A generalization of the cylinder conjecture for divisible codes

TL;DR

This paper generalizes the cylinder conjecture to divisible codes over finite fields, providing new proofs, counterexamples, and classifications for small field sizes, advancing understanding in combinatorial coding theory.

Contribution

It extends the cylinder conjecture to divisible linear codes, offers a reduction theorem, and provides proofs for small finite fields, correcting previous errors.

Findings

Proved the generalized conjecture for small q values

Identified instances where the conjecture does not hold

Corrected a flawed proof for q=5 and proved for q=7

Abstract

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 $q = 7$.