Linear secret-sharing scheme for four players
First, calculate the parity-check matrix $H$, whose rows generate the null space of $M^T$. $H$ is a generator matrix for the dual code $C^\bot$ of the linear code $C := \{ x \, M^T \}$ associated to your secret-sharing scheme. The access structure of your scheme is given by the minimal codewords $\mathbf x = (x_0, x_1, x_2, x_3, x_4)$ of $C^\bot$ that have $x_0 = 1$; specifically, $\{P_i : x_i \neq 0\}$ is a minimal reconstructing set if and only if there is a minimal codeword $\mathbf x$ in $C^\bot$ with $x_0 =
 1$ (Massey, 1993).