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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Volume 24, Number 3, Pages 155–163
DOI: https://doi.org/10.21538/0134-4889-2018-24-3-155-163
(Mi timm1559)
 

This article is cited in 1 scientific paper (total in 1 paper)

Codes in distance-regular graphs with $\theta_2~= -1$

M. S. Nirova

Kabardino-Balkar State University, Nal'chik
Full-text PDF (186 kB) Citations (1)
References:
Abstract: If a distance-regular graph $\Gamma$ of diameter 3 contains a maximal 1-code $C$ that is both locally regular and last subconstituent perfect, then $\Gamma$ has intersection array $\{a(p+1),cp,a+1;1,c,ap\}$ or $\{a(p+1),(a+1)p,c;1,c,ap\}$, where $a=a_3$, $c=c_2$, and $p=p^3_{33}$ (Juri$\check{\mathrm{s}}$i$\acute{\mathrm{c}}$ and Vidali). In first case, $\Gamma$ has eigenvalue $\theta_2=-1$ and the graph $\Gamma_3$ is pseudogeometric for $GQ(p+1,a)$. In the second case, $\Gamma$ is a Shilla graph. We study graphs with intersection array $\{a(p+1),cp,a+1;1,c,ap\}$ in which any two vertices at distance 3 are in a maximal 1-code. In particular, we find four new infinite families of intersection arrays: $\{a(a-2),(a-1)(a-3),a+1;1,a-1,a(a-3)\}$ for $a\ge 5$, $\{a(2a+3),2(a-1)(a+1),a+1;1,a-1,2a(a+1)\}$ for $a$ not congruent to $1$ modulo $3$, $\{a(2a-3),2(a-1)(a-2),a+1;1,a-1,2a(a-2)\}$ for even $a$ not congruent to $1$ modulo $3$, and $\{a(3a-4),(a-1)(3a-5),a+1;1,a-1,a(3a-5)\}$ for even $a$ congruent to 0 or 2 modulo 5.
Keywords: distance-regular graph, maximal code.
Received: 26.06.2018
Bibliographic databases:
Document Type: Article
UDC: 519.17
MSC: 05C25
Language: Russian
Citation: M. S. Nirova, “Codes in distance-regular graphs with $\theta_2~= -1$”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 3, 2018, 155–163
Citation in format AMSBIB
\Bibitem{Nir18}
\by M.~S.~Nirova
\paper Codes in distance-regular graphs with $\theta_2~= -1$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 3
\pages 155--163
\mathnet{http://mi.mathnet.ru/timm1559}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-3-155-163}
\elib{https://elibrary.ru/item.asp?id=35511284}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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