIT, 1114–1118 (2006)
Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE. Trans. Inf. Theory 53, 1183–1188 (2007)
La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331(1)-042331 (11) (2009)
Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021–0035 (2013)
Kai, X., Zhu, S., Tang, Y.: Quantum negacyclic codes. Phys. Rev. A 88 (1), 012326(1)-012326 (5) (2013)
Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)
La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60(3), 1528–1535 (2014)
Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224–5228 (2015)
Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic codes and new quantum codes. Quantum Inf. Process. 16, 1–16 (2017)
Qian, J., Zhang, L.: Improved constructions for nonbinary quantum BCH codes. Int J Theor Phys. 56(4), 1–9 (2017)
Yuan, J., Zhu, S., Kai, X., Li, P.: On the construction of quantum constacyclic codes. Des. Codes Cryptogr. 85(1), 179–190 (2017)
Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)
Aydin, N., Siap, I., Ray-Chaudhuri, D.J.: The structure of 1-generator quasi-twisted codes and new linear codes. Des., Codes Cryptogr. 24, 313–326 (2001)
Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE Trans. Inf. Theory 36, 880–884 (1990)
Li, R., Zuo, F., Liu, Y.: A study of skew symmetric q2-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)
Lin, X.: Quantum cyclic and constacyclic codes. IEEE Trans. Inf. Theory 50(3), 547–549 (2004)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting codes. Cambridge University Press, Cambridge (2003)
MacWilliams, F.J., Sloane, N.J.A.: The theory of Error-Correcting codes.s North-Holland (1977)
Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des Codes Cryptogr. https://doi.org/10.1007/s10623-017-0441-6 (2017)
Y. Edel’s homepage: http://www.mathi.uni-heidelberg.de/yves/matritzen/QTBCH/QTBCH-Index.html
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The authors would like to thank the anonymous referees and the chief editor Prof. Claude Carlet for their very meticulous reading and valuable comments.
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Guo, G., Li, R., Liu, Y. et al. A family of negacyclic BCH codes of length \(n=\frac {q^{2m}-1}{2}\). Cryptogr. Commun. 12, 187–203 (2020). https://doi.org/10.1007/s12095-019-00387-1
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DOI: https://doi.org/10.1007/s12095-019-00387-1