Intersection numbers for subspace designs

Abstract

Intersection numbers for subspace designs are introduced and $q$-analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative $q$-analog of the Fano plane for any prime power $q$. It is shown that its existence implies the existence of a $2$-$(7,3,q^4{)}_q$ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.

References in corpus (1)

• Derived and Residual Subspace Designs

Cited by in corpus (16)

• Derived and Residual Subspace Designs
• Large sets of subspace designs
• The order of the automorphism group of a binary $q$-analog of the Fano plane is at most two
• On the automorphism group of a binary $q$-analog of the Fano plane
• On putative q-Analogues of the Fano Plane and Related Combinatorial Structures
• A New Approach to the Main Problem of Subspace Coding
• Higher incidence matrices and tactical decomposition matrices
• Constructions of new matroids and designs over GF(q)
• Designs in finite classical polar spaces
• On $\alpha$-points of $q$-analogs of the Fano plane
• Generalized vector space partitions
• A subspace code of size $333$ in the setting of a binary $q$-analog of the Fano plane
• The degree of functions in the Johnson and q-Johnson schemes
• New and Updated Semidefinite Programming Bounds for Subspace Codes
• Weighted Subspace Designs from $q$-Polymatroids
• Necessary conditions for the existence of 3-designs over finite fields with nontrivial automorphism groups