nach oben

Designs, Codes and Cryptography

Erschienen in:

12.07.2022

The build-up construction over a commutative non-unital ring

verfasst von: Adel Alahmadi, Amani Alkathiry, Alaa Altassan, Alexis Bonnecaze, Hatoon Shoaib, Patrick Solé

Erschienen in: Designs, Codes and Cryptography | Ausgabe 12/2022

Einloggen, um Zugang zu erhalten

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

search-config

KI-gestützte Suche

Aus

Abstract

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as

$$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^2=b,\, \,ab=0 \rangle . \end{aligned}$$

We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size $2^n$ (called here quasi self-dual codes or QSD) up to the length $n=6.$ In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to $n=6.$

Vorheriger Artikel Parallel sub-code construction for constant-dimension codes

Nächster Artikel A gapless code-based hash proof system based on RQC and its applications

Alahmadi A., Alkathiry A., Altassan A., Bonnecaze A., Shoaib H., Solé P.: The build-up construction of quasi self-dual codes over a non-unital ring. J. Algebra Appl. (2020). https://doi.org/10.1142/S0219498822501432.

Alahmadi A., Altassan A., Basaffar W., Bonnecaze A., Shoaib H., Solé P.: Type IV codes over a non-unital ring. J. Algebra Appl. (2021). https://doi.org/10.1142/S0219498822501420.

Alahmadi A., Altassan A., Basaffar W., Bonnecaze A., Shoaib H., Solé P.: Quasi Type IV codes over a non-unital ring. J. AAECC 32, 217–228 (2021).MathSciNetCrossRefMATH

Conway J.H., Sloane N.J.A.: Self-dual codes over the integers modulo four. J. Comb. Theory Ser. A 62, 30–45 (1993).CrossRefMATH

Dougherty S., Leroy A.: Euclidean self-dual codes over non-commutative Frobenius rings. Appl. Algebra Eng. Commun. Comput. 27(3), 185–203 (2016).MathSciNetCrossRefMATH

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

Fine B.: Classification of finite rings of order $p^2$. Math. Mag. 66(4), 248–252 (1993).MathSciNetCrossRefMATH

Gaborit P.: Mass formulas for self-dual codes over $ {\mathbb{Z}}_4$ and ${\mathbb{F}}_q+u{\mathbb{F}}_q$ rings. IEEE Trans. Inf. Theory 42, 1222–1228 (1996).CrossRef

Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A.: Patrick Solé: The $Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).CrossRefMATH

10.

Han S., Lee H., Lee Y.: Construction of self-dual codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2,$ Bull. Korean Math Soc. 49, 135–143 (2012).CrossRef

11.

Hou X.D.: On the number of inequivalent binary self-orthogonal codes. Trans. Inf. Theory 53, 2459–2479 (2007).MathSciNetCrossRefMATH

12.

http://magma.maths.usyd.edu.au/magma/

13.

Hufman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRef

14.

Kim J.-L., Lee Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory A 105, 79–95 (2004).MathSciNetCrossRefMATH

15.

Kim J.-L., Lee Y.: An efficient construction of self-dual codes. Bull. Korean Math. Soc. 52, 915–923 (2015).MathSciNetCrossRefMATH

16.

MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).MATH

17.

Raghavendran R.: A class of finite rings. Compos. Math. 21, 195–229 (1969).MathSciNetMATH

18.

Shi M., Alahmadi A., Solé P.: Codes and Rings: Theory and Practice. Academic Press, Cambridge (2017).MATH

19.

Solé P.: Codes over Rings. World Scientific, Singapore (2008).MATH

Titel: The build-up construction over a commutative non-unital ring
verfasst von: Adel Alahmadi
Amani Alkathiry
Alaa Altassan
Alexis Bonnecaze
Hatoon Shoaib
Patrick Solé
Publikationsdatum: 12.07.2022
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 12/2022
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01044-0

Springer Professional

The build-up construction over a commutative non-unital ring

Abstract

Premium Partner

Springer Professional

Abstract

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

Weitere Artikel der Ausgabe 12/2022

Steganography from perfect codes on Cayley graphs over Gaussian integers, Eisenstein–Jacobi integers and Lipschitz integers

Decoding algorithms of monotone codes and azinv codes and their unified view

A gapless code-based hash proof system based on RQC and its applications

Construction of asymmetric Chudnovsky-type algorithms for multiplication in finite fields

Equidistant permutation group codes

A new construction of linear codes with one-dimensional hull

Premium Partner