Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion
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2025
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| author | Chen, Bocong Huang, Jing Wu, Hao |
| author_facet | Chen, Bocong Huang, Jing Wu, Hao |
| contents | 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 $φ(m)/2$ $K$-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 $2h+1\equiv \pm 2^e \pmod n$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion Chen, Bocong Huang, Jing Wu, Hao Combinatorics 51E21, 51E20, 20B25, 11T71, 94B15 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 $φ(m)/2$ $K$-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 $2h+1\equiv \pm 2^e \pmod n$ 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. |
| title | Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion |
| topic | Combinatorics 51E21, 51E20, 20B25, 11T71, 94B15 |
| url | https://arxiv.org/abs/2512.19371 |