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Cryptography and Communications

Erschienen in:

07.10.2019

Complete classification for simple root cyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \)

verfasst von: Yuan Cao, Yonglin Cao

Erschienen in: Cryptography and Communications | Ausgabe 2/2020

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Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over \(\mathbb {Z}_{4}\) and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \).

Vorheriger Artikel On the minimum weights of binary linear complementary dual codes

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Abualrub, T., Siap, I.: Cyclic codes over the ring \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}\) and \(\mathbb {Z}_{2}+u\mathbb {Z}_{2}+u^{2}\mathbb {Z}_{2}\). Des. Codes Cryptogr. 42, 273–287 (2007)MathSciNetCrossRef

Calderbank, A.R., Hammons, A.R. Jr, Kumar, P.V., Sloane, N.J.A., Solé, P.: A linear construction for certain Kerdock and Preparata codes. Bull. Am. Math. Soc. 29(2), 218–222 (1993)MathSciNetCrossRef

Calderbank, A.R., Hammons, A.R. Jr, Kumar, P.V., Sloane, N.J.A., Solé, P.: The \(\mathbb {Z}_{4}\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)MathSciNetCrossRef

Cao, Y.: On constacyclic codes over finite chain rings. Finite Fields Appl. 24, 124–135 (2013)MathSciNetCrossRef

Cao, Y., Cao, Y., Li, Q.: The concatenated structure of cyclic codes over \(\mathbb {Z}_{p^{2}}\). J. Appl. Math. Comput. 52, 363–385 (2016)MathSciNetCrossRef

Cao, Y., Cao, Y., Li, Q.: Concatenated structure of cyclic codes over \(\mathbb {Z}_{4}\) of length 4n. Appl. Algebra in Engrg. Comm. Comput. 10, 279–302 (2016)MathSciNetCrossRef

Cao, Y., Li, Q.: Cyclic codes of odd length over \(\mathbb {Z}_{4}[u]/\langle u^{k}\rangle \). Cryptogr. Commun. 9, 599–624 (2017)MathSciNetCrossRef

Cao, Y., Cao, Y.: Negacyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \) of oddly even length and their Gray images. Finite Fields Appl. 52, 67–93 (2018)MathSciNetCrossRef

Cao, Y., Gao, Y.: Repeate root cyclic \(\mathbb {F}_{q}\)-linear codes over \(\mathbb {F}_{q^{l}}\). Finite Fields Appl. 31, 202–227 (2015)MathSciNetCrossRef

10.

Cao, Y., Cao, Y.: Complete classification for simple root cyclic codes over local rings \(\mathbb {Z}_{p^{s}}[v]/\langle v^{2}-pv\rangle \), https://www.researchgate.net/publication/320620031

11.

Dinh, H.Q.: Complete distance of all negacyclic codes of length 2^s over \(\mathbb {Z}_{2^{a}}\). IEEE Trans. Inf. Theory 53, 147–161 (2007)CrossRef

12.

Dinh, H.Q., Dhompongsa, S., Sriboonchitta, S: On constacyclic codes of length 4p^s over \(\mathbb {F}_{p^{m}}+u \mathbb {F}_{p^{m}}\). Discrete Math. 340, 832–849 (2017)MathSciNetCrossRef

13.

Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)MathSciNetCrossRef

14.

Harada, M.: Binary extremal self-dual codes of length 60 and related codes. Des. Codes Cryptogr. 86, 1085–1094 (2018)MathSciNetCrossRef

15.

Martínez-Moro, E., Szabo, S., Yildiz, B.: Linear codes over \(\frac {\mathbb {Z}_{4}[x]}{\langle x^{2}+2x\rangle }\). Int. J. Information and Coding Theory 3(1), 78–96 (2015)MathSciNetCrossRef

16.

Norton, G., Sălăgean-Mandache, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra in Engrg. Comm. Comput. 10, 489–506 (2000)MathSciNetCrossRef

17.

Shi, M., Zhu, S., Yang, S.: A class of optimal p-ary codes from one-weight codes over F_p[u]/〈u^m〉. J. Franklin Inst. 350(5), 729–737 (2013)CrossRef

18.

Shi, M., Qian, L., Sok, L., Aydin, N., Solé, P.: On constacyclic codes over \(\mathbb {Z}_{4}[u]/\langle u^{2}-1\rangle \) and their Gray images. Finite Fields Appl. 45(3), 86–95 (2017)MathSciNetCrossRef

19.

Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016)MathSciNetCrossRef

20.

Shi, M., Solé, P., Wu, B.: Cyclic codes and the weight enumerators over \(\mathbb {F}_{2} +v\mathbb {F}_{2} +v^{2}\mathbb {F}_{2}\). Applied and Computational Mathematics 12(2), 247–255 (2013)MathSciNetMATH

21.

Wan, Z.-X.: Cyclic codes over Galois rings. Algebra Colloq. 6(3), 291–304 (1999)MathSciNetMATH

22.

Wan, Z.-X.: Lectures on finite fields and Galois rings. World Scientific Pub Co Inc (2003)

23.

Wood, J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999)MathSciNetCrossRef

24.

Yankov, N., Lee, M.H.: New binary self-dual codes of lengths 50–60. Des. Codes Cryptogr. 73, 983–996 (2014)MathSciNetCrossRef

25.

Yildiz, B., Karadeniz, S.: Linear codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\): Macwilliams identities, projections, and formally self-dual codes. Finite Fields Appl. 27, 24–40 (2014)MathSciNetCrossRef

26.

Yildiz, B., Aydin, N.: Cyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) and \(\mathbb {Z}_{4}\) images. International Journal of Information and Coding Theory 2(4), 226–237 (2014)MathSciNetCrossRef

Titel: Complete classification for simple root cyclic codes over the local ring
verfasst von: Yuan Cao
Yonglin Cao
Publikationsdatum: 07.10.2019
Verlag: Springer US
Erschienen in: Cryptography and Communications / Ausgabe 2/2020
Print ISSN: 1936-2447
Elektronische ISSN: 1936-2455
DOI: https://doi.org/10.1007/s12095-019-00403-4

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