Certain Diagonal Equations and Conflict-Avoiding Codes of Prime Lengths

TL;DR

This paper explores the construction of optimal conflict-avoiding codes of prime length and weight 3 by analyzing the solvability of twisted Fermat equations over finite fields, providing new methods for code construction.

Contribution

It introduces a number theoretical approach linking the size of optimal CAC to the solvability of twisted Fermat equations over finite fields, extending to general finite extensions.

Findings

Existence of solutions for twisted Fermat equations over large finite fields.

New constructions of optimal conflict-avoiding codes of prime lengths.

Connection between algebraic equations and combinatorial code optimality.

Abstract

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 $3.$