We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length $p$ and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form $g^2 X^{\ell} + g Y^{\ell} + 1 = 0$ over the finite field $\mathbb{F}_{p}$ for some primitive root $g$ modulo $p.$ We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field $\mathbb{F}_q$ of $\mathbb{F}_{p}.$ We show that for $q$ greater than a lower bound of the order of magnitude $O(\ell^2)$ there exists a generator $g$ of $\mathbb{F}_{q}^{\times}$ such that the equation in question is solvable over $\mathbb{F}_{q}.$ Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight $3.$

翻译：我们从数论角度研究最优冲突避免码的构造问题。对素数长度$p$、重量为3的最优冲突避免码的规模确定，可转化为以下形式的扭曲费马方程在有限域$\mathbb{F}_{p}$上可解性问题：对于模$p$的某个本原根$g$，方程$g^2 X^{\ell} + g Y^{\ell} + 1 = 0$的求解。我们将该扭曲费马方程问题的求解推广至更一般情形，允许基域为$\mathbb{F}_{p}$的任意有限扩张域$\mathbb{F}_{q}$。研究表明，当$q$大于量级为$O(\ell^2)$的下界时，存在$\mathbb{F}_{q}^{\times}$的生成元$g$使得所讨论方程在$\mathbb{F}_{q}$上可解。基于上述结果，我们为素数长度且重量为$3$的最优冲突避免码构造提供了新结论。