Subspaces in $d$-bounded distance-regular graphs and their applications

Abstract

Let $\Gamma$ be a $d$-bounded distance-regular graph with diameter $d\ge 3$. For $x\in V\left(\Gamma \right)$, let $P\left(x\right)$ be the set of all subspaces containing $x$ in $\Gamma$. Suppose that $0\le t\le i+t,j+t\le i+j+t\le d_1\le d$, and suppose that $\Delta$ and ${\Delta }^{\ast }$ are subspaces with diameter $i+t$ and diameter $d_1$ in $P\left(x\right)$, respectively. Let $\Delta \subseteq {\Delta }^{\ast }$; we give the number of subspaces ${\Delta }^{\prime }$ with diameter $j+t$ and ${\Delta }^{\prime }\subseteq {\Delta }^{\ast }$ in $P\left(x\right)$ such that $d(\Delta \cap {\Delta }^{\prime })=t$ and $d(\Delta +{\Delta }^{\prime })=i+j+t$. Using the subspaces in $P\left(x\right)$, we construct a new Cartesian authentication code. We also compute its size parameters and its probabilities of successful impersonation attack and of successful substitution attack.