Summary: Let \(F^{(n)}_q\) be an \(n\)-dimensional vector space over a finite field \(F_q\), and \(P\) and \(Q\) be \(m\)-dimensional and \(r\)-dimensional subspaces, where \(\dim(P\cap Q)= i\). The number of all \(s\)-dimensional subspaces \(R\) such that \(\dim(P\cap R)= j\) and \(\dim(Q\cap R)= k\) in \(F^{(n)}_q\) is computed. Furthermore, an authentication code is constructed using the subspaces in \(F^{(n)}_q\).