A remark on generalized Fisher inequalities of Delsarte's \(t\)-designs (Q1362998)

Let \({\mathcal X}= (X,\{R_0,R_1,\dots, R_d\})\) be an association scheme with \(d\) classes, and let \(m_i\) \((i=0,1,\dots, d)\) be the multiplicities of \(\mathcal X\). P. Delsarte defined algebraically the notion of \(t\)-designs in \(Q\)-polynomial association schemes and proved that for any \(t\)-design \(\mathcal B\) of \(\mathcal X\), we have \(|{\mathcal B}|\geq m_0+ m_1+\cdots+ m_{[t/2]}= M_t\). \(\mathcal B\) is called a tight \(t\)-design if \(|{\mathcal B}|= M_t\). This paper considers the following classical association schemes over the field \(\text{GF}(q)\): (0) association scheme of bilinear forms; (1) association scheme of Hermitian forms; (2) association scheme of alternating bilinear forms; (3) association scheme of quadratic forms. P. Delsarte, D. Stanton and A. Munemasa gave the notion of \(t\)-design with index \(\lambda\) in classical schemes, which is equivalent to the Delsarte notion of \(t\)-design. It is easy to see that \(|{\mathcal B}|\) can be expressed in terms of \(q\), \(t\) and \(\lambda\) and that \(|{\mathcal B}|\) is, as a function of \(\lambda\), minimized when \(\lambda=1\). The value of \(|{\mathcal B}|\) that corresponds to \(\lambda=1\) is denoted by \(N_t\). In this paper, it is proved for (0) \(M_t<N_t\), establishing the nonexistence of tight \(t\)-designs, and for (1)--(3) \(M_t>N_t\) on the assumption that the dimension of the vector space is large relative to \(t\) and that \(t\geq 2\), giving the non-trivial lower bound of the index \(\lambda\geq M_t/N_t>1\).