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Cryptography and Communications

Erschienen in:

21.04.2022

On the construction of self-dual cyclic codes over \(\mathbb {Z}_{4}\) with arbitrary even length

verfasst von: Yuan Cao, Yonglin Cao, San Ling, Guidong Wang

Erschienen in: Cryptography and Communications | Ausgabe 5/2022

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Abstract

Self-dual codes over the ring \(\mathbb {Z}_{4}\) are related to combinatorial designs and unimodular lattices. First, we discuss briefly how to construct self-dual cyclic codes over \(\mathbb {Z}_{4}\) of arbitrary even length. Then we focus on solving one key problem of this subject: for any positive integers k and m such that m is even, we give a direct and effective method to construct all distinct Hermitian self-dual cyclic codes of length 2^k over the Galois ring GR(4,m). This then allows us to provide explicit expressions to accurately represent all these Hermitian self-dual cyclic codes in terms of binomial coefficients. In particular, several numerical examples are presented to illustrate our applications.

Vorheriger Artikel Symbolic dynamics and rotation symmetric Boolean functions

Nächster Artikel Multi-user BBB security of public permutations based MAC

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Abualrub, T., Oehmke, R.: On the generators of \(\mathbb {Z}_{4}\) cyclic codes of length 2^e. IEEE Trans. Inform. Theory 49, 2126–2133 (2003)MathSciNetMATHCrossRef

Bachoc, C.: Application of coding theory to the construction of modular lattices. J. Combin. Theory Ser. A 78, 92–119 (1997)MathSciNetMATHCrossRef

Blackford, T., Ray-Chaudhuri, D.K.: A transform approach to permutation groups of cyclic codes over Galois rings. IEEE Trans. Inform. Theory 46 (7), 2350–2358 (2000)MathSciNetMATHCrossRef

Blackford, T.: Cyclic codes over \(\mathbb {Z}_{4}\) of oddly even length. Discret. Appl. Math. 128, 27–46 (2003)MathSciNetMATHCrossRef

Blackford, T.: Negacyclic codes over \(\mathbb {Z}_{4}\) of even length. IEEE Trans. Inform. Theory 49(6), 1417–1424 (2003)MathSciNetMATHCrossRef

Bonnecaze, A., Rains, E., Solé, P.: 3-colored 5-designs and Z₄,-codes. J. Statist. Plann. Inference 86(2), 349–368 (2000). Special issue in honor of Professor Ralph Stanton. MR 1768278 (2001g:05021)MathSciNetMATHCrossRef

Calderbank, A.R., Sloane, N.J.A.: Modular and p-adic cyclic codes. Des. Codes Cryptogr. 6(1), 21–35 (1995)MathSciNetMATHCrossRef

Calderbank, A.R., Sloane, N.J.A.: Double circulant codes over \(\mathbb {Z}_{4}\) and even unimodular lattices. J. Alg. Combin. 6, 119–131 (1997)MathSciNetMATHCrossRef

Cao, Y., Cao, Y., Dougherty, S.T., Ling, S.: Construction and enumeration for self-dual cyclic codes over \(\mathbb {Z}_{4}\) of oddly even length. Des. Codes Cryptogr. 87, 2419–2446 (2019)MathSciNetMATHCrossRef

10.

Cao, Y., Cao, Y., Fu, F.-W., Wang, G.: Self-dual cyclic codes over \(\mathbb {Z}_{4}\) of length 4n. Appl. Algebra in Engrg. Comm. Comput. 33, 21–51 (2022)MathSciNetMATHCrossRef

11.

Cao, Y., Cao, Y., Ling, S., Wang, G.: An explicit expression for Euclidean self-dual cyclic codes of length 2^k over Galois ring GR(4,m). Finite Fields Appl. 72, 101817 (2021)MATHCrossRef

12.

Cao, Y., Cao, Y., Dinh, H.Q., Jitman, S.: An explicit representation and enumeration for self-dual cyclic codes over \(\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}\) of length 2^s. Discrete Math. 342, 2077–2091 (2019)MathSciNetMATHCrossRef

13.

Cao, Y., Cao, Y., Dinh, H.Q., Jitman, S.: An efficient method to construct self-dual cyclic codes of length p^s over \(\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}\). Discret. Math. 343, 111868 (2020)MATHCrossRef

14.

Castagnoli, G., Massey, J.L., Schoeller, P.A., von Seemann, N.: On repeated-root cyclic codes. IEEE Trans. Inform. Theory 37(2), 337–342 (1991)MathSciNetMATHCrossRef

15.

Dinh, H.Q., López-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inform. Theory 50(8), 1728–1744 (2004)MathSciNetMATHCrossRef

16.

Dinh, H.Q.: Negacyclic codes of length 2^s over Galois rings. IEEE Trans. Inform. Theory 51(12), 4252–4262 (2005)MathSciNetMATHCrossRef

17.

Dinh, H.Q.: Complete distances of all negacyclic codes of length 2^s over \(\mathbb {Z}_{2^{a}}\). IEEE Trans. Inform. Theory 53(1), 147–161 (2007)MathSciNetMATHCrossRef

18.

Dinh, H.Q.: On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions. Finite Fields Appl. 14(1), 22–40 (2008)MathSciNetMATHCrossRef

19.

Dinh, H.Q.: Constacyclic codes of length 2^s over Galois extension rings of \(\mathbb {F}_{2} + u\mathbb {F}_{2}\). IEEE Trans. Inform. Theory 55(4), 1730–1740 (2009)MathSciNetMATHCrossRef

20.

Dinh, H.Q.: Constacyclic codes of length p^s over \(\mathbb {F}_{p^{m}} + u\mathbb {F}_{p^{m}}\). J. Algebra 324(5), 940–950 (2010)MathSciNetMATHCrossRef

21.

Dougherty, S.T., Park, Y.H.: On modular cyclic codes. Finite Fields Appl. 13, 31–57 (2007)MathSciNetMATHCrossRef

22.

Dougherty, S.T., Ling, S.: Cyclic codes over \(\mathbb {Z}_{4}\) of even length. Des. Codes Cryptogr. 39, 127–153 (2006)MathSciNetMATHCrossRef

23.

Gaborit, P., Natividad, A.M., Solé, P.: Eisenstein lattices, Galois rings and quaternary codes. Int. J. Number Theory 2, 289–303 (2006)MathSciNetMATHCrossRef

24.

Hammons, Jr.A. R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The Z₄-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301–319 (1994)MathSciNetMATHCrossRef

25.

Harada, M.: Self-dual \(\mathbb {Z}_{4}\)-codes and Hadamard matrices. Discret. Math. 245, 273–278 (2002)MathSciNetMATHCrossRef

26.

Harada, M., Kitazume, M., Munemasa, A., Venkov, B.: On some self-dual codes and unimodular lattices in dimension 48. Eur. J. Combin. 26, 543–557 (2005)MathSciNetMATHCrossRef

27.

Harada, M., Miezaki, T.: An optimal odd unimodular lattice in dimension 72. Arch. Math. 97(6), 529–533 (2011)MathSciNetMATHCrossRef

28.

Harada, M., Solé, P., Gaborit, P.: Self-dual codes over \(\mathbb {Z}_{4}\) and unimodular lattices: a survey. In: Algebras and Combinatorics, Hong Kong, 1997, pp 255–275. Springer, Singapore (1999)

29.

Jitman, S., Ling, S., Sangwisut, E.: On self-dual cyclic codes of length p^a over GR(p²,s). Adv. Math. Commun. 10, 255–273 (2016)MathSciNetMATHCrossRef

30.

Kai, X., Zhu, S.: On the distance of cyclic codes of length 2^e over \(\mathbb {Z}_{4}\). Discret. Math. 310(1), 12–20 (2010)CrossRef

31.

Kaya, A., Yildiz, B.: Various constructions for self-dual codes over rings and new binary self-dual codes. Discret. Math. 339, 460–469 (2016)MathSciNetMATHCrossRef

32.

Kiah, H.M., Leung, K.H., Ling, S.: Cyclic codes over GR(p²,m) of length p^k. Finite Fields Appl. 14, 834–846 (2008)MathSciNetMATHCrossRef

33.

Kiah, H.M., Leung, K.H., Ling, S.: A note on cyclic codes over GR(p²,m) of length p^k. Des. Codes Cryptogr. 63, 105–112 (2012)MathSciNetMATHCrossRef

34.

Kiermaier, M.: There is no self-dual Z₄,-linear codes whose Gray image has the parameter (72, 2³⁶, 16). IEEE Trans. Inform. Theory 59(6), 3384–3386 (2013)MathSciNetMATHCrossRef

35.

Kanwar, P., López-Permouth, S.R.: Cyclic codes over the integers modulo p^m. Finite Fields Appl. 3(4), 334–352 (1997)MathSciNetMATHCrossRef

36.

Liu, H., Youcef, M.: Some repeated-root constacyclic codes over Galois rings. IEEE Trans. Inform. Theory 63(10), 6247–6255 (2017)MathSciNetMATHCrossRef

37.

López-Permouth, S.R., Szabo, S.: On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Adv. Math. Commun. 3(4), 409–420 (2009)MathSciNetMATHCrossRef

38.

Massey, J.L., Costello, D.J., Justesen, J.: Polynomial weights and code constructions. IEEE Trans. Inform. Theory 19, 101 (1973)MathSciNetMATHCrossRef

39.

Norton, G.H., Ana Sălăgean, A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Engrg. Comm. Comput. 10(6), 489–506 (2000)MathSciNetMATHCrossRef

40.

Pless, V.S., Qian, Z.: Cyclic codes and quadratic residue codes over \(\mathbb {Z}_{4}\). IEEE Trans. Inform. Theory 42(5), 1594–1600 (1996)MathSciNetMATHCrossRef

41.

Roth, R.M., Seroussi, G.: On cyclic MDS codes of length q over GF(q). IEEE Trans. Inform. Theory 32(2), 284–285 (1986)MathSciNetMATHCrossRef

42.

Sălăgean, A.: Repeated-root cyclic and negacyclic codes over a finite chain ring. Discret. Appl. Math. 154(2), 413–419 (2006)MathSciNetMATHCrossRef

43.

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, 86–95 (2017)MathSciNetMATHCrossRef

44.

Vega, G., Wolfmann, J.: Some families of \(\mathbb {Z}_{4}\)-cyclic codes. Finite Fields Appl. 10(4), 530–539 (2004)MathSciNetMATHCrossRef

45.

van Lint, J.H.: Repeated-root cyclic codes. IEEE Trans. Inform. Theory 37(2), 343–345 (1991)MathSciNetMATHCrossRef

46.

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

47.

Wolfmann, J.: Negacyclic and cyclic codes over \(\mathbb {Z}_{4}\). IEEE Trans. Inform. Theory 45(7), 2527–2532 (1999)MathSciNetMATHCrossRef

48.

Wolfmann, J.: Binary images of cyclic codes over \(\mathbb {Z}_{4}\). IEEE Trans. Inform. Theory 47(5), 1773–1779 (2001)MathSciNetMATHCrossRef

49.

Zimmermann, K.-H.: On generalizations of repeated-root cyclic codes. IEEE Trans. Inform. Theory 42(2), 641–649 (1996)MathSciNetMATHCrossRef

Titel: On the construction of self-dual cyclic codes over with arbitrary even length
verfasst von: Yuan Cao
Yonglin Cao
San Ling
Guidong Wang
Publikationsdatum: 21.04.2022
Verlag: Springer US
Erschienen in: Cryptography and Communications / Ausgabe 5/2022
Print ISSN: 1936-2447
Elektronische ISSN: 1936-2455
DOI: https://doi.org/10.1007/s12095-022-00579-2

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