Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion
Bocong Chen, Jing Huang, Hao Wu
TL;DR
This work analyzes q+1-arcs in PG(3,q) with a regular cyclic symmetry, embedding the problem into diagonal monomial models over the extension field K=F_{q^2} and establishing a spectral rigidity principle. The key descent result shows that a cyclic monomial model M_a is K-projectively equivalent to a PG(3,q)-arc iff a ≡ ±2^e mod n with gcd(e,m)=1, yielding φ(m)/2 equivalence classes of regular cyclic pairs. Applied to the BCH family, the authors prove an exact MDS criterion: the code is MDS iff 2h+1 ≡ ±2^e mod n for some e coprime to m, sharply constraining h modulo n. The introduced spectral-rigidity framework provides a portable method to reduce projective equivalence questions to arithmetic on exponent data for diagonal cyclic configurations.
Abstract
Let $q=2^m$ with $m\ge 3$ and set $n:=q+1$. We investigate $(q+1)$-arcs $\mathcal A\subset \mathrm{PG}(3,q)$ that admit a regular cyclic subgroup $C\le \mathrm{PGL}(4,q)$ of order $n$. Over $K=\mathbb{F}_{q^2}$, such an action can be conjugated to a diagonal one, producing explicit cyclic monomial models \[ \mathcal M_a = \{[1:t:t^a:t^{a+1}]:t\in U_n\}\subset \mathrm{PG}(3,K), \qquad U_n=\{u\in K^\times:u^n=1\}, \] with $a\in(\mathbb{Z}/n\mathbb{Z})^\times$. We develop a spectral rigidity principle to obtain a precise descent criterion: $\mathcal M_a$ is $K$-projectively equivalent to a $(q+1)$-arc defined over $\mathbb{F}_q$ if and only if $a\equiv \pm 2^e \pmod n$ for some integer $e$ with $\gcd(e,m)=1$. Consequently, regular cyclic pairs $(\mathcal A,C)$ fall into exactly $\varphi(m)/2$ $K$-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 $2h+1\equiv \pm 2^e \pmod n$ 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 $\mathrm{PG}(r,K)$, providing a portable reduction of projective equivalence questions to explicit congruences on exponent data.