# Additive perfect codes in Doob graphs

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2018-11-29

AUTHORS ABSTRACT

The Doob graph D(m, n) is the Cartesian product of m>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>0$$\end{document} copies of the Shrikhande graph and n copies of the complete graph of order 4. Naturally, D(m, n) can be represented as a Cayley graph on the additive group (Z42)m×(Z22)n′×Z4n′′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$$\end{document}, where n′+n′′=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n'+n''=n$$\end{document}. A set of vertices of D(m, n) is called an additive code if it forms a subgroup of this group. We construct a 3-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive 1-perfect codes in D(m,n′+n′′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(m,n'+n'')$$\end{document} are sufficient. Additionally, two quasi-cyclic additive 1-perfect codes are constructed in D(155,0+31)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(155,0+31)$$\end{document} and D(2667,0+127)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(2667,0+127)$$\end{document}. More... »

PAGES

1857-1869

### References to SciGraph publications

• 2015-03-26. Perfect codes in Doob graphs in DESIGNS, CODES AND CRYPTOGRAPHY
• 2016-12-31. There is exactly one Z2Z4-cyclic 1-perfect code in DESIGNS, CODES AND CRYPTOGRAPHY
• ### Journal

TITLE

Designs, Codes and Cryptography

ISSUE

8

VOLUME

87

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10623-018-0586-y

DOI

http://dx.doi.org/10.1007/s10623-018-0586-y

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1110267010

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