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 ${ {\mathbb {F}}_{q}}$ 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 $q = 7$ .