The Grassmann variety of all projective \(k\)-dimensional subspaces in projective \(n\)-space embeds, using the Plücker mapping, into \(N\)-dimensional projective space, where \(N=\displaystyle{\left(\begin{array}{c}{n+1}\\ {k+1}\end{array}\right)}-1\). In the case \(k=1\), \(n=3\), the resulting algebraic variety in \({\mathbb{P}}^{5}\) is the famous Klein quadric. We study the “next case”, namely \(k=1,n=4\). Here (and in all the higher cases) the resulting variety is an intersection of quadrics. In the case specified, after giving a basis for the ideal of polynomials defining the variety, we list the points of the variety over the field of two elements. This is to obtain new information, which we present, on the binary linear code corresponding to the resulting projective system in \({\mathbb{P}}^{9}\). For example, we find the full list of higher weights for this code.