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arXiv · 2011.02923

A generalization of the cylinder conjecture for divisible codes

Sascha Kurz·Sam Mattheus
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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 $q = 7$.

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Sascha Kurz, Sam Mattheus. 2020-11-05. A generalization of the cylinder conjecture for divisible codes. https://doi.org/10.1109/tit.2021.3134201

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