On the Structure of the $q$-Fano Plane
Tuvi Etzion
TL;DR
This paper investigates the structure of the $q$-Fano plane, a combinatorial design, revealing extensive structural knowledge but leaving its existence unresolved, especially for the case $q=2$.
Contribution
The paper provides a detailed structural analysis of the $q$-Fano plane for any prime power $q$, highlighting what is known and emphasizing the unresolved existence question.
Findings
Most of the $q$-Fano plane's structure is understood.
The existence of the $q$-Fano plane remains open.
Special focus on the case $q=2$, which is still unresolved.
Abstract
The -Fano plane is the -analog of the Steiner system . For any given prime power it is not known whether the -Fano plane exists or not. We consider the structure of the -Fano plane for any given and conclude that most of its structure is known. Even so, we were unable to determine whether it exists or not. A special attention is given for the case which was considered by most researchers before.
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Taxonomy
TopicsCoding theory and cryptography · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models