# A new distance-regular graph of diameter 3 on 1024 vertices

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2019-01-24

AUTHORS ABSTRACT

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on 210\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{10}$$\end{document} vertices, with new intersection array {33,30,15;1,2,15}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{33,30,15;1,2,15\}$$\end{document}. The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters (210,495,238,240)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2^{10}, 495,238, 240)$$\end{document}. Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed. More... »

PAGES

2091-2101

### References to SciGraph publications

• 2015-03-26. Perfect codes in Doob graphs in DESIGNS, CODES AND CRYPTOGRAPHY
• 2012. Spectra of Graphs in NONE
• 2014-11. On automorphisms of a distance-regular graph with intersection array {33, 30, 15; 1, 2, 15} in DOKLADY MATHEMATICS
• 2003-09. Designs in Additive Codes over GF(4) in DESIGNS, CODES AND CRYPTOGRAPHY
• 1989. Distance-Regular Graphs in NONE
• ### Journal

TITLE

Designs, Codes and Cryptography

ISSUE

9

VOLUME

87

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10623-019-00609-w

DOI

http://dx.doi.org/10.1007/s10623-019-00609-w

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1111653008

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