A generalization of the cylinder conjecture for divisible codes

We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over Fq and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of q. In particular, we correct a flawed proof for the original cylinder conjecture for q = 5 and present the first proof for q = 7.