Abstract
A new class of binary quantum codes from cyclic codes over \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \), \(u^4=0\), is introduced. The generator polynomials of \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes of length (r, s) are obtained through the factorization of \(x^r-1\) and \(x^s-1\) into pairwise coprime monic polynomials over \(\mathbb {Z}_2\), where r and s are odd positive integers. A minimal spanning set for these codes is obtained. Under some restricted conditions, the structure of the duals of \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4\rangle \)-cyclic codes is also determined. Necessary and sufficient conditions for a \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic code of this restricted class to contain its dual or to be self-orthogonal are obtained. A new Gray map is defined, and the binary quantum codes are obtained by using the Calderbank-Shor-Steane construction on self-orthogonal or dual containing \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes. Some examples of binary quantum codes with good parameters constructed from \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes are given.
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References
Abualrub T., Siap I.: Cyclic codes over the rings \(\mathbb{Z}_2+u\mathbb{Z}_2\) and \(\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2\). Des. Codes Cryptogr. 42(3), 273–287 (2007).
Aydogdu I., Siap I.: The structure of \(\mathbb{Z}_2\mathbb{Z}_{2^s}\)-additive codes: bounds on the minimum distance. Appl. Math. Inf. Sci. 7(6), 2271–2278 (2013).
Aydogdu I., Siap I.: On \(\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\)-additive codes. Linear Multilinear Algebra 63(10), 2089–2102 (2015).
Aydogdu I., Abualrub T., Siap I.: On \(\mathbb{Z}_2\mathbb{Z}_2[u]\)-additive codes. Int. J. Comput. Math. 92(9), 1806–1814 (2015).
Aydogdu I., Abualrub T., Siap I.: \(\mathbb{Z}_2\mathbb{Z}_2[u]\)-cyclic and constacyclic codes. IEEE Trans. Inf. Theory 63(8), 4883–4893 (2017).
Aydogdu I., Siap I., Ten-Valls R.: On the structure of \(\mathbb{Z}_2\mathbb{Z}_2[u^3]\)-linear and cyclic codes. Finite Fields Appl. 48, 241–260 (2017).
Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: \(\mathbb{Z}_2\mathbb{Z}_4\)-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54(2), 167–179 (2010).
Borges J., Fernández-Córdoba C., Ten-Valls R.: \(\mathbb{Z}_2\)-double cyclic codes. Des. Codes Cryptogr. 86(3), 463–479 (2018).
Bosma W., Cannon J.: Handbook of Magma Functions. University of Sydney, Sydney (1995).
Calderbank A.R., Shor P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (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).
Dinh H.Q., López-Permouth S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50(8), 1728–1744 (2004).
Edel Y.: Some good quantum twisted codes. https://www.mathi.uni-heidelberg.de/~yves/Matrizen/QTBCH/QTBCHIndex.html. (2020). Last accessed 7 Sept 2021.
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de (1995). Last accessed 7 Sept 2021.
Guenda K., Gulliver T.A.: Quantum codes over rings. Int. J. Quantum Inf. 12(4), 1–11 (2014).
Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z}_4\)-linearity of Kerdock, Preperata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).
Kai X., Zhu S.: Quaternary construction of quantum codes from cyclic codes over \(\mathbb{F}_4+u\mathbb{F}_4\). Int. J. Quantum Inf. 9(2), 689–700 (2011).
Ketkar A., Klappenecker A., Kumar S., Sarvepalli P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006).
Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010).
Qian J., Ma W., Wang X.: Quantum error-correcting codes from quasi-cyclic codes. Int. J. Quantum Inf. 6, 1263–1269 (2008).
Qian J., Ma W., Guo W.: Quantum codes from cyclic codes over finite rings. Int. J. Quantum Inf. 7, 1277–1283 (2009).
Sari M., Siap I.: Quantum codes over a class of finite chain rings. Quantum Inf. Comput. 16(1–2), 0039–0049 (2016).
Shor P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A. 52(4), 2493–2496 (1995).
Singh A.K., Kewat P.K.: On cyclic codes over the ring \(\mathbb{Z}_p[u]/\langle u^k\rangle \). Des. Codes Cryptogr. 74, 1–13 (2015).
Steane A.M.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2577 (1996).
Steane A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54(6), 4741–4751 (1996).
Tang Y., Zhu S., Kai X., Ding J.: New quantum codes from dual-containing cyclic codes over finite rings. Quantum Inf. Process. 15(11), 4489–4500 (2016).
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions that greatly improved the presentation of the paper. The first author would like to thank Ministry of Human Resource Development (MHRD), India for providing financial support.
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Biswas, S., Bhaintwal, M. Quantum codes from \(\mathbb {Z}_2\mathbb {Z}_2[u]/\langle u^4 \rangle \)-cyclic codes. Des. Codes Cryptogr. 90, 343–366 (2022). https://doi.org/10.1007/s10623-021-00978-1
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DOI: https://doi.org/10.1007/s10623-021-00978-1