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

M. S. Nirova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 155-163

Résumé

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.

Détail

Keywords: distance-regular graph
Mots-clés : maximal code.