On the image of code polynomials under theta map

The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for g ≤ 3 and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for g ≥ 5. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for g ≥ 4 and consequently we get the non surjectivity of the graded rings for the remaining case g = 4.