[Paper Review] Small complete caps from nodal cubics

Nurdagül Anbar, Daniele Bartoli+2 more|arXiv (Cornell University)|May 14, 2013

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper constructs the smallest known complete caps in affine spaces $AG(N,q)$ for $N \equiv 0 \pmod{4}$ by leveraging bicovering arcs derived from nodal cubic curves over $\mathbb{F}_q$. It proves that when $m = m_1m_2$ with $m_1, m_2$ coprime divisors of $q-1$, $(m,6)=1$, and $m \leq \sqrt[4]{q}/3.5$, such arcs yield complete caps of size at most $\frac{(m_1+m_2)(q-1)}{m_1m_2} q^{(N-2)/2}$, significantly improving prior bounds for infinitely many $q$. The construction relies on maximal 3-independent subsets in factor groups of the group of rational points on nodal cubics.

ABSTRACT

Bicovering arcs in Galois affine planes of odd order are a powerful tool for constructing complete caps in spaces of higher dimensions. In this paper we investigate whether some arcs contained in nodal cubic curves are bicovering. For $m_1$, $m_2$ coprime divisors of $q-1$, bicovering arcs in $AG(2,q)$ of size $k\le (q-1)\frac{m_1+m_2}{m_1m_2}$ are obtained, provided that $(m_1m_2,6)=1$ and $m_1m_2

Motivation & Objective

• To construct smaller complete caps in high-dimensional affine spaces $AG(N,q)$ for $N \equiv 0 \pmod{4}$, especially when $q$ is odd.
• To investigate whether arcs contained in nodal cubic curves over $\mathbb{F}_q$ are bicovering, a key requirement for lifting to higher-dimensional caps.
• To improve upon existing constructions of complete caps by exploiting composite index subgroups $K$ of the group of rational points on nodal cubics.
• To establish that maximal $3$-independent subsets in $G/K$ with $m = m_1m_2$, $(m_1,m_2)=1$, $(m,6)=1$, and $m \leq \sqrt[4]{q}/3.5$ yield bicovering arcs.
• To demonstrate that the resulting caps achieve sizes significantly smaller than the previously known $q^N/3$ bound for infinitely many $q$.

Proposed method

• Leverages the group structure of rational points on nodal cubic curves over $\mathbb{F}_q$, using subgroups $K$ of index $m$ in the group $G$ of non-singular points.
• Constructs bicovering arcs as unions of cosets of $K$ corresponding to maximal $3$-independent subsets in the factor group $G/K$, ensuring no three points are collinear and all external points are covered by at least two secants.
• Applies algebraic geometry and function field theory to verify that such arcs are bicovering when $m = m_1m_2$, $(m_1,m_2)=1$, $(m,6)=1$, and $m \leq \sqrt[4]{q}/3.5$, using a sufficient condition from prior work.
• Uses the lifting method: a bicovering arc of size $k$ in $AG(2,q)$ generates a complete cap of size $k q^{(N-2)/2}$ in $AG(N,q)$ for $N \equiv 0 \pmod{4}$.
• Employs known results on maximal $3$-independent subsets in abelian groups, particularly when $G/K \cong \mathbb{Z}_{m_1} \times \mathbb{Z}_{m_2}$, to construct such subsets of size at most $m_1 + m_2$, minimizing arc size.
• Derives quantitative bounds by optimizing $m_1, m_2$ such that $\frac{m_1 + m_2}{m_1 m_2}$ is minimized under the constraints $m_1 m_2 \leq \sqrt[4]{q}/3.5$ and $(m_1 m_2, 6) = 1$.

Experimental results

Research questions

• RQ1 Can arcs contained in nodal cubic curves over $\mathbb{F}_q$ be bicovering under specific group-theoretic conditions on the index $m$ of a subgroup $K$ of the rational point group $G$?
• RQ2 What is the minimal size of a bicovering arc in $AG(2,q)$ that can be lifted to a complete cap in $AG(N,q)$ for $N \equiv 0 \pmod{4}$, when $m = m_1 m_2$ with $m_1, m_2$ coprime and $m \leq \sqrt[4]{q}/3.5$?
• RQ3 How does the size of the resulting complete cap in $AG(N,q)$ compare to prior constructions, especially for large $q$ or $h$ such that $q = p^h$?
• RQ4 Under what conditions on $q$ and the factorization of $q-1$ can the ratio $\frac{m_1 + m_2}{m_1 m_2}$ be made significantly smaller than $1/3$ or $2p/q^{1/8}$?
• RQ5 Can the use of composite indices $m = m_1 m_2$ instead of prime indices improve the size of bicovering arcs and thus of complete caps in higher dimensions?

Key findings

• For $m = m_1 m_2$ with $m_1, m_2$ coprime divisors of $q-1$, $(m,6)=1$, and $m \leq \sqrt[4]{q}/3.5$, the union of cosets corresponding to a maximal $3$-independent subset in $G/K$ forms a bicovering arc in $AG(2,q)$ of size at most $\frac{(m_1 + m_2)(q - 1)}{m_1 m_2}$.
• This construction yields a complete cap in $AG(N,q)$ of size at most $\frac{(m_1 + m_2)(q - 1)}{m_1 m_2} q^{(N-2)/2}$ for all $N \equiv 0 \pmod{4}$, $N \geq 4$, which is the smallest known size for infinitely many $q$.
• When $p$ is large and $m = m_1 m_2$ with $m_1, m_2$ coprime and $m \leq \sqrt[4]{q}/3.5$, the ratio $\frac{m_1 + m_2}{m_1 m_2}$ can be significantly smaller than both $1/3$ and $2p/q^{1/8}$, leading to improved cap sizes.
• For $h \leq 8$, the construction improves upon the prior $q^N/3$ bound when $p-1$ has a composite divisor $m < \sqrt[4]{p}/3.5$ with $(m,6)=1$, yielding caps of size approximately $\frac{2^{s_2} + 2^{s_1}3^k}{q^{1/8}} q^{N/2 - 1/8}$.
• For $h > 8$, particularly when $q = p^{12}$ and $p \equiv 1 \pmod{12}$ with $(p^2 + 1)/2$ composite, the construction achieves caps of size less than $2q^{N/2}/p$, improving on the previous best bound of $2q^{N/2}/p^{\lfloor(\lceil h/4\rceil - 1)/2\rfloor}$.
• The method generalizes and improves upon prior results for elliptic and cuspidal cubics by allowing composite $m$, enabling smaller maximal $3$-independent subsets than the $m/3$ bound for prime $m$.