From the generating matrix of a linear code one can construct a sequence of generalized star configurations which are strongly connected to the generalized Hamming weights and the underlying matroid of the code. When the code is MDS, the matrix is generic and we obtain the usual star configurations. In our main result, we show that the degree of a generalized star configuration as a projective scheme is determined by the Tutte polynomial of the code. In the process, we obtain preliminary results on the primary decomposition of the defining ideals of these schemes. Additionally, we conjecture that these ideals have linear minimal free resolutions and prove partial results in this direction.