nach oben

Designs, Codes and Cryptography

Erschienen in:

18.07.2017

On two-weight \(\mathbb {Z}_{2^k}\)-codes

verfasst von: Minjia Shi, Zahra Sepasdar, Adel Alahmadi, Patrick Solé

Erschienen in: Designs, Codes and Cryptography | Ausgabe 6/2018

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We determine the possible homogeneous weights of regular projective two-weight codes over \(\mathbb {Z}_{2^k}\) of length \(n>3\), with dual Krotov distance \(d^{\lozenge }\) at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When \(k=2\), we characterize the parameters of such codes as those of the inverse Gray images of \(\mathbb {Z}_4\)-linear Hadamard codes, which have been characterized by their types by several authors.

Vorheriger Artikel A recursive construction for simple t-designs using resolutions

Nächster Artikel On data complexity of distinguishing attacks versus message recovery attacks on stream ciphers

Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2011).MATH

Byrne E., Greferath M., Honold T.: Ring geometries, two-weight codes and strongly regular graphs. Des. Codes Cryptogr. 48, 1–16 (2008).MathSciNetCrossRefMATH

Byrne E., Kiermaier M., Sneyd A.: Properties of codes with two homogeneous weights. Finite Fields Appl. 18, 711–727 (2012).MathSciNetCrossRefMATH

Calderbank R.: On uniformly packed \([n,n-k,4]\) codes over \(GF(q)\) and a class of caps in \(PG(k-1,q)\). J. Lond. Math. Soc. (2) 26, 365–384 (1982).MathSciNetCrossRefMATH

Carlet C.: \({\mathbb{Z}}_{2^k}\)-linear codes. IEEE Trans. Inf. Theory 44, 1543–1547 (1998).CrossRefMATH

Constantinescu I., Heise W.: A metric for codes over residue class rings of integers. Probl. Inf. Transm. 33, 208–213 (1997).MathSciNetMATH

Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3, 47–64 (1972).MathSciNetCrossRefMATH

Delsarte P.: An algebraic approach to the association schemes of Coding Theory. Philips Research Reports Supplement No. 10 (1973)

Hammons R., Kumar V.P., 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).MathSciNetCrossRefMATH

10.

Honold T.: A characterization of finite Frobenius rings. Arch. Math. 76, 406–415 (2001).MathSciNetCrossRefMATH

11.

Honold T.: Two-intersection sets in projective Hjemslev spaces. In: Edelmayer A. (ed.) Proceedings of MTNS 2010, Budapest, pp. 1807–1813 (2010)

12.

Krotov D.: \({\mathbb{Z}}_4-\)linear Hadamard and extended perfect codes. In: Proceedings of the International Workshop on Coding and Cryptography WCC 2001, Paris, January 2001, pp. 329–334. Electronic Notes in Discrete Mathematics, vol. 6, pp. 107–112 (2001)

13.

Krotov D.: On \({\mathbb{Z}}_{2^k}-\)dual binary codes. IEEE Trans. Inf. Theory 53, 1532–1537 (2007).CrossRefMATH

14.

Ling S., Xing C.P.: Coding Theory: A First Course. Cambridge University Press, New York (2004).CrossRef

15.

Phelps K.T., Rifà J., Villanueva M.: On the additive \(({\mathbb{Z}}_4-\)linear and non-\({\mathbb{Z}}_4-\)linear ) Hadamard codes. Rank and Kernel. IEEE Trans. Inf. Theory 52, 316–319 (2006).MathSciNetCrossRefMATH

16.

Shi M.J., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over \({\mathbb{Z}}_4,\). J. Syst. Sci. Complex. 27, 795–810 (2014).MathSciNetCrossRefMATH

Titel: On two-weight -codes
verfasst von: Minjia Shi
Zahra Sepasdar
Adel Alahmadi
Patrick Solé
Publikationsdatum: 18.07.2017
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 6/2018
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-017-0390-0

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 6/2018

On some constacyclic codes over , their images, and new codes

A framework for constructing partial geometric difference sets

Conditional cube attack on round-reduced River Keyak

Miklós–Manickam–Singhi conjectures on partial geometries

A recursive construction for simple t-designs using resolutions

New optimal binary sequences with period 4p via interleaving Ding–Helleseth–Lam sequences