Spaces of Hankel matrices over finite fields
Let \(M_n(F)\) denote the linear space of \(n\times n\) matrices over a field \(F\), let \(C_n(F)= \{(a_{ij})^{n- 1}_{i, j= 0}: a_{ij}= a_{k\ell}\) if \(i+ j= k+ \ell\}\) denote its \(2n- 1\)-dimensional subspace consisting of Hankel matrices. A subspace \(W\subset M_n(F)\) is a \(\underline k\)-space if \(\text{rank }A\geq k\) for all \(0\neq A\in W\). Let \(f_F(n, k)\) denote the maximal dimension of a \(\underline k\)-space \(W\subset C_n(F)\), and let \(g_F(n, k)\) denote the maximal dimension of a \(\underline k\)-space \(W\subset M_n(F)\). The author proves the theorem: Let \(1\leq k\leq n\), then (1) \(f_F(n, k)= 2(n- k)+ 1\) if \(F\) is an algebraically closed field, (2) \(f_F(n, k)= 2n- k\) if \(F\) is a finite field. Then he gives a simple proof for a theorem proved by \textit{R. M. Roth} [IEEE Trans. Inf. Theory 37, No. 2, 328-336 (1991; Zbl 0721.94012)]: If \(F\) is a finite field, then \(g_F(n, k)= n(n- k+ 1)\) for \(1\leq k\leq n\).
scientific article; zbMATH DE number 753862