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：我们从若干理论角度研究最佳避免冲突代码(CAC)的构建问题。确定最佳CAC(PI长度为$p$和重量为3的最佳CAC)的大小,是根据某些扭曲的Fermat方程式的可溶性制定的,这些方程式的形式为$g=2 X ⁇ ell}+ g Y ⁇ ell}+ 1=0美元,用于某些原始根块$\mathbb{F ⁇ p}。我们通过允许基础字段成为任何限定的扩展字段$\mathbb{F ⁇ q$\mathb{F ⁇ }。我们显示,对于超过重量值为$O(ell2)的较低约束的美元来说,有一台发电机$g$(mathb{F}qp}$,这样,在更一般情况下解决扭曲的Fermat方程式的问题,我们通过允许基字段成为任何有限的扩展字段$\mathbb{F ⁇ q$\q$\q$\q$。我们可以用我们的成果为最佳的CAprimal press 3 imal maxy maxal maxyal max 3 we can prial max max max cmax maxitals.
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