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Designs, Codes and Cryptography

Erschienen in:

01.09.2014

Codes over an infinite family of rings with a Gray map

verfasst von: Yasemin Cengellenmis, Abdullah Dertli, S. T. Dougherty

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2014

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Abstract

Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.

Vorheriger Artikel On the sharpness of Bruen’s bound for intersection sets in Desarguesian affine spaces

Nächster Artikel Constructions of almost difference sets from finite fields

Bannai E., Dougherty S.T., Harada M., Oura M.: Type II codes, even unimodular lattices, and invariant rings. IEEE-IT 45(4), 1194–1205 (1999).

Cengellenmis Y.: On the Cyclic Codes over \({\mathbb{F}}_3 + v {\mathbb{F}}_3\). Int. J. Algebra 4(6), 253–259 (2010).

Cengellenmis Y., Dougherty S.T.: Cyclic codes over \(A_k\). In: Proceedings of ACCT2012, Pomorie, Bulgaria.

Dougherty S.T., Fernandez-Cordoba C.: Codes over \(Z_{2^k}\), Gray maps and self-dual codes. Adv. Math. Commun. 5(4), 571–588 (2011).

Dougherty S.T., Liu H.: Independence of vectors in codes over rings. Des. Codes Cryptogr. 51, 55–68 (2009).

Dougherty S.T., Shiromoto K.: Maximum distance codes over rings of order 4. IEEE-IT 47(1), 400–404 (2001).

Dougherty S.T., Harada M., Gaborit P., Solé P.: Type II Codes Over \({\mathbb{F}}_2 + u {\mathbb{F}}_2\). IEEE Trans. Inf. Theory 45(1), 32–45 (1999).

Dougherty S.T., Gaborit P., Harada M., Munemasa A., Solé P.: Type IV self-dual codes over rings. IEEE-IT 45(7), 2345–2360 (1999).

Dougherty S.T., Kim J.L., Kulosman H.: MDS codes over finite principal ideal rings. Des. Codes Cryptogr. 50, 77–92 (2009).

10.

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

11.

Dougherty S.T., Yildiz B., Karadeniz S.: Codes over \(R_k\), Gray maps and their binary images. Finite Fields Appl. 17(3), 205–219 (2011).

12.

Dougherty S.T., Yildiz B., Karadeniz S.: Cyclic codes over \(R_k\), Gray maps and their binary images. Des. Codes Cryptogr. 63(1), (2012).

13.

Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z}}_4\)-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).

14.

Nebe G., Rains E.M., Sloane N.J.A.: Self-Dual Codes and Invariant Theory. Springer, Berlin (2006).

15.

Rains E., Sloane N.J.A.: Self-Dual Codes in Handbook of Coding Theory. Elsevier, Amsterdam (1998).

16.

Shaska T., Wijesiri S.: Codes over rings of size four, Hermitian lattices and corresponding theta functions. Proc. Am. Math. Soc. 136(3), 849–857 (2008).

17.

Shiromoto K.: Singleton bounds for codes over finite rings. J. Algebraic Comb. 12(1), 95–99 (2000).

18.

Sloane N.J.A., Thompson J.G.: Cyclic self-dual codes. IEEE Trans. Inf. Theory IT-29 5, 364–366 (1983).

19.

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

Titel: Codes over an infinite family of rings with a Gray map
verfasst von: Yasemin Cengellenmis
Abdullah Dertli
S. T. Dougherty
Publikationsdatum: 01.09.2014
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2014
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-012-9787-y

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