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
$q = 7$
.