Certain Diagonal Equations and Conflict-Avoiding Codes of Prime Lengths

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 $3.$