Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion

TL;DR

This paper characterizes regular cyclic $(q+1)$-arcs in projective 3-space over finite fields, establishing spectral rigidity principles and criteria for descent and MDS properties, with applications to coding theory.

Contribution

It introduces a spectral rigidity principle and explicit descent criterion for cyclic arcs, linking algebraic parameters to geometric and coding-theoretic properties.

Findings

Classification of cyclic $(q+1)$-arcs up to projective equivalence.

A criterion for when these arcs are defined over the base field.

Resolution of the MDS condition for a family of BCH codes.

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.