2025/12/22 by Bocong Chen, Jing Huang, Chen, Bocong +3
Mathematics · #11T71 #20B25 #51E20 #51E21 #94B15 #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2512.19371
openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
Let q=2m with m≥ 3 and set n:=q+1. We investigate (q+1)-arcs \mathcal A⊂ PG(3,q) that admit a regular cyclic subgroup C≤ PGL(4,q) of order n. Over K=\mathbbFq2, such an action can be conjugated to a diagonal one, producing explicit cyclic monomial models \mathcal Ma = \[1:t:ta:ta+1]:t∈ Un\⊂ PG(3,K), Un=\u∈ K^×:un=1\, with a∈(ℤ/nℤ)^×. We develop a spectral rigidity principle to obtain a precise descent criterion: \mathcal Ma is K-projectively equivalent to a (q+1)-arc defined over \mathbbFq if and only if a≡ ± 2e \pmod n for some integer e with gcd(e,m)=1. Consequently, regular cyclic pairs (\mathcal A,C) fall into exactly φ(m)/2 K-projective equivalence classes. As an immediate coding-theoretic application, we resolve the remaining AMDS/MDS dichotomy for the BCH family \mathcal C(q,q+1,3,h) studied by Xu et al.: \mathcal C(q,q+1,3,h) is MDS if and only if 2h+1≡ ± 2e \pmod n for some e with gcd(e,m)=1. The underlying spectral rigidity step is formulated in a general setting for diagonal regular cyclic pairs in PG(r,K), providing a portable reduction of projective equivalence questions to explicit congruences on exponent data.