A characterization of Hermitian varieties as codewords

2017/06/19 by Angela Aguglia, Daniele Bartoli, Aguglia, A. +5
Mathematics · #51E20 #94B05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1706.06578

openalex publication_date 2017/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces PG(r,q²). In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of PG(r,q²) of the same size as a non-singular Hermitian variety of PG(r,q²), having the same intersection sizes with the hyperplanes of PG(r,q²). In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of PG(2,q²) is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in PG(3,q²), q=p^h, as well as in PG(r,q²), q=p prime, or q=p², p prime, and r≥ 4.

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