An Isometric Invariant of Quadratic Spaces over Finite Fields

2021/05/28 by Semin Yoo, Yoo, Semin
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2105.14057

openalex publication_date 2021/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_q be the finite field with an odd prime power q. In this paper, we construct a new isometric invariant of combinatorial type on (\mathbbFⁿ_q,dot_n), where dot_n(x):=x₁²+⋯+x_n². Additionally, using counts from our new invariant, we give a new proof of Minkowski's formula on the size of spheres over finite fields. We also show which types of quadratic subspaces can be embedded in (\mathbbF_qⁿ,dot_n).

Related

• Incidences between quadratic subspaces over finite fields
• Isometries of quadratic spaces
• Isometries of virtual quadratic spaces
• Graphs associated with orthogonal collections of k-planes over finite\n fields
• Combinatorics of Euclidean spaces over finite fields